Pressure
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about pressure in the physical sciences. For the psychological meaning, see Peer pressure. For the Billy Joel song, see Pressure (Billy Joel song). For the Paramore song, see Pressure (Paramore song). For Under Pressure, see Under Pressure (disambiguation).
Pressure (symbol: p or sometimes P) is the force per unit area applied to an object in a direction perpendicular to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.
Contents
[hide]
* 1 Definition
o 1.1 Formula
o 1.2 Units
o 1.3 Examples
o 1.4 Scalar nature
* 2 Types
o 2.1 Explosion or deflagration pressures
o 2.2 Negative pressures
o 2.3 Stagnation pressure
o 2.4 Surface pressure
o 2.5 Pressure of an ideal gas
* 3 See also
* 4 Notes
* 5 External links
[edit] Definition
Pressure is an effect which occurs when a force is applied on a surface. The symbol of pressure is p (lower case).[1]
[edit] Formula
Conjugate variables
of thermodynamics
Pressure Volume
(Stress) (Strain)
Temperature Entropy
Chem. potential Particle no.
Mathematically:
p = \frac{F}{A}\ \mbox{or}\ p = \frac{dF}{dA}
where:
p is the pressure,
F is the normal force,
A is the area.
Pressure is a scalar quantity, and has SI units of pascals; 1 Pa = 1 N/m2, and has EES units of psi; 1 psi = 1 lb/in2.
Pressure is transmitted to solid boundaries or across arbitrary sections of fluid normal to these boundaries or sections at every point. It is a fundamental parameter in thermodynamics and it is conjugate to volume.
[edit] Units
Mercury column
The SI unit for pressure is the pascal (Pa), equal to one newton per square meter (N·m-2 or kg·m-1·s-2). This special name for the unit was added in 1971; before that, pressure in SI was expressed simply as N/m2.
Non-SI measures such as pound per square inch (psi) and bar are used in parts of the world. The cgs unit of pressure is the barye (ba), equal to 1 dyn·cm-2. Pressure is sometimes expressed in grams-force/cm2, or as kg/cm2 and the like without properly identifying the force units. But using the names kilogram, gram, kilogram-force, or gram-force (or their symbols) as units of force is expressly forbidden in SI. The technical atmosphere (symbol: at) is 1 kgf/cm2. In US Customary units, it is 14.696 psi.
Some meteorologists prefer the hectopascal (hPa) for atmospheric air pressure, which is equivalent to the older unit millibar (mbar). Similar pressures are given in kilopascals (kPa) in most other fields, where the hecto prefix is rarely used. The unit inch of mercury (inHg, see below) is still used in the United States. Oceanographers usually measure underwater pressure in decibars (dbar) because an increase in pressure of 1 dbar is approximately equal to an increase in depth of 1 meter. Scuba divers often use a manometric rule of thumb: the pressure exerted by ten metres depth of water is approximately equal to one atmosphere.
The standard atmosphere (atm) is an established constant. It is approximately equal to typical air pressure at earth mean sea level and is defined as follows:
standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25 hPa.
Because pressure is commonly measured by its ability to displace a column of liquid in a manometer, pressures are often expressed as a depth of a particular fluid (e.g., inches of water). The most common choices are mercury (Hg) and water; water is nontoxic and readily available, while mercury's high density allows for a shorter column (and so a smaller manometer) to measure a given pressure. The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation p = ρgh. Fluid density and local gravity can vary from one reading to another depending on local factors, so the height of a fluid column does not define pressure precisely. When millimeters of mercury or inches of mercury are quoted today, these units are not based on a physical column of mercury; rather, they have been given precise definitions that can be expressed in terms of SI units. The water-based units still depend on the density of water, a measured, rather than defined, quantity. These manometric units are still encountered in many fields. Blood pressure is measured in millimeters of mercury in most of the world, and lung pressures in centimeters of water are still common.
Gauge pressure is often given in units with 'g' appended, eg 'kPag' or 'psig', and units for measurements of absolute pressure are sometimes given a suffix of 'a', to avoid confusion, for example 'kPaa', 'psia'.
Presently or formerly popular pressure units include the following:
* atmosphere (atm)
* manometric units:
o centimeter, inch, and millimeter of mercury (torr)
o millimeter, centimeter, meter, inch, and foot of water
* customary units:
o kip, ton-force (short), ton-force (long), pound-force, ounce-force, and poundal per square inch
o pound-force, ton-force (short), and ton-force (long)
* non-SI metric units:
o bar, decibar, millibar
o kilogram-force, or kilopond, per square centimetre (technical atmosphere)
o gram-force and tonne-force (metric ton-force) per square centimetre
o barye (dyne per square centimetre)
o kilogram-force and tonne-force per square metre
o sthene per square metre (pieze)
Pressure Units
pascal
(Pa)
bar
(bar) technical atmosphere
(at)
atmosphere
(atm)
torr
(Torr) pound-force per
square inch
(psi)
1 Pa ≡ 1 N/m2 10−5 1.0197×10−5 9.8692×10−6 7.5006×10−3 145.04×10−6
1 bar 100,000 ≡ 106 dyn/cm2 1.0197 0.98692 750.06 14.5037744
1 at 98,066.5 0.980665 ≡ 1 kgf/cm2 0.96784 735.56 14.223
1 atm 101,325 1.01325 1.0332 ≡ 1 atm 760 14.696
1 torr 133.322 1.3332×10−3 1.3595×10−3 1.3158×10−3 ≡ 1 Torr; ≈ 1 mmHg 19.337×10−3
1 psi 6,894.76 68.948×10−3 70.307×10−3 68.046×10−3 51.715 ≡ 1 lbf/in2
Example reading: 1 Pa = 1 N/m2 = 10−5 bar = 10.197×10−6 at = 9.8692×10−6 atm, etc.
[edit] Examples
As an example of varying pressures, a finger can be pressed against a wall without making any lasting impression; however, the same finger pushing a thumbtack can easily damage the wall. Although the force applied to the surface is the same, the thumbtack applies more pressure because the point concentrates that force into a smaller area. Pressure is transmitted to solid boundaries or across arbitrary sections of fluid normal to these boundaries or sections at every point. Unlike stress, pressure is defined as a scalar quantity.
Another example is of a common knife. If we try and cut a fruit with the flat side it obviously won't cut. But if we take the thin side, it will cut smoothly. The reason is, the flat side has a greater surface area(less pressure) and so it does not cut the fruit. When we take the thin side, the surface area is reduced and so it cuts the fruit easily and quickly. This is one example of a practical application of Pressure.
The gradient of pressure is called the force density. For gases, pressure is sometimes measured not as an absolute pressure, but relative to atmospheric pressure; such measurements are called gauge pressure (also sometimes spelled gage pressure).[2] An example of this is the air pressure in an automobile tire, which might be said to be "220 kPa/32psi", but is actually 220 kPa/32 psi above atmospheric pressure. Since atmospheric pressure at sea level is about 100 kPa/14.7 psi, the absolute pressure in the tire is therefore about 320 kPa/46.7 psi. In technical work, this is written "a gauge pressure of 220 kPa/32 psi". Where space is limited, such as on pressure gauges, name plates, graph labels, and table headings, the use of a modifier in parentheses, such as "kPa (gauge)" or "kPa (absolute)", is permitted. In non-SI technical work, a gauge pressure of 32 psi is sometimes written as "32 psig" and an absolute pressure as "32 psia", though the other methods explained above that avoid attaching characters to the unit of pressure are preferred.[3]
Gauge pressure is the relevant measure of pressure wherever one is interested in the stress on storage vessels and the plumbing components of fluidics systems. However, whenever equation-of-state properties, such as densities or changes in densities, must be calculated, pressures must be expressed in terms of their absolute values. For instance, if the atmospheric pressure is 100 kPa, a gas (such as helium) at 200 kPa (gauge) (300 kPa [absolute]) is 50 % denser than the same gas at 100 kPa (gauge) (200 kPa [absolute]). Focusing on gauge values, one might erroneously conclude the first sample had twice the density of the second one.
[edit] Scalar nature
In a static gas, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant random motion. Because we are dealing with an extremely large number of molecules and because the motion of the individual molecules is random in every direction, we do not detect any motion. If we enclose the gas within a container, we detect a pressure in the gas from the molecules colliding with the walls of our container. We can put the walls of our container anywhere inside the gas, and the force per unit area (the pressure) is the same. We can shrink the size of our "container" down to an infinitely small point, and the pressure has a single value at that point. Therefore, pressure is a scalar quantity, not a vector quantity. It has magnitude but no direction sense associated with it. Pressure acts in all directions at a point inside a gas. At the surface of a gas, the pressure force acts perpendicular (at right angle) to the surface.
A closely related quantity is the stress tensor σ, which relates the vector force F to the vector area A via
\mathbf{F}=\mathbf{\sigma A}\,
This tensor may be divided up into a scalar part (pressure) and a traceless tensor part shear. The shear tensor gives the force in directions parallel to the surface, usually due to viscous or frictional forces. The stress tensor is sometimes called the pressure tensor, but in the following, the term "pressure" will refer only to the scalar pressure.
[edit] Types
[edit] Explosion or deflagration pressures
Explosion or deflagration pressures are the result of the ignition of explosive gases, mists, dust/air suspensions, in unconfined and confined spaces.
[edit] Negative pressures
While pressures are generally positive, there are several situations in which negative pressures may be encountered:
* When dealing in relative (gauge) pressures. For instance, an absolute pressure of 80 kPa may be described as a gauge pressure of -21 kPa (i.e., 21 kPa below an atmospheric pressure of 101 kPa).
* When attractive forces (e.g., Van der Waals forces) between the particles of a fluid exceed repulsive forces. Such scenarios are generally unstable since the particles will move closer together until repulsive forces balance attractive forces. Negative pressure exists in the transpiration pull of plants.
* The Casimir effect can create a small attractive force due to interactions with vacuum energy; this force is sometimes termed 'vacuum pressure' (not to be confused with the negative gauge pressure of a vacuum).
* Depending on how the orientation of a surface is chosen, the same distribution of forces may be described either as a positive pressure along one surface normal, or as a negative pressure acting along the opposite surface normal.
* In the cosmological constant.
[edit] Stagnation pressure
Stagnation pressure is the pressure a fluid exerts when it is forced to stop moving. Consequently, although a fluid moving at higher speed will have a lower static pressure, it may have a higher stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by the Mach number of the fluid. In addition, there can be differences in pressure due to differences in the elevation (height) of the fluid. See Bernoulli's equation (note: Bernoulli's equation only applies for incompressible flow).
The pressure of a moving fluid can be measured using a Pitot tube, or one of its variations such as a Kiel probe or Cobra probe, connected to a manometer. Depending on where the inlet holes are located on the probe, it can measure static pressure or stagnation pressure.
[edit] Surface pressure
There is a two-dimensional analog of pressure – the lateral force per unit length applied on a line perpendicular to the force.
Surface pressure is denoted by π and shares many similar properties with three-dimensional pressure. Properties of surface chemicals can be investigated by measuring pressure/area isotherms, as the two-dimensional analog of Boyle's law, πA = k, at constant temperature.
\pi = \frac{F}{l}.
[edit] Pressure of an ideal gas
In an ideal gas, molecules have no volume and do not interact. Pressure varies linearly with temperature, volume, and quantity according to the ideal gas law,
p=\frac{nRT}{V}
where:
p is the absolute pressure of the gas
n is the amount of substance (in mole)
T is the temperature (in kelvin)
V is the volume
R is the ideal gas constant.
Wednesday, March 11, 2009
♦maGn!tic f0rce♦
Magnetic field
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Magnetic field (disambiguation).
Electromagnetism
Electricity · Magnetism
[show]Electrostatics
Electric charge · Coulomb's law · Electric field · Electric flux · Gauss's law · Electric potential · Electrostatic induction · Electric dipole moment ·
[hide]Magnetostatics
Ampère’s law · Electric current · Magnetic field · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss’s law for magnetism ·
[show]Electrodynamics
Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potential · Maxwell tensor · Eddy current ·
[show]Electrical Network
Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides ·
[show]Covariant formulation
Electromagnetic tensor · EM Stress-energy tensor · Four-current · Four-potential ·
[show]Scientists
Ampère · Coulomb · Faraday · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Weber ·
This box: view • talk • edit
Magnetic field lines shown by iron filings. The high permeability of individual iron filings causes the magnetic field to be larger at the ends of the filings. This causes individual filings to attract each other, forming elongated clusters that trace out the appearance of lines. It would not be expected that these "lines" be precisely accurate field lines for this magnet; rather, the magnetization of the iron itself would be expected to alter the field somewhat.
A magnetic field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles (such as permanent magnets). When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field. Magnetic fields surround and are created by electric currents, magnetic dipoles, and changing electric fields. Magnetic fields also have their own energy, with an energy density proportional to the square of the field intensity.
There are some notable specific instances of the magnetic field. For the physics of magnetic materials, see magnetism and magnet, and more specifically ferromagnetism, paramagnetism, and diamagnetism. For constant magnetic fields, such as are generated by stationary dipoles and steady currents, see magnetostatics. A changing electric field (which is mathematically identical to a moving electric field) also results in a magnetic field (see electromagnetism).
The magnetic field forms one aspect of electromagnetism. (See also relativistic electromagnetism.) In a simplified form the magnetic field can be thought of as the relativistic part of an electric field. More precisely, magnetic fields are a necessary consequence of the existence of electric fields and special relativity. A pure electric field in one reference frame will be viewed as a combination of both an electric field and a magnetic field in a moving reference frame. Together, the electric and magnetic fields make up the electromagnetic field, which is best known for underlying light and other electromagnetic waves.
Contents
[hide]
* 1 B and H
o 1.1 Alternative names for B and H
o 1.2 Units
* 2 Permanent magnets, magnetic poles, and compasses
* 3 Visualizing the magnetic field
o 3.1 Magnetic B field lines
+ 3.1.1 Magnetic B field lines always form loops
* 4 Elementary effects of the magnetic field, B
o 4.1 Force due to a magnetic field on a moving charge
+ 4.1.1 Force on a charged particle
+ 4.1.2 Force on current-carrying wire
+ 4.1.3 Direction of force
o 4.2 Torque on a magnetic dipole
o 4.3 Force on a magnetic dipole due to a non-uniform B
o 4.4 Electric force due to a changing B
* 5 Elementary sources of magnetic fields
o 5.1 Electrical currents (moving charges)
+ 5.1.1 Magnetic field of a steady current
o 5.2 Magnetic dipoles
o 5.3 Changing electric field
o 5.4 Magnetic monopole (hypothetical)
* 6 Definition and mathematical properties of B
o 6.1 Maxwell's equations
* 7 Measuring the magnetic B field
o 7.1 Hall effect
o 7.2 SQUID magnetometer
* 8 Magnetization
* 9 The H field
o 9.1 Comparison of the H and B fields
o 9.2 Uses of the H field
+ 9.2.1 Energy stored in magnetic fields
+ 9.2.2 Magnetic circuits
o 9.3 History of B and H
* 10 Special relativity and electromagnetism
o 10.1 Moving magnet and conductor problem
o 10.2 Electric and magnetic fields different aspects of the same phenomenon
* 11 Magnetic field shape descriptions
* 12 Important uses and examples of magnetic field
o 12.1 Earth's magnetic field
o 12.2 Rotating magnetic fields
* 13 See also
* 14 References
* 15 Notes
* 16 External links
[edit] B and H
Unfortunately, the term magnetic field is used for two separate vector fields which are denoted \mathbf{H} and \mathbf{B}. Although the term "magnetic field" was historically reserved for \mathbf{H}, with \mathbf{B} being termed the "magnetic induction", \mathbf{B} is now understood to be the more fundamental entity. Modern writers vary in their usage of \mathbf{B} as the magnetic field.[1] This article follows the convention of referring to \mathbf{B} as the magnetic field and will discuss the more fundamental \mathbf{B} magnetic field, before treating the \mathbf{H} field. But the reader is cautioned that the literature is inconsistent. A technical paper may fail to make a distinction between the magnetic field and magnetic induction, knowing that the audience may know the difference, but as can be seen in the case of a textbook such as Jackson, the distinction is made precisely.
See History of B and H below for further discussion.
[edit] Alternative names for B and H
The vector field \mathbf{H} is known among electrical engineers as the magnetic field intensity or magnetic field strength and is also known among physicists as auxiliary magnetic field or magnetizing field. The vector field \mathbf{B} is known among electrical engineers as magnetic flux density or magnetic induction or simply magnetic field, as used by physicists.
[edit] Units
Main articles: Tesla (unit), Gauss (unit), and Oersted
The magnetic field \mathbf{B} has the SI units of teslas (T), equivalent to webers per square meter (Wb/m²) or volt seconds per square meter (V s/m²).[2][3][4] In cgs units, \mathbf{B} has units of gauss (G). The vector field \mathbf{H} is measured in amperes per meter (A/m) in SI or oersteds (Oe) in cgs units.
[edit] Permanent magnets, magnetic poles, and compasses
The direction of the magnetic field near the poles of a magnet is revealed by placing compasses nearby. As seen here, the magnetic field points towards a magnet's south pole and away from its north pole.
Main article: Magnet
Permanent magnets are objects that produce their own persistent magnetic fields. All permanent magnets have both a north and a south pole. (Magnetic poles always come in north-south pairs.) Like poles repel and opposite poles attract. (See Force on a magnetic dipole due to a non-uniform B below.) The magnetism in a permanent magnet arises from properties of the atoms (in particular the electrons) that compose it. Each atom acts like a little individual magnet. If these magnets line up, they combine to create a macroscopic magnetic effect. For more details about what happens both microscopically and macroscopically, see the article ferromagnetism.
If allowed to twist freely, a magnet will turn to point in the direction of the local magnetic field. (See Torque on a magnetic dipole below.) A compass uses this effect to indicate the direction of the local magnetic field. A small magnet is mounted such that it is free to turn (in a given plane) and its north pole is marked. By definition, the direction of the local magnetic field is the direction that the north pole of a compass (or of any magnet) would tend to point.
A compass placed near the north pole of a magnet will point away from that pole---like poles repel. The opposite occurs if we place the compass near a magnet's south pole. The magnetic field points away from a magnet near its north pole and towards a magnet near its south pole. Not all magnetic fields are describable in terms of poles, though. A straight current-carrying wire, for instance, produces a magnetic field that points neither towards nor away from the wire, but encircles it instead.
[edit] Visualizing the magnetic field
Mapping out the strength and direction of the magnetic field as a function of location is simple in principle. First, measure the strength and direction of the magnetic field at a large number of points. Then mark each location with an arrow (called a vector) pointing in the direction of the local magnetic field with a length proportional to the strength of the magnetic field. This has the unfortunate consequence, though, of cluttering up a graph even for a small number of points. An alternative method of visualizing the magnetic field which greatly simplifies the diagram while containing the same information is to 'connect' the arrows to form "magnetic field lines".
[edit] Magnetic B field lines
Various physical phenomena have the effect of displaying magnetic field lines. For example, iron filings placed in a magnetic field will line up in such a way as to visually show the orientation of the magnetic field (see figure at top). Another place where magnetic fields are visually displayed is in the polar auroras, in which visible streaks of light line up with the local direction of Earth's magnetic field (due to plasma particle dipole interactions). In these phenomena, lines or curves appear that follow along the direction of the local magnetic field.
These field lines provide us with a way to depict or draw the magnetic field (or any other vector field). Technically, field lines are a set of lines through space whose direction at any point is the direction of the local magnetic field, and whose density is proportional to the magnitude of the local magnetic field. Note that when a magnetic field is depicted with field lines, it is not meant to imply that the field is only nonzero along the drawn-in field lines.[5] Rather, the field is typically smooth and continuous everywhere, and can be estimated at any point (whether on a field line or not) by looking at the direction and density of the field lines nearby. The choice of which field lines to draw in such a depiction is arbitrary, apart from the requirement that they be spaced out so that their density approximates the magnitude of the local field. The level of detail at which the magnetic field is depicted can be increased by increasing the number of lines.
Field lines are also a good tool for visualizing magnetic forces. When dealing with magnetic fields in ferromagnetic substances like iron, and in plasmas, the magnetic forces can be understood by imagining that the field lines exert a tension, (like a rubber band) along their length, and a pressure perpendicular to their length on neighboring field lines. The 'unlike' poles of magnets attract because they are linked by many field lines, while 'like' poles repel because the field lines between them don't meet, but run parallel, pushing on each other.
They can also be represented mathematically in terms of "Euler potentials," though that representation is nonlinear and has other problems.[citation needed]
[edit] Magnetic B field lines always form loops
Field lines are a useful way to represent any vector field and often reveal sophisticated properties of fields quite simply. One important property of the magnetic \mathbf{B} field that can be verified with field lines is that magnetic field lines always make complete loops. Magnetic field lines neither start nor end (although they can extend to or from infinity). To date no exception to this rule has been found. (See magnetic monopole below.)
Since magnetic field lines always come in loops, magnetic poles always come in N and S pairs. [6] If a magnetic field line enters a magnet somewhere it has to leave the magnet somewhere else; it is not allowed to have an end point. For this reason as well, cutting a magnet in half will result in two separate magnets each with both a north and a south pole.
[edit] Elementary effects of the magnetic field, B
The myriad of effects that a magnetic field has on different materials and particles can be broken down into four elementary effects that affect either the elementary charges or magnetic dipoles of the particles that make up that material. These effects are:
* Sideways force on a moving charge or current
* Torque on a magnetic dipole
* Force on a magnetic dipole due to a non-uniform B
* Force on a charge due to a changing B
Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H
[edit] Force due to a magnetic field on a moving charge
Main article: Lorentz force
[edit] Force on a charged particle
Beam of electrons moving in a circle. Lighting is caused by excitation of atoms of gas in a bulb.
A charged particle moving in a magnetic \mathbf{B} field will feel a sideways force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the Lorentz force, and is given by
\mathbf{F} = q (\mathbf{v} \times \mathbf{B}),
where
F is the force (in newtons)
q is the electric charge of the particle (in coulombs)
v is the instantaneous velocity of the particle (in meters per second)
B is the magnetic field (in teslas).
The force is always perpendicular to both the velocity of the particle and the magnetic field that created it. Neither a stationary particle nor one moving in the direction of the magnetic field lines will experience a force. For that reason, charged particles move in a circle (or more generally, in a helix) around magnetic field lines; this is called cyclotron motion. Because the magnetic field is always perpendicular to the motion, the magnetic fields can do no work on a charged particle; a magnetic field alone cannot speed up or slow down a charged particle. It can and does, however, change the particle's direction, even to the extent that a force applied in one direction can cause the particle to drift in a perpendicular direction. (See above figure.)
[edit] Force on current-carrying wire
The force on a current carrying wire is similar to that of a moving charge as expected since a charge carrying wire is a collection of moving charges. A current carrying wire will feel a sideways force in the presence of a magnetic field. The Lorentz force on a macroscopic current is often referred to as the Laplace force.
The right-hand rule: For a conventional current or moving positive charge in the direction of the thumb of the right hand and the magnetic field along the direction of the fingers (pointing away from palm) the force on the current will be in a direction out of the palm. The direction of the force is reversed for a negative charge.
[edit] Direction of force
The direction of force on a positive charge or a current is determined by the right-hand rule. See the figure on the right. Using the right hand and pointing the thumb in the direction of the moving positive charge or positive current and the fingers in the direction of the magnetic field the resulting force on the charge will point outwards from the palm. The force on a negative charged particle is in the opposite direction. If both the speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these will produce the same current.) On the other hand, a magnetic field combined with an electric field can distinguish between these, see Hall effect below.
An alternative, similar trick to the right hand rule is Fleming's left hand rule.
[edit] Torque on a magnetic dipole
A magnet placed in a magnetic field will feel a torque that will try to align the magnet with the magnetic field. The torque on a magnet due to an external magnetic field is easy to observe by placing two magnets near each other while allowing one to rotate. This magnetic torque is the basis for how compasses work. It is used to define the direction of the magnetic field (see above).
The magnetic torque also provides the driving torque for simple electric motors. A magnet (called a rotor) placed on a rotating shaft will feel a strong torque if like poles are placed near its own poles. If the magnet that caused the rotation—called the stator—is constantly being flipped such that it always has like poles close to the rotor then the rotor will generate a torque that is transferred to the shaft. The polarity of the rotor can easily be flipped if it is an electromagnet by flipping the direction of the current through its coils.
See Rotating magnetic fields below for an example using this effect with electromagnets.
[edit] Force on a magnetic dipole due to a non-uniform B
The most commonly experienced effect of the magnetic field is the force between two magnets: Like poles repel and opposites attract. It is tempting, therefore, to describe the force between two magnets as a force between magnetic poles. Unfortunately, the idea of "poles" does not accurately reflect what happens inside a magnet (see ferromagnetism). The best of these models, called the "Gilbert model", predicts completely wrong magnetic fields and forces inside magnets even though it produces both the correct force between magnets, and the correct \mathbf{B} field outside the magnets.)
The more physically accurate picture is that a magnetic dipole experiences a force, when placed in a non-uniform external magnetic field such that it will move to maximize the magnetic field in the direction of its magnetic moment. A magnet, therefore, experiences no magnetic force from a uniform magnetic field, no matter how strong it is. (It may experience a torque though.) The south pole of one magnet is attracted to the north pole of another because the magnetic field is stronger nearer to the pole and in the direction of the magnetic moment of the attracted magnet.
Mathematically the force on a magnetic dipole having a magnetic moment \mathbf{m} is:
\mathbf{F} = \mathbf{\nabla} \left(\mathbf{m}\cdot\mathbf{B}\right).
The force on a magnet due to a non-uniform magnetic field therefore can be determined by summing up all of the forces on the elementary dipoles that make up the magnet.
[edit] Electric force due to a changing B
Main articles: Faraday's law of induction and Magnetic flux
If the magnetic field in an area is varying with time it generates an electric field that forms closed loops around that area. A conducting wire that forms a closed loop around the area will have an induced voltage generated by this changing magnetic field. This effect is represented mathematically as Faraday's Law and forms the basis of many generators. Care must be taken to understand that the changing magnetic field is a source for an extended electric field. The changing magnetic field does not only create an electric field at that location; rather it generates an electric field that forms closed loops around the location where the magnetic field is changing.
Mathematically, Faraday's law is most often represented in terms of the change of magnetic flux with time. The magnetic flux is the property of a closed loop (say of a coil of wire) and is the product of the area times the magnetic field that is normal to that area. Engineers and physicists often use magnetic flux as a convenient physical property of a loop(s). They then express the magnetic \mathbf{B} field as the magnetic flux per unit area. It is for this reason that the \mathbf{B} field is often referred to as the "magnetic flux density". This approach has the benefit of making certain calculations easier such as in magnetic circuits. It is typically not used outside of electrical circuits, though, because the magnetic \mathbf{B} field truly is the more 'fundamental' quantity in that it directly connects all of electrodynamics in the simplest manner.
[edit] Elementary sources of magnetic fields
There are three elementary ways to create a magnetic \mathbf{B} field.
* Electrical currents (moving charges)
* Magnetic dipoles
* Changing electric field
These sources are thought to affect the virtual particles that compose the field.
[edit] Electrical currents (moving charges)
All moving charges produce a magnetic field. [7] The magnetic field of a moving charge is very complicated but is well known. (See Jefimenko's equations.) It forms closed loops around a line that is pointing in the direction the charge is moving. The magnetic field of a current on the other hand is much easier to calculate.
[edit] Magnetic field of a steady current
Main article: Biot-Savart law
Current (I) through a wire produces a magnetic field (\mathbf{B}) around the wire. The field is oriented according to the right hand grip rule.
The magnetic field generated by a steady current (a continual flow of charges, for example through a wire, which is constant in time and in which charge is neither building up nor depleting at any point), is described by the Biot-Savart law.[8] This is a consequence of Ampere's law, one of the four Maxwell's equations that describe electricity and magnetism. The magnetic field lines generated by a current carrying wire form concentric circles around the wire. The direction of the magnetic field of the loops is determined by the right hand grip rule. (See figure to the right.) The strength of the magnetic field decreases with distance from the wire.
A current carrying wire can be bent in a loop such that the field is concentrated (and in the same direction) inside of the loop. The field will be weaker outside of the loop. Stacking many such loops to form a solenoid (or long coil) can greatly increase the magnetic field in the center and decrease the magnetic field outside of the solenoid. Such devices are called electromagnets and are extremely important in generating strong and well controlled magnetic fields. An infinitely long solenoid will have a uniform magnetic field inside of the loops and no magnetic field outside. A finite length electromagnet will produce essentially the same magnetic field as a uniform permanent magnet of the same shape and size. An electromagnet has the advantage, though, that you can easily vary the strength (even creating a field in the opposite direction) simply by controlling the input current. One important use is to continually switch the polarity of a stationary electromagnet to force a rotating permanent magnet to continually rotate using the fact that opposite poles attract and like poles repel. This can be used to create an important type of electrical motor.
[edit] Magnetic dipoles
Main article: Magnetic dipole
See also: Spin magnetic moment
Magnetic field lines around a ”magnetostatic dipole” the magnetic dipole itself is in the center and is seen from the side.
The magnetic field due to a permanent magnet is well known. (See the first figure of article.) But, what causes the magnetic field of a permanent magnet? The answer again is that the magnetic field is essentially created due to currents. But this time it is due to the cumulative effect of many small 'currents' of electrons 'orbiting' the nuclei of the magnetic material. Alternatively it is due to the structure of the electron itself which, in some sense, can be thought of as forming a tiny loop of current. (The true nature of the electron's magnetic field is relativistic in nature, but this model often works.) Both of these tiny loops are modeled in terms of what is called the magnetic dipole. The dipole moment of that dipole can be defined as the current times the area of the loop, then an equation for the magnetic field due to that magnetic dipole can be derived. (See the above image for what that magnetic field looks like.) Magnetic field of a larger magnet can be calculated by adding up the magnetic fields of many magnetic dipoles.
[edit] Changing electric field
Main article: Ampere's Law
The final elementary source of magnetic fields is a changing electric field. Just as a changing magnetic field generates an electric field so does a changing electric field generate a magnetic field. (These two effects bootstrap together to form electromagnetic waves, such as light.) Similar to the way magnetic field lines form close loops around a current a time varying electric field generates a magnetic field that forms closed loops around the region where the electric field is changing. The strength of this magnetic field is proportional to the time rate of the change of the electric field (which is called the displacement current). [9] The fact that a changing electric field creates a magnetic field is known as Maxwell's correction to Ampere's Law.
[edit] Magnetic monopole (hypothetical)
Main article: Magnetic monopole
A magnetic monopole is a hypothetical particle (or class of particles) that has, as its name suggests, only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to electric charge.
Modern interest in this concept stems from particle theories, notably Grand Unified Theories and superstring theories, that predict either the existence or the possibility of magnetic monopoles. These theories and others have inspired extensive efforts to search for monopoles. Despite these efforts, no magnetic monopole has been observed to date.[10]
[edit] Definition and mathematical properties of B
There are several different but physically equivalent ways to define the magnetic \mathbf{B} field. In principle any of the above effects due to the magnetic field or any of the sources of the magnetic field can be used to define its magnitude and the direction. Its direction at a given point can be thought of as being the direction that a hypothetical freely rotating small test dipole would rotate to point if it were placed at that point. Its magnitude is defined (in SI units) in terms of the voltage induced per unit area on a current carrying loop in a uniform magnetic field normal to the loop when the magnetic field is reduced to zero in a unit amount of time. The SI unit of magnetic field is the tesla.
The magnetic field vector is a pseudovector (also called an axial vector). (This is a technical statement about how the magnetic field behaves when you reflect the world in a mirror.) This fact is apparent from many of the definitions and properties of the field; for example, the magnitude of the field is proportional to the torque on a dipole, and torque is a well-known pseudovector.
[edit] Maxwell's equations
Main article: Maxwell's equations
As a vector field, the magnetic field has two important mathematical properties that relates the magnetic field to its sources. These two properties, along with the two corresponding properties of the electric field, make up Maxwell's Equations. Maxwell's Equations together with the Lorentz force law form a complete description of classical electrodynamics including both electricity and magnetism.
The first property is that a magnetic \mathbf{B} field line never starts nor ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of the magnetic \mathbf{B} is zero. (Such vector fields are called solenoidal vector fields.) This property is called Gauss' law for magnetism and is equivalent to the statement that there are no magnetic charges or magnetic monopoles.
The second mathematical property of the magnetic field is that it always loops around the source that creates it. This source could be a current, a magnet, or a changing electric field, but it is always within the loops of magnetic field they create. Mathematically, this fact is described by the Ampère-Maxwell equation.
[edit] Measuring the magnetic B field
Main article: Magnetometer
Devices used to measure the local magnetic field are called magnetometers. Important classes of magnetometers include using a rotating coil, Hall effect magnetometers, NMR magnetometer, SQUID magnetometer, and a fluxgate magnetometer. The magnetic fields of distant astronomical objects can be determined by noting their effects on local charged particles. For instance, electrons spiraling around a field line will produce synchotron radiation which is detectable in radio waves.
[edit] Hall effect
Main article: Hall effect
When a current carrying conductor is placed in a transverse magnetic field the sideways Lorentz force on the charge carriers results in a charge separation in a direction perpendicular to both the current and the magnetic field. The resultant voltage, due to that charge separation, is proportional to the applied magnetic field. This is known as the Hall effect. The Hall effect is often used to measure the magnitude of a magnetic field as well as to find the sign of the dominant charge carriers in semiconductors (negative electrons or positive holes).
[edit] SQUID magnetometer
Main article: SQUID
See also: superconductivity
Superconductors are materials with both distinctive electric properties (perfect conductivity) and magnetic properties (such as the Meissner effect, in which many superconductors can perfectly expel magnetic fields). Due to these properties, loops of superconducting material broken up by Josephson junctions can function as very sensitive magnetometers, called SQUIDs. SQUID magnetometers are used in a Scanning SQUID microscope to create a 2D map of the magnetic field.
[edit] Magnetization
See also: Magnetization
The magnetization of all magnetic materials is due to the accumulated effect of many tiny magnetic dipole moments that occur on the atomic level. In non-magnetized materials, these magnetic dipoles are aligned randomly such that the net magnetic moment cancels producing no net magnetic field. But, if the magnetic dipoles of the material becomes aligned a net magnetization and magnetic field is produced. The magnetization field \mathbf{M} represents how strongly a region is magnetized and is defined as the volume density of the net magnetic dipole moment in that region of material.
An equivalent way to represent magnetization is to add all of the currents of the dipole moments that produce the magnetization. The resultant current is called bound current and is the source of the magnetic field due to the magnet. Mathematically, the curl of \mathbf{M} equals the bound current. This is similar to the magnetic \mathbf{B} field. Unlike \mathbf{B}, though, magnetization must begin and end at the poles. (There is no magnetization outside of the material.) Therefore, the divergence of \mathbf{M} must be non-zero near the poles of a magnet.
Most materials produce a magnetization in response to an applied \mathbf{B} field. Typically the response is very weak, though. Paramagnetic materials) produce a magnetization in the same direction as the applied magnetic field. Diamagnetic materials produce a magnetization that opposes the magnetic field. Ferromagnetic materials can have a magnetization independent of an applied \mathbf{B} field with a complex relationship between the two fields.
[edit] The H field
The term 'magnetic field' is also used for the magnetic \mathbf{H} field. The magnetic \mathbf{H} field is used in situations where magnetization is present. Outside of magnetizable materials the \mathbf{H} field differs from the \mathbf{B} field only by a multiplicative constant. Inside of a magnetic material they can be very different. The \mathbf{H} field is defined as:
\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M}) (SI units)
\mathbf{B}=\mathbf{H}+4\pi\mathbf{M} (cgs units),
where \mathbf{M} is magnetization density of any magnetic material. In SI units, \mathbf{H} is measured in amperes per meter (A/m); in cgs units, it is measured in oersteds (Oe).
[edit] Comparison of the H and B fields
When magnetic materials are present, the total magnetic field \mathbf{B} is caused by two different types of currents: free current and bound current. Free currents are the ordinary currents in wires and other conductors, that can be directly controlled and measured. Bound currents are the result of the tiny circular currents inside atoms that are responsible for the magnetization \mathbf{M} of magnetic materials. It is important to distinguish between these two sources of magnetic field for two main reasons. First, free currents are easy to measure and calculate unlike bound currents. Second, in calculating the energy of a magnetic field it is the free currents that do the work, not the bound currents.
Unlike the \mathbf{B} fields which loops around both bound and free currents the \mathbf{H} field loops only around free current. (Mathematically, the curl of \mathbf{H} is equal to the free current (and the free current only).) The portion of the \mathbf{H} due to the bound currents does not form loops at all but field lines starting near the north magnetic pole and ending near the south pole. (By subtracting the magnetization from the B field the bound current sources are essentially converted to Gilbert-like magnetic charge distributions at the poles. More precisely, these 'magnetic charges' are calculated as -\mathbf{\nabla}\cdot\mathbf{M}=\mathbf{\nabla}\cdot\mathbf{H}.)
The magnetic \mathbf{H} field is the same as the magnetic \mathbf{B} field to a multiplicative constant outside of magnetic materials, but is completely different from the magnetic \mathbf{B} field inside a magnetic material. The advantage of the hybrid \mathbf{H} field is that its sources are treated so differently that they can often be isolated from the other. For example, a line integral of the magnetic \mathbf{H} field in a closed loop will yield the total free current in the loop (not including the bound current). Similarly, a surface integral of \mathbf{H} over any closed surface will pick out the 'magnetic charges' within that closed surface.
[edit] Uses of the H field
[edit] Energy stored in magnetic fields
In asking how much energy does it take to create a specific magnetic field using a particular current it is important to distinguish between free and bound currents. It is the free current that we directly 'push' on to create the magnetic field. The bound currents create a magnetic field that the free current has to work against without doing any of the work.
It is not surprising, therefore, that the \mathbf{H} field is important in magnetic energy calculations since it treats the two sources differently. In general the incremental amount of work per unit volume δW needed to cause a small change of magnetic field \delta\mathbf{B} is:
\delta W = \mathbf{H}\cdot\delta\mathbf{B}
The energy density needed, assuming a linear relationship between \mathbf{H} and \mathbf{B} is:
u = \frac{\mathbf{H}\cdot\mathbf{B}}{2}.
If there are no magnetic materials around then we can replace \mathbf{H} with \frac{\mathbf{B}}{\mu_o},
u = \frac{\mathbf{B}\cdot\mathbf{B}}{2\mu_o}.
[edit] Magnetic circuits
Main article: Magnetic circuits
A second use for \mathbf{H} is in magnetic circuits where inside a linear material \mathbf{B} = \mu \mathbf{H}. Here, μ is the permeability of the material. This is similar in form to Ohm's Law \mathbf{J} = \sigma \mathbf{E}, where \mathbf{J} is the current density, σ is the conductance and \mathbf{E} is the Electric field. Extending this analogy we derive the counterpoint to the macroscopic Ohm's law ( \frac{V}{R}=I) as:
\Phi = \frac F R_m,
where \Phi = \int \mathbf{B}\cdot d\mathbf{A} is the magnetic flux in the circuit, F = \int \mathbf{H}\cdot d\mathbf{l} is the magnetomotive force applied to the circuit, and Rm is the reluctance of the circuit. Here the reluctance Rm is a quantity similar in nature to resistance for the flux.
Using this analogy it is straight-forward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of circuit theory.
[edit] History of B and H
The modern understanding that the magnetic \mathbf{B} field is the more fundamental field with the \mathbf{H} being an auxiliary field was not easy to arrive at. Indeed, largely because of mathematical similarities to the electric field, the \mathbf{H} was developed first and was thought at first to be the more fundamental of the two. A brief history of this important transition in thought is instructional in giving insight into the nature of both \mathbf{H} and \mathbf{B}.
Perhaps the earliest description of a magnetic field was performed by Petrus Peregrinus and published in his “Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete” and is dated 1269 A.D. Petrus Peregrinus mapped out the magnetic field on the surface of a spherical magnet. Noting that the resulting field lines crossed at two points he named those points 'poles' in analogy to Earth's poles. Almost three centuries later, near the end of the sixteenth century, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth itself was a magnet. William Gilbert's great work De Magnete was published in 1600 A.D. and helped to establish the study of magnetism as a science.
The modern distinction between the magnetic \mathbf{B} and \mathbf{H} fields does not become important until Siméon-Denis Poisson (1781–1840) developed one of the first mathematical theories of magnetism. Poisson's model, developed in 1824, assumed that magnetism was due to magnetic charges. In analogy to electric charges, these magnetic charges produce a magnetic \mathbf{H} field. In modern notation, Poisson's model was exactly analogous to electrostatics with the magnetic \mathbf{H} field replacing the electric field \mathbf{E} field and the magnetic \mathbf{B} field replacing the auxiliary \mathbf{D} field.
Poisson's model was, unfortunately, incorrect. Magnetism is not due to magnetic charges. Nor is magnetism created by the magnetic \mathbf{H} field polarizing magnetic charge in a material. The model, however, was remarkably successful for being fundamentally wrong. It predicts the correct relationship between the \mathbf{H} field and the \mathbf{B} field, even though it wrongly places \mathbf{H} as the fundamental field with \mathbf{B} as the auxiliary field. It predicts the correct forces between magnets.
It even predicts the correct energy stored in the magnetic fields. By the definition of magnetization, in this model, and in analogy to the physics of springs, the work done per unit volume, in stretching and twisting the bonds between magnetic charge to increment the magnetization by \mu_o\delta\mathbf{M} is W = \mathbf{H}\cdot\mu_o\delta\mathbf{M}. In this model, \mathbf{B} = \mu_o\left(\mathbf{H}+\mathbf{M}\right) is an effective magnetization which includes \mathbf{H} term to account for the energy of setting up the magnetic field in a vacuum. Therefore the total energy density increment needed to increment the magnetic field is W = \mathbf{H}\cdot\delta\mathbf{B}. This is the correct result, but it is derived from an incorrect model.
In retrospect the success of this model is due largely to the remarkable coincidence that from the 'outside' the field of an electric dipole has the exact same form as that of a magnetic dipole. It is therefore only for the physics of magnetism 'inside' of magnetic material where the simpler model of magnetic charges fails. It is also important to note that this model is still useful in many situations dealing with magnetic material. One example of its utility is the concept of magnetic circuits.
The formation of the correct theory of magnetism begins with a series of revolutionary discoveries in 1820, four years before Poisson's model was developed. (The first clue that something was amiss, though, was that unlike electrical charges magnetic poles cannot be separated from each other or form magnetic currents.) The revolution began when Hans Christian Oersted discovered that an electrical current generates a magnetic field that encircles the wire. In a quick succession that discovery was followed by Andre Marie Ampere showing that parallel wires having currents in the same direction attract, and by Jean-Baptiste Biot and Felix Savart developing the correct equation, the Biot-Savart Law, for the magnetic field of a current carrying wire. In 1825, Ampere extended this revolution by publishing his Ampere's Law which provided a more mathematically subtle and correct description of the magnetic field generated by a current than the Biot-Savart Law.
Subsequent development in the nineteenth century interlinked magnetic and electric phenomena even tighter, until the concept of magnetic charge was not needed. Magnetism became an electric phenomenon with even the magnetism of permanent magnets being due to small loops of current in their interior. This development was aided greatly by Michael Faraday, who in 1831 showed that a changing magnetic field generates an encircling electric field. The final blow to magnetic charge was delivered by James Clerk Maxwell in a series of three great works that established Maxwell's equations which formed a complete foundation of classical electrodynamics. Maxwell's equations have only two sources for the electric and magnetic fields: electric charge, and electric current. Though the original source for the magnetic \mathbf{H} field was rejected, the magnetic \mathbf{H} field still had a prominent role in Maxwell's equations. But, now it was as an auxiliary field to the fundamental magnetic \mathbf{B} field.
Although the classical theory of electrodynamics was essentially complete with Maxwell's equations, the twentieth century saw a number of improvements and extensions to the theory. Albert Einstein in his great paper of 1905 that established relativity, showed that both the electric and magnetic fields were part of the same phenomena viewed from different reference frames. Finally, the emergent field of quantum mechanics was merged with electrodynamics to form quantum electrodynamics or QED.
[edit] Special relativity and electromagnetism
Main articles: Classical electromagnetism and special relativity and Relativistic electromagnetism
Magnetic fields played an important role in helping to develop the theory of special relativity.
[edit] Moving magnet and conductor problem
Main article: Moving magnet and conductor problem
Imagine a moving conducting loop that is passing by a stationary magnet. Such a conducting loop will have a current generated in it as it passes through the magnetic field. But why? It is answering this seemingly innocent question that led Albert Einstein to develop his theory of special relativity.
A stationary observer would see an unchanging magnetic field and a moving conducting loop. Since the loop is moving all of the charges that make up the loop are also moving. Each of these charges will have a sideways, Lorentz force, acting on it which generates the current. Meanwhile, an observer on the moving reference frame would see a changing magnetic field and stationary charges. (The loop is not moving in this observers reference frame. The magnet is.) This changing magnetic field generates an electric field.
The stationary observer claims there is only a magnetic field that creates a magnetic force on a moving charge. The moving observer claims that there is both a magnetic and an electric field but all of the force is due to the electric field. Which is true? Does the electric field exist or not? The answer, according to special relativity, is that both observers are right from their reference frame. A pure magnetic field in one reference can be a mixture of magnetic and electric field in another reference frame.
[edit] Electric and magnetic fields different aspects of the same phenomenon
Main article: Electromagnetic tensor
According to special relativity, electric and magnetic forces are part of a single physical phenomenon, electromagnetism; an electric force perceived by one observer will be perceived by another observer in a different frame of reference as a mixture of electric and magnetic forces. A magnetic force can be considered as simply the relativistic part of an electric force when the latter is seen by a moving observer.
More specifically, rather than treating the electric and magnetic fields as separate fields, special relativity shows that they naturally mix together into a rank-2 tensor, called the electromagnetic tensor. This is analogous to the way that special relativity "mixes" space and time into spacetime, and mass, momentum and energy into four-momentum.
[edit] Magnetic field shape descriptions
Schematic quadrupole magnet ("four-pole") magnetic field. There are four steel pole tips, two opposing magnetic north poles and two opposing magnetic south poles.
* An azimuthal magnetic field is one that runs east-west.
* A meridional magnetic field is one that runs north-south. In the solar dynamo model of the Sun, differential rotation of the solar plasma causes the meridional magnetic field to stretch into an azimuthal magnetic field, a process called the omega-effect. The reverse process is called the alpha-effect.[11]
* A dipole magnetic field is one seen around a bar magnet or around a charged elementary particle with nonzero spin.
* A quadrupole magnetic field is one seen, for example, between the poles of four bar magnets. The field strength grows linearly with the radial distance from its longitudinal axis.
* A solenoidal magnetic field is similar to a dipole magnetic field, except that a solid bar magnet is replaced by a hollow electromagnetic coil magnet.
* A toroidal magnetic field occurs in a doughnut-shaped coil, the electric current spiraling around the tube-like surface, and is found, for example, in a tokamak.
* A poloidal magnetic field is generated by a current flowing in a ring, and is found, for example, in a tokamak.
* A radial magnetic field is one in which the field lines are directed from the center outwards, similar to the spokes in a bicycle wheel. An example can be found in a loudspeaker transducers (driver).[12]
* A helical magnetic field is corkscrew-shaped, and sometimes seen in space plasmas such as the Orion Molecular Cloud.[13]
[edit] Important uses and examples of magnetic field
[edit] Earth's magnetic field
A sketch of Earth's magnetic field representing the source of Earth's magnetic field as a magnet. The north pole of earth is near the top of the diagram, the south pole near the bottom. Notice that the south pole of that magnet is deep in Earth's interior below Earth's North Magnetic Pole. Earth's magnetic field is produced in the outer liquid part of its core due to a dynamo that produce electrical currents there.
Main article: Earth's magnetic field
See also: North Magnetic Pole and South Magnetic Pole
Because of Earth's magnetic field, a compass placed anywhere on Earth will turn so that the "north pole" of the magnet inside the compass points roughly north, toward Earth's north magnetic pole in northern Canada. This is the traditional definition of the "north pole" of a magnet, although other equivalent definitions are also possible. One confusion that arises from this definition is that if Earth itself is considered as a magnet, the south pole of that magnet would be the one nearer the north magnetic pole, and vice-versa. (Opposite poles attract and the north pole of the compass magnet is attracted to the north magnetic pole.) The north magnetic pole is so named not because of the polarity of the field there but because of its geographical location.
The figure to the right is a sketch of Earth's magnetic field represented by field lines. The magnetic field at any given point does not point straight toward (or away) from the poles and has a significant up/down component for most locations. (In addition, there is an East/West component as Earth's magnetic poles do not coincide exactly with Earth's geological pole.) The magnetic field is as if there were a magnet deep in Earth's interior.
Earth's magnetic field is probably due to a dynamo that produces electric currents in the outer liquid part of its core. Earth's magnetic field is not constant: Its strength and the location of its poles vary. The poles even periodically reverse direction, in a process called geomagnetic reversal.
[edit] Rotating magnetic fields
Main article: Alternator
The rotating magnetic field is a key principle in the operation of alternating-current motors. A permanent magnet in such a field will rotate so as to maintain its alignment with the external field. This effect was conceptualized by Nikola Tesla, and later utilized in his, and others', early AC (alternating-current) electric motors. A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents. This inequality would cause serious problems in standardization of the conductor size and so, in order to overcome it, three-phase systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees will create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's electrical power supply systems.
Because magnets degrade with time, synchronous motors and induction motors use short-circuited rotors (instead of a magnet) following the rotating magnetic field of a multicoiled stator. The short-circuited turns of the rotor develop eddy currents in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force.
In 1882, Nikola Tesla identified the concept of the rotating magnetic field. In 1885, Galileo Ferraris independently researched the concept. In 1888, Tesla gained U.S. Patent 381,968 for his work. Also in 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Magnetic field (disambiguation).
Electromagnetism
Electricity · Magnetism
[show]Electrostatics
Electric charge · Coulomb's law · Electric field · Electric flux · Gauss's law · Electric potential · Electrostatic induction · Electric dipole moment ·
[hide]Magnetostatics
Ampère’s law · Electric current · Magnetic field · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss’s law for magnetism ·
[show]Electrodynamics
Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potential · Maxwell tensor · Eddy current ·
[show]Electrical Network
Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides ·
[show]Covariant formulation
Electromagnetic tensor · EM Stress-energy tensor · Four-current · Four-potential ·
[show]Scientists
Ampère · Coulomb · Faraday · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Weber ·
This box: view • talk • edit
Magnetic field lines shown by iron filings. The high permeability of individual iron filings causes the magnetic field to be larger at the ends of the filings. This causes individual filings to attract each other, forming elongated clusters that trace out the appearance of lines. It would not be expected that these "lines" be precisely accurate field lines for this magnet; rather, the magnetization of the iron itself would be expected to alter the field somewhat.
A magnetic field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles (such as permanent magnets). When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field. Magnetic fields surround and are created by electric currents, magnetic dipoles, and changing electric fields. Magnetic fields also have their own energy, with an energy density proportional to the square of the field intensity.
There are some notable specific instances of the magnetic field. For the physics of magnetic materials, see magnetism and magnet, and more specifically ferromagnetism, paramagnetism, and diamagnetism. For constant magnetic fields, such as are generated by stationary dipoles and steady currents, see magnetostatics. A changing electric field (which is mathematically identical to a moving electric field) also results in a magnetic field (see electromagnetism).
The magnetic field forms one aspect of electromagnetism. (See also relativistic electromagnetism.) In a simplified form the magnetic field can be thought of as the relativistic part of an electric field. More precisely, magnetic fields are a necessary consequence of the existence of electric fields and special relativity. A pure electric field in one reference frame will be viewed as a combination of both an electric field and a magnetic field in a moving reference frame. Together, the electric and magnetic fields make up the electromagnetic field, which is best known for underlying light and other electromagnetic waves.
Contents
[hide]
* 1 B and H
o 1.1 Alternative names for B and H
o 1.2 Units
* 2 Permanent magnets, magnetic poles, and compasses
* 3 Visualizing the magnetic field
o 3.1 Magnetic B field lines
+ 3.1.1 Magnetic B field lines always form loops
* 4 Elementary effects of the magnetic field, B
o 4.1 Force due to a magnetic field on a moving charge
+ 4.1.1 Force on a charged particle
+ 4.1.2 Force on current-carrying wire
+ 4.1.3 Direction of force
o 4.2 Torque on a magnetic dipole
o 4.3 Force on a magnetic dipole due to a non-uniform B
o 4.4 Electric force due to a changing B
* 5 Elementary sources of magnetic fields
o 5.1 Electrical currents (moving charges)
+ 5.1.1 Magnetic field of a steady current
o 5.2 Magnetic dipoles
o 5.3 Changing electric field
o 5.4 Magnetic monopole (hypothetical)
* 6 Definition and mathematical properties of B
o 6.1 Maxwell's equations
* 7 Measuring the magnetic B field
o 7.1 Hall effect
o 7.2 SQUID magnetometer
* 8 Magnetization
* 9 The H field
o 9.1 Comparison of the H and B fields
o 9.2 Uses of the H field
+ 9.2.1 Energy stored in magnetic fields
+ 9.2.2 Magnetic circuits
o 9.3 History of B and H
* 10 Special relativity and electromagnetism
o 10.1 Moving magnet and conductor problem
o 10.2 Electric and magnetic fields different aspects of the same phenomenon
* 11 Magnetic field shape descriptions
* 12 Important uses and examples of magnetic field
o 12.1 Earth's magnetic field
o 12.2 Rotating magnetic fields
* 13 See also
* 14 References
* 15 Notes
* 16 External links
[edit] B and H
Unfortunately, the term magnetic field is used for two separate vector fields which are denoted \mathbf{H} and \mathbf{B}. Although the term "magnetic field" was historically reserved for \mathbf{H}, with \mathbf{B} being termed the "magnetic induction", \mathbf{B} is now understood to be the more fundamental entity. Modern writers vary in their usage of \mathbf{B} as the magnetic field.[1] This article follows the convention of referring to \mathbf{B} as the magnetic field and will discuss the more fundamental \mathbf{B} magnetic field, before treating the \mathbf{H} field. But the reader is cautioned that the literature is inconsistent. A technical paper may fail to make a distinction between the magnetic field and magnetic induction, knowing that the audience may know the difference, but as can be seen in the case of a textbook such as Jackson, the distinction is made precisely.
See History of B and H below for further discussion.
[edit] Alternative names for B and H
The vector field \mathbf{H} is known among electrical engineers as the magnetic field intensity or magnetic field strength and is also known among physicists as auxiliary magnetic field or magnetizing field. The vector field \mathbf{B} is known among electrical engineers as magnetic flux density or magnetic induction or simply magnetic field, as used by physicists.
[edit] Units
Main articles: Tesla (unit), Gauss (unit), and Oersted
The magnetic field \mathbf{B} has the SI units of teslas (T), equivalent to webers per square meter (Wb/m²) or volt seconds per square meter (V s/m²).[2][3][4] In cgs units, \mathbf{B} has units of gauss (G). The vector field \mathbf{H} is measured in amperes per meter (A/m) in SI or oersteds (Oe) in cgs units.
[edit] Permanent magnets, magnetic poles, and compasses
The direction of the magnetic field near the poles of a magnet is revealed by placing compasses nearby. As seen here, the magnetic field points towards a magnet's south pole and away from its north pole.
Main article: Magnet
Permanent magnets are objects that produce their own persistent magnetic fields. All permanent magnets have both a north and a south pole. (Magnetic poles always come in north-south pairs.) Like poles repel and opposite poles attract. (See Force on a magnetic dipole due to a non-uniform B below.) The magnetism in a permanent magnet arises from properties of the atoms (in particular the electrons) that compose it. Each atom acts like a little individual magnet. If these magnets line up, they combine to create a macroscopic magnetic effect. For more details about what happens both microscopically and macroscopically, see the article ferromagnetism.
If allowed to twist freely, a magnet will turn to point in the direction of the local magnetic field. (See Torque on a magnetic dipole below.) A compass uses this effect to indicate the direction of the local magnetic field. A small magnet is mounted such that it is free to turn (in a given plane) and its north pole is marked. By definition, the direction of the local magnetic field is the direction that the north pole of a compass (or of any magnet) would tend to point.
A compass placed near the north pole of a magnet will point away from that pole---like poles repel. The opposite occurs if we place the compass near a magnet's south pole. The magnetic field points away from a magnet near its north pole and towards a magnet near its south pole. Not all magnetic fields are describable in terms of poles, though. A straight current-carrying wire, for instance, produces a magnetic field that points neither towards nor away from the wire, but encircles it instead.
[edit] Visualizing the magnetic field
Mapping out the strength and direction of the magnetic field as a function of location is simple in principle. First, measure the strength and direction of the magnetic field at a large number of points. Then mark each location with an arrow (called a vector) pointing in the direction of the local magnetic field with a length proportional to the strength of the magnetic field. This has the unfortunate consequence, though, of cluttering up a graph even for a small number of points. An alternative method of visualizing the magnetic field which greatly simplifies the diagram while containing the same information is to 'connect' the arrows to form "magnetic field lines".
[edit] Magnetic B field lines
Various physical phenomena have the effect of displaying magnetic field lines. For example, iron filings placed in a magnetic field will line up in such a way as to visually show the orientation of the magnetic field (see figure at top). Another place where magnetic fields are visually displayed is in the polar auroras, in which visible streaks of light line up with the local direction of Earth's magnetic field (due to plasma particle dipole interactions). In these phenomena, lines or curves appear that follow along the direction of the local magnetic field.
These field lines provide us with a way to depict or draw the magnetic field (or any other vector field). Technically, field lines are a set of lines through space whose direction at any point is the direction of the local magnetic field, and whose density is proportional to the magnitude of the local magnetic field. Note that when a magnetic field is depicted with field lines, it is not meant to imply that the field is only nonzero along the drawn-in field lines.[5] Rather, the field is typically smooth and continuous everywhere, and can be estimated at any point (whether on a field line or not) by looking at the direction and density of the field lines nearby. The choice of which field lines to draw in such a depiction is arbitrary, apart from the requirement that they be spaced out so that their density approximates the magnitude of the local field. The level of detail at which the magnetic field is depicted can be increased by increasing the number of lines.
Field lines are also a good tool for visualizing magnetic forces. When dealing with magnetic fields in ferromagnetic substances like iron, and in plasmas, the magnetic forces can be understood by imagining that the field lines exert a tension, (like a rubber band) along their length, and a pressure perpendicular to their length on neighboring field lines. The 'unlike' poles of magnets attract because they are linked by many field lines, while 'like' poles repel because the field lines between them don't meet, but run parallel, pushing on each other.
They can also be represented mathematically in terms of "Euler potentials," though that representation is nonlinear and has other problems.[citation needed]
[edit] Magnetic B field lines always form loops
Field lines are a useful way to represent any vector field and often reveal sophisticated properties of fields quite simply. One important property of the magnetic \mathbf{B} field that can be verified with field lines is that magnetic field lines always make complete loops. Magnetic field lines neither start nor end (although they can extend to or from infinity). To date no exception to this rule has been found. (See magnetic monopole below.)
Since magnetic field lines always come in loops, magnetic poles always come in N and S pairs. [6] If a magnetic field line enters a magnet somewhere it has to leave the magnet somewhere else; it is not allowed to have an end point. For this reason as well, cutting a magnet in half will result in two separate magnets each with both a north and a south pole.
[edit] Elementary effects of the magnetic field, B
The myriad of effects that a magnetic field has on different materials and particles can be broken down into four elementary effects that affect either the elementary charges or magnetic dipoles of the particles that make up that material. These effects are:
* Sideways force on a moving charge or current
* Torque on a magnetic dipole
* Force on a magnetic dipole due to a non-uniform B
* Force on a charge due to a changing B
Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H
[edit] Force due to a magnetic field on a moving charge
Main article: Lorentz force
[edit] Force on a charged particle
Beam of electrons moving in a circle. Lighting is caused by excitation of atoms of gas in a bulb.
A charged particle moving in a magnetic \mathbf{B} field will feel a sideways force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the Lorentz force, and is given by
\mathbf{F} = q (\mathbf{v} \times \mathbf{B}),
where
F is the force (in newtons)
q is the electric charge of the particle (in coulombs)
v is the instantaneous velocity of the particle (in meters per second)
B is the magnetic field (in teslas).
The force is always perpendicular to both the velocity of the particle and the magnetic field that created it. Neither a stationary particle nor one moving in the direction of the magnetic field lines will experience a force. For that reason, charged particles move in a circle (or more generally, in a helix) around magnetic field lines; this is called cyclotron motion. Because the magnetic field is always perpendicular to the motion, the magnetic fields can do no work on a charged particle; a magnetic field alone cannot speed up or slow down a charged particle. It can and does, however, change the particle's direction, even to the extent that a force applied in one direction can cause the particle to drift in a perpendicular direction. (See above figure.)
[edit] Force on current-carrying wire
The force on a current carrying wire is similar to that of a moving charge as expected since a charge carrying wire is a collection of moving charges. A current carrying wire will feel a sideways force in the presence of a magnetic field. The Lorentz force on a macroscopic current is often referred to as the Laplace force.
The right-hand rule: For a conventional current or moving positive charge in the direction of the thumb of the right hand and the magnetic field along the direction of the fingers (pointing away from palm) the force on the current will be in a direction out of the palm. The direction of the force is reversed for a negative charge.
[edit] Direction of force
The direction of force on a positive charge or a current is determined by the right-hand rule. See the figure on the right. Using the right hand and pointing the thumb in the direction of the moving positive charge or positive current and the fingers in the direction of the magnetic field the resulting force on the charge will point outwards from the palm. The force on a negative charged particle is in the opposite direction. If both the speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these will produce the same current.) On the other hand, a magnetic field combined with an electric field can distinguish between these, see Hall effect below.
An alternative, similar trick to the right hand rule is Fleming's left hand rule.
[edit] Torque on a magnetic dipole
A magnet placed in a magnetic field will feel a torque that will try to align the magnet with the magnetic field. The torque on a magnet due to an external magnetic field is easy to observe by placing two magnets near each other while allowing one to rotate. This magnetic torque is the basis for how compasses work. It is used to define the direction of the magnetic field (see above).
The magnetic torque also provides the driving torque for simple electric motors. A magnet (called a rotor) placed on a rotating shaft will feel a strong torque if like poles are placed near its own poles. If the magnet that caused the rotation—called the stator—is constantly being flipped such that it always has like poles close to the rotor then the rotor will generate a torque that is transferred to the shaft. The polarity of the rotor can easily be flipped if it is an electromagnet by flipping the direction of the current through its coils.
See Rotating magnetic fields below for an example using this effect with electromagnets.
[edit] Force on a magnetic dipole due to a non-uniform B
The most commonly experienced effect of the magnetic field is the force between two magnets: Like poles repel and opposites attract. It is tempting, therefore, to describe the force between two magnets as a force between magnetic poles. Unfortunately, the idea of "poles" does not accurately reflect what happens inside a magnet (see ferromagnetism). The best of these models, called the "Gilbert model", predicts completely wrong magnetic fields and forces inside magnets even though it produces both the correct force between magnets, and the correct \mathbf{B} field outside the magnets.)
The more physically accurate picture is that a magnetic dipole experiences a force, when placed in a non-uniform external magnetic field such that it will move to maximize the magnetic field in the direction of its magnetic moment. A magnet, therefore, experiences no magnetic force from a uniform magnetic field, no matter how strong it is. (It may experience a torque though.) The south pole of one magnet is attracted to the north pole of another because the magnetic field is stronger nearer to the pole and in the direction of the magnetic moment of the attracted magnet.
Mathematically the force on a magnetic dipole having a magnetic moment \mathbf{m} is:
\mathbf{F} = \mathbf{\nabla} \left(\mathbf{m}\cdot\mathbf{B}\right).
The force on a magnet due to a non-uniform magnetic field therefore can be determined by summing up all of the forces on the elementary dipoles that make up the magnet.
[edit] Electric force due to a changing B
Main articles: Faraday's law of induction and Magnetic flux
If the magnetic field in an area is varying with time it generates an electric field that forms closed loops around that area. A conducting wire that forms a closed loop around the area will have an induced voltage generated by this changing magnetic field. This effect is represented mathematically as Faraday's Law and forms the basis of many generators. Care must be taken to understand that the changing magnetic field is a source for an extended electric field. The changing magnetic field does not only create an electric field at that location; rather it generates an electric field that forms closed loops around the location where the magnetic field is changing.
Mathematically, Faraday's law is most often represented in terms of the change of magnetic flux with time. The magnetic flux is the property of a closed loop (say of a coil of wire) and is the product of the area times the magnetic field that is normal to that area. Engineers and physicists often use magnetic flux as a convenient physical property of a loop(s). They then express the magnetic \mathbf{B} field as the magnetic flux per unit area. It is for this reason that the \mathbf{B} field is often referred to as the "magnetic flux density". This approach has the benefit of making certain calculations easier such as in magnetic circuits. It is typically not used outside of electrical circuits, though, because the magnetic \mathbf{B} field truly is the more 'fundamental' quantity in that it directly connects all of electrodynamics in the simplest manner.
[edit] Elementary sources of magnetic fields
There are three elementary ways to create a magnetic \mathbf{B} field.
* Electrical currents (moving charges)
* Magnetic dipoles
* Changing electric field
These sources are thought to affect the virtual particles that compose the field.
[edit] Electrical currents (moving charges)
All moving charges produce a magnetic field. [7] The magnetic field of a moving charge is very complicated but is well known. (See Jefimenko's equations.) It forms closed loops around a line that is pointing in the direction the charge is moving. The magnetic field of a current on the other hand is much easier to calculate.
[edit] Magnetic field of a steady current
Main article: Biot-Savart law
Current (I) through a wire produces a magnetic field (\mathbf{B}) around the wire. The field is oriented according to the right hand grip rule.
The magnetic field generated by a steady current (a continual flow of charges, for example through a wire, which is constant in time and in which charge is neither building up nor depleting at any point), is described by the Biot-Savart law.[8] This is a consequence of Ampere's law, one of the four Maxwell's equations that describe electricity and magnetism. The magnetic field lines generated by a current carrying wire form concentric circles around the wire. The direction of the magnetic field of the loops is determined by the right hand grip rule. (See figure to the right.) The strength of the magnetic field decreases with distance from the wire.
A current carrying wire can be bent in a loop such that the field is concentrated (and in the same direction) inside of the loop. The field will be weaker outside of the loop. Stacking many such loops to form a solenoid (or long coil) can greatly increase the magnetic field in the center and decrease the magnetic field outside of the solenoid. Such devices are called electromagnets and are extremely important in generating strong and well controlled magnetic fields. An infinitely long solenoid will have a uniform magnetic field inside of the loops and no magnetic field outside. A finite length electromagnet will produce essentially the same magnetic field as a uniform permanent magnet of the same shape and size. An electromagnet has the advantage, though, that you can easily vary the strength (even creating a field in the opposite direction) simply by controlling the input current. One important use is to continually switch the polarity of a stationary electromagnet to force a rotating permanent magnet to continually rotate using the fact that opposite poles attract and like poles repel. This can be used to create an important type of electrical motor.
[edit] Magnetic dipoles
Main article: Magnetic dipole
See also: Spin magnetic moment
Magnetic field lines around a ”magnetostatic dipole” the magnetic dipole itself is in the center and is seen from the side.
The magnetic field due to a permanent magnet is well known. (See the first figure of article.) But, what causes the magnetic field of a permanent magnet? The answer again is that the magnetic field is essentially created due to currents. But this time it is due to the cumulative effect of many small 'currents' of electrons 'orbiting' the nuclei of the magnetic material. Alternatively it is due to the structure of the electron itself which, in some sense, can be thought of as forming a tiny loop of current. (The true nature of the electron's magnetic field is relativistic in nature, but this model often works.) Both of these tiny loops are modeled in terms of what is called the magnetic dipole. The dipole moment of that dipole can be defined as the current times the area of the loop, then an equation for the magnetic field due to that magnetic dipole can be derived. (See the above image for what that magnetic field looks like.) Magnetic field of a larger magnet can be calculated by adding up the magnetic fields of many magnetic dipoles.
[edit] Changing electric field
Main article: Ampere's Law
The final elementary source of magnetic fields is a changing electric field. Just as a changing magnetic field generates an electric field so does a changing electric field generate a magnetic field. (These two effects bootstrap together to form electromagnetic waves, such as light.) Similar to the way magnetic field lines form close loops around a current a time varying electric field generates a magnetic field that forms closed loops around the region where the electric field is changing. The strength of this magnetic field is proportional to the time rate of the change of the electric field (which is called the displacement current). [9] The fact that a changing electric field creates a magnetic field is known as Maxwell's correction to Ampere's Law.
[edit] Magnetic monopole (hypothetical)
Main article: Magnetic monopole
A magnetic monopole is a hypothetical particle (or class of particles) that has, as its name suggests, only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to electric charge.
Modern interest in this concept stems from particle theories, notably Grand Unified Theories and superstring theories, that predict either the existence or the possibility of magnetic monopoles. These theories and others have inspired extensive efforts to search for monopoles. Despite these efforts, no magnetic monopole has been observed to date.[10]
[edit] Definition and mathematical properties of B
There are several different but physically equivalent ways to define the magnetic \mathbf{B} field. In principle any of the above effects due to the magnetic field or any of the sources of the magnetic field can be used to define its magnitude and the direction. Its direction at a given point can be thought of as being the direction that a hypothetical freely rotating small test dipole would rotate to point if it were placed at that point. Its magnitude is defined (in SI units) in terms of the voltage induced per unit area on a current carrying loop in a uniform magnetic field normal to the loop when the magnetic field is reduced to zero in a unit amount of time. The SI unit of magnetic field is the tesla.
The magnetic field vector is a pseudovector (also called an axial vector). (This is a technical statement about how the magnetic field behaves when you reflect the world in a mirror.) This fact is apparent from many of the definitions and properties of the field; for example, the magnitude of the field is proportional to the torque on a dipole, and torque is a well-known pseudovector.
[edit] Maxwell's equations
Main article: Maxwell's equations
As a vector field, the magnetic field has two important mathematical properties that relates the magnetic field to its sources. These two properties, along with the two corresponding properties of the electric field, make up Maxwell's Equations. Maxwell's Equations together with the Lorentz force law form a complete description of classical electrodynamics including both electricity and magnetism.
The first property is that a magnetic \mathbf{B} field line never starts nor ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of the magnetic \mathbf{B} is zero. (Such vector fields are called solenoidal vector fields.) This property is called Gauss' law for magnetism and is equivalent to the statement that there are no magnetic charges or magnetic monopoles.
The second mathematical property of the magnetic field is that it always loops around the source that creates it. This source could be a current, a magnet, or a changing electric field, but it is always within the loops of magnetic field they create. Mathematically, this fact is described by the Ampère-Maxwell equation.
[edit] Measuring the magnetic B field
Main article: Magnetometer
Devices used to measure the local magnetic field are called magnetometers. Important classes of magnetometers include using a rotating coil, Hall effect magnetometers, NMR magnetometer, SQUID magnetometer, and a fluxgate magnetometer. The magnetic fields of distant astronomical objects can be determined by noting their effects on local charged particles. For instance, electrons spiraling around a field line will produce synchotron radiation which is detectable in radio waves.
[edit] Hall effect
Main article: Hall effect
When a current carrying conductor is placed in a transverse magnetic field the sideways Lorentz force on the charge carriers results in a charge separation in a direction perpendicular to both the current and the magnetic field. The resultant voltage, due to that charge separation, is proportional to the applied magnetic field. This is known as the Hall effect. The Hall effect is often used to measure the magnitude of a magnetic field as well as to find the sign of the dominant charge carriers in semiconductors (negative electrons or positive holes).
[edit] SQUID magnetometer
Main article: SQUID
See also: superconductivity
Superconductors are materials with both distinctive electric properties (perfect conductivity) and magnetic properties (such as the Meissner effect, in which many superconductors can perfectly expel magnetic fields). Due to these properties, loops of superconducting material broken up by Josephson junctions can function as very sensitive magnetometers, called SQUIDs. SQUID magnetometers are used in a Scanning SQUID microscope to create a 2D map of the magnetic field.
[edit] Magnetization
See also: Magnetization
The magnetization of all magnetic materials is due to the accumulated effect of many tiny magnetic dipole moments that occur on the atomic level. In non-magnetized materials, these magnetic dipoles are aligned randomly such that the net magnetic moment cancels producing no net magnetic field. But, if the magnetic dipoles of the material becomes aligned a net magnetization and magnetic field is produced. The magnetization field \mathbf{M} represents how strongly a region is magnetized and is defined as the volume density of the net magnetic dipole moment in that region of material.
An equivalent way to represent magnetization is to add all of the currents of the dipole moments that produce the magnetization. The resultant current is called bound current and is the source of the magnetic field due to the magnet. Mathematically, the curl of \mathbf{M} equals the bound current. This is similar to the magnetic \mathbf{B} field. Unlike \mathbf{B}, though, magnetization must begin and end at the poles. (There is no magnetization outside of the material.) Therefore, the divergence of \mathbf{M} must be non-zero near the poles of a magnet.
Most materials produce a magnetization in response to an applied \mathbf{B} field. Typically the response is very weak, though. Paramagnetic materials) produce a magnetization in the same direction as the applied magnetic field. Diamagnetic materials produce a magnetization that opposes the magnetic field. Ferromagnetic materials can have a magnetization independent of an applied \mathbf{B} field with a complex relationship between the two fields.
[edit] The H field
The term 'magnetic field' is also used for the magnetic \mathbf{H} field. The magnetic \mathbf{H} field is used in situations where magnetization is present. Outside of magnetizable materials the \mathbf{H} field differs from the \mathbf{B} field only by a multiplicative constant. Inside of a magnetic material they can be very different. The \mathbf{H} field is defined as:
\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M}) (SI units)
\mathbf{B}=\mathbf{H}+4\pi\mathbf{M} (cgs units),
where \mathbf{M} is magnetization density of any magnetic material. In SI units, \mathbf{H} is measured in amperes per meter (A/m); in cgs units, it is measured in oersteds (Oe).
[edit] Comparison of the H and B fields
When magnetic materials are present, the total magnetic field \mathbf{B} is caused by two different types of currents: free current and bound current. Free currents are the ordinary currents in wires and other conductors, that can be directly controlled and measured. Bound currents are the result of the tiny circular currents inside atoms that are responsible for the magnetization \mathbf{M} of magnetic materials. It is important to distinguish between these two sources of magnetic field for two main reasons. First, free currents are easy to measure and calculate unlike bound currents. Second, in calculating the energy of a magnetic field it is the free currents that do the work, not the bound currents.
Unlike the \mathbf{B} fields which loops around both bound and free currents the \mathbf{H} field loops only around free current. (Mathematically, the curl of \mathbf{H} is equal to the free current (and the free current only).) The portion of the \mathbf{H} due to the bound currents does not form loops at all but field lines starting near the north magnetic pole and ending near the south pole. (By subtracting the magnetization from the B field the bound current sources are essentially converted to Gilbert-like magnetic charge distributions at the poles. More precisely, these 'magnetic charges' are calculated as -\mathbf{\nabla}\cdot\mathbf{M}=\mathbf{\nabla}\cdot\mathbf{H}.)
The magnetic \mathbf{H} field is the same as the magnetic \mathbf{B} field to a multiplicative constant outside of magnetic materials, but is completely different from the magnetic \mathbf{B} field inside a magnetic material. The advantage of the hybrid \mathbf{H} field is that its sources are treated so differently that they can often be isolated from the other. For example, a line integral of the magnetic \mathbf{H} field in a closed loop will yield the total free current in the loop (not including the bound current). Similarly, a surface integral of \mathbf{H} over any closed surface will pick out the 'magnetic charges' within that closed surface.
[edit] Uses of the H field
[edit] Energy stored in magnetic fields
In asking how much energy does it take to create a specific magnetic field using a particular current it is important to distinguish between free and bound currents. It is the free current that we directly 'push' on to create the magnetic field. The bound currents create a magnetic field that the free current has to work against without doing any of the work.
It is not surprising, therefore, that the \mathbf{H} field is important in magnetic energy calculations since it treats the two sources differently. In general the incremental amount of work per unit volume δW needed to cause a small change of magnetic field \delta\mathbf{B} is:
\delta W = \mathbf{H}\cdot\delta\mathbf{B}
The energy density needed, assuming a linear relationship between \mathbf{H} and \mathbf{B} is:
u = \frac{\mathbf{H}\cdot\mathbf{B}}{2}.
If there are no magnetic materials around then we can replace \mathbf{H} with \frac{\mathbf{B}}{\mu_o},
u = \frac{\mathbf{B}\cdot\mathbf{B}}{2\mu_o}.
[edit] Magnetic circuits
Main article: Magnetic circuits
A second use for \mathbf{H} is in magnetic circuits where inside a linear material \mathbf{B} = \mu \mathbf{H}. Here, μ is the permeability of the material. This is similar in form to Ohm's Law \mathbf{J} = \sigma \mathbf{E}, where \mathbf{J} is the current density, σ is the conductance and \mathbf{E} is the Electric field. Extending this analogy we derive the counterpoint to the macroscopic Ohm's law ( \frac{V}{R}=I) as:
\Phi = \frac F R_m,
where \Phi = \int \mathbf{B}\cdot d\mathbf{A} is the magnetic flux in the circuit, F = \int \mathbf{H}\cdot d\mathbf{l} is the magnetomotive force applied to the circuit, and Rm is the reluctance of the circuit. Here the reluctance Rm is a quantity similar in nature to resistance for the flux.
Using this analogy it is straight-forward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of circuit theory.
[edit] History of B and H
The modern understanding that the magnetic \mathbf{B} field is the more fundamental field with the \mathbf{H} being an auxiliary field was not easy to arrive at. Indeed, largely because of mathematical similarities to the electric field, the \mathbf{H} was developed first and was thought at first to be the more fundamental of the two. A brief history of this important transition in thought is instructional in giving insight into the nature of both \mathbf{H} and \mathbf{B}.
Perhaps the earliest description of a magnetic field was performed by Petrus Peregrinus and published in his “Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete” and is dated 1269 A.D. Petrus Peregrinus mapped out the magnetic field on the surface of a spherical magnet. Noting that the resulting field lines crossed at two points he named those points 'poles' in analogy to Earth's poles. Almost three centuries later, near the end of the sixteenth century, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth itself was a magnet. William Gilbert's great work De Magnete was published in 1600 A.D. and helped to establish the study of magnetism as a science.
The modern distinction between the magnetic \mathbf{B} and \mathbf{H} fields does not become important until Siméon-Denis Poisson (1781–1840) developed one of the first mathematical theories of magnetism. Poisson's model, developed in 1824, assumed that magnetism was due to magnetic charges. In analogy to electric charges, these magnetic charges produce a magnetic \mathbf{H} field. In modern notation, Poisson's model was exactly analogous to electrostatics with the magnetic \mathbf{H} field replacing the electric field \mathbf{E} field and the magnetic \mathbf{B} field replacing the auxiliary \mathbf{D} field.
Poisson's model was, unfortunately, incorrect. Magnetism is not due to magnetic charges. Nor is magnetism created by the magnetic \mathbf{H} field polarizing magnetic charge in a material. The model, however, was remarkably successful for being fundamentally wrong. It predicts the correct relationship between the \mathbf{H} field and the \mathbf{B} field, even though it wrongly places \mathbf{H} as the fundamental field with \mathbf{B} as the auxiliary field. It predicts the correct forces between magnets.
It even predicts the correct energy stored in the magnetic fields. By the definition of magnetization, in this model, and in analogy to the physics of springs, the work done per unit volume, in stretching and twisting the bonds between magnetic charge to increment the magnetization by \mu_o\delta\mathbf{M} is W = \mathbf{H}\cdot\mu_o\delta\mathbf{M}. In this model, \mathbf{B} = \mu_o\left(\mathbf{H}+\mathbf{M}\right) is an effective magnetization which includes \mathbf{H} term to account for the energy of setting up the magnetic field in a vacuum. Therefore the total energy density increment needed to increment the magnetic field is W = \mathbf{H}\cdot\delta\mathbf{B}. This is the correct result, but it is derived from an incorrect model.
In retrospect the success of this model is due largely to the remarkable coincidence that from the 'outside' the field of an electric dipole has the exact same form as that of a magnetic dipole. It is therefore only for the physics of magnetism 'inside' of magnetic material where the simpler model of magnetic charges fails. It is also important to note that this model is still useful in many situations dealing with magnetic material. One example of its utility is the concept of magnetic circuits.
The formation of the correct theory of magnetism begins with a series of revolutionary discoveries in 1820, four years before Poisson's model was developed. (The first clue that something was amiss, though, was that unlike electrical charges magnetic poles cannot be separated from each other or form magnetic currents.) The revolution began when Hans Christian Oersted discovered that an electrical current generates a magnetic field that encircles the wire. In a quick succession that discovery was followed by Andre Marie Ampere showing that parallel wires having currents in the same direction attract, and by Jean-Baptiste Biot and Felix Savart developing the correct equation, the Biot-Savart Law, for the magnetic field of a current carrying wire. In 1825, Ampere extended this revolution by publishing his Ampere's Law which provided a more mathematically subtle and correct description of the magnetic field generated by a current than the Biot-Savart Law.
Subsequent development in the nineteenth century interlinked magnetic and electric phenomena even tighter, until the concept of magnetic charge was not needed. Magnetism became an electric phenomenon with even the magnetism of permanent magnets being due to small loops of current in their interior. This development was aided greatly by Michael Faraday, who in 1831 showed that a changing magnetic field generates an encircling electric field. The final blow to magnetic charge was delivered by James Clerk Maxwell in a series of three great works that established Maxwell's equations which formed a complete foundation of classical electrodynamics. Maxwell's equations have only two sources for the electric and magnetic fields: electric charge, and electric current. Though the original source for the magnetic \mathbf{H} field was rejected, the magnetic \mathbf{H} field still had a prominent role in Maxwell's equations. But, now it was as an auxiliary field to the fundamental magnetic \mathbf{B} field.
Although the classical theory of electrodynamics was essentially complete with Maxwell's equations, the twentieth century saw a number of improvements and extensions to the theory. Albert Einstein in his great paper of 1905 that established relativity, showed that both the electric and magnetic fields were part of the same phenomena viewed from different reference frames. Finally, the emergent field of quantum mechanics was merged with electrodynamics to form quantum electrodynamics or QED.
[edit] Special relativity and electromagnetism
Main articles: Classical electromagnetism and special relativity and Relativistic electromagnetism
Magnetic fields played an important role in helping to develop the theory of special relativity.
[edit] Moving magnet and conductor problem
Main article: Moving magnet and conductor problem
Imagine a moving conducting loop that is passing by a stationary magnet. Such a conducting loop will have a current generated in it as it passes through the magnetic field. But why? It is answering this seemingly innocent question that led Albert Einstein to develop his theory of special relativity.
A stationary observer would see an unchanging magnetic field and a moving conducting loop. Since the loop is moving all of the charges that make up the loop are also moving. Each of these charges will have a sideways, Lorentz force, acting on it which generates the current. Meanwhile, an observer on the moving reference frame would see a changing magnetic field and stationary charges. (The loop is not moving in this observers reference frame. The magnet is.) This changing magnetic field generates an electric field.
The stationary observer claims there is only a magnetic field that creates a magnetic force on a moving charge. The moving observer claims that there is both a magnetic and an electric field but all of the force is due to the electric field. Which is true? Does the electric field exist or not? The answer, according to special relativity, is that both observers are right from their reference frame. A pure magnetic field in one reference can be a mixture of magnetic and electric field in another reference frame.
[edit] Electric and magnetic fields different aspects of the same phenomenon
Main article: Electromagnetic tensor
According to special relativity, electric and magnetic forces are part of a single physical phenomenon, electromagnetism; an electric force perceived by one observer will be perceived by another observer in a different frame of reference as a mixture of electric and magnetic forces. A magnetic force can be considered as simply the relativistic part of an electric force when the latter is seen by a moving observer.
More specifically, rather than treating the electric and magnetic fields as separate fields, special relativity shows that they naturally mix together into a rank-2 tensor, called the electromagnetic tensor. This is analogous to the way that special relativity "mixes" space and time into spacetime, and mass, momentum and energy into four-momentum.
[edit] Magnetic field shape descriptions
Schematic quadrupole magnet ("four-pole") magnetic field. There are four steel pole tips, two opposing magnetic north poles and two opposing magnetic south poles.
* An azimuthal magnetic field is one that runs east-west.
* A meridional magnetic field is one that runs north-south. In the solar dynamo model of the Sun, differential rotation of the solar plasma causes the meridional magnetic field to stretch into an azimuthal magnetic field, a process called the omega-effect. The reverse process is called the alpha-effect.[11]
* A dipole magnetic field is one seen around a bar magnet or around a charged elementary particle with nonzero spin.
* A quadrupole magnetic field is one seen, for example, between the poles of four bar magnets. The field strength grows linearly with the radial distance from its longitudinal axis.
* A solenoidal magnetic field is similar to a dipole magnetic field, except that a solid bar magnet is replaced by a hollow electromagnetic coil magnet.
* A toroidal magnetic field occurs in a doughnut-shaped coil, the electric current spiraling around the tube-like surface, and is found, for example, in a tokamak.
* A poloidal magnetic field is generated by a current flowing in a ring, and is found, for example, in a tokamak.
* A radial magnetic field is one in which the field lines are directed from the center outwards, similar to the spokes in a bicycle wheel. An example can be found in a loudspeaker transducers (driver).[12]
* A helical magnetic field is corkscrew-shaped, and sometimes seen in space plasmas such as the Orion Molecular Cloud.[13]
[edit] Important uses and examples of magnetic field
[edit] Earth's magnetic field
A sketch of Earth's magnetic field representing the source of Earth's magnetic field as a magnet. The north pole of earth is near the top of the diagram, the south pole near the bottom. Notice that the south pole of that magnet is deep in Earth's interior below Earth's North Magnetic Pole. Earth's magnetic field is produced in the outer liquid part of its core due to a dynamo that produce electrical currents there.
Main article: Earth's magnetic field
See also: North Magnetic Pole and South Magnetic Pole
Because of Earth's magnetic field, a compass placed anywhere on Earth will turn so that the "north pole" of the magnet inside the compass points roughly north, toward Earth's north magnetic pole in northern Canada. This is the traditional definition of the "north pole" of a magnet, although other equivalent definitions are also possible. One confusion that arises from this definition is that if Earth itself is considered as a magnet, the south pole of that magnet would be the one nearer the north magnetic pole, and vice-versa. (Opposite poles attract and the north pole of the compass magnet is attracted to the north magnetic pole.) The north magnetic pole is so named not because of the polarity of the field there but because of its geographical location.
The figure to the right is a sketch of Earth's magnetic field represented by field lines. The magnetic field at any given point does not point straight toward (or away) from the poles and has a significant up/down component for most locations. (In addition, there is an East/West component as Earth's magnetic poles do not coincide exactly with Earth's geological pole.) The magnetic field is as if there were a magnet deep in Earth's interior.
Earth's magnetic field is probably due to a dynamo that produces electric currents in the outer liquid part of its core. Earth's magnetic field is not constant: Its strength and the location of its poles vary. The poles even periodically reverse direction, in a process called geomagnetic reversal.
[edit] Rotating magnetic fields
Main article: Alternator
The rotating magnetic field is a key principle in the operation of alternating-current motors. A permanent magnet in such a field will rotate so as to maintain its alignment with the external field. This effect was conceptualized by Nikola Tesla, and later utilized in his, and others', early AC (alternating-current) electric motors. A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents. This inequality would cause serious problems in standardization of the conductor size and so, in order to overcome it, three-phase systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees will create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's electrical power supply systems.
Because magnets degrade with time, synchronous motors and induction motors use short-circuited rotors (instead of a magnet) following the rotating magnetic field of a multicoiled stator. The short-circuited turns of the rotor develop eddy currents in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force.
In 1882, Nikola Tesla identified the concept of the rotating magnetic field. In 1885, Galileo Ferraris independently researched the concept. In 1888, Tesla gained U.S. Patent 381,968 for his work. Also in 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.
Electricity
By Keiji Oenoki and Hector Judez [physics@amersol.edu.pe]
We often experience the power of nature. One of the most fearsome natural phenomena is lightning. Some people might be curious why and how it occurs. Today, we know that lightning is a flow of electricity. In this chapter we are going to study about characteristics of electricity.
1. Net Electrical Charge
2. Conductors and Insulators
3. Charged Objects
1. Method of Charging
4. Unit of Charge: The Coulomb
5. Coulomb's Law
6. Chapter Quiz
Section
Section 1. Net Electrical Charge
Electrons around an atom Matters are made of atoms. An atom is basically composed of three different components -- electrons, protons, and neutrons. An electron can be removed easily from an atom.
Normally, an atom is electrically neutral, which means that there are equal numbers of protons and electrons. Positive charge of protons is balanced by negative charge of electrons. It has no net electrical charge.
When atoms gain or lose electrons, they are called "ions."
* A positive ion is a cation that misses electron(s).
* A negative ion is an anion that gains extra electron(s).
Important!
repell and attract What is charge? Objects that exert electric forces are said to have charge. Charge is the source of electrical force. There are two kinds of electrical charges, positive and negative. Same charges (+ and +, or - and -) repel and opposite charges (+ and -) attract each other.
The law of consevation of charge says that charge is neither created nor destroyed. This state ment brings us new question. What do you think about this statement?
Section
Section 2. Conductors and Insulators
Important!
Substances can be classified into three types -- insulators, conductors, and semiconductors.
Insulators are materials which allow very little electrical charges and heat energy to flow. Plastics, glass, dry air and wood are examples of insulators.
Conductors are materials which electrical charges and heat energy can be transmitted very easily. Almost all metals such as gold, silver, copper, iron, and lead are good conductors.
Semiconductors are materials which allow the electrical charges to flow better than insulators, but less than conductors. Examples are silicon and germanium.
Example Problem 1.
Which one of these is a conductor?
(a) dry air (b) lead (c) silicon (d) glass
(e.g. "a" )
Section
Section 3. Charged Objects
Net charges
When two objects are rubbed together, some electrons from one object move to another object. For example, when a plastic bar is rubbed with fur, electrons will move from the fur to the plastic stick. Therefore, plastic bar will be negatively charged and the fur will be positively charged.
a. Methods of Charging
Charging by Induction
(Electroscope) When you bring a negatively charged object close to another object, electrons in the second object will be repelled from the first object. Therefore, that end will have a negative charge. This process is called charging by induction.
When a negatively charged object touches a neutral body, electrons will spread on both objects and make both objects negatively charged. This process is called charging by conduction. The other case, positively charged object touching the neutral body, is just the same in principle. Charge by Conduction
(Electroscope)
How does lightning start and end?
Section
Section 4. Unit of Electrical Charge: The Coulomb " C "
Important!
The symbol for electric charge is written q, -q or Q. The unit of electric charge is coulomb "C". The charge of one electron is equal to the charge of one proton, which is 1.6 * 10-19C. This number is given a symbol "e".
Example Problem 2.
How many electrons are there in 1 C of charge?
(e.g. "1.00*10^10")
Section
Section 5. Coulomb's Law
Important!
The magnitude of force that a particle exerts on another particle is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Direction of the force between particles
where:
F is the force between the two particles,
q1 is the net charge on particle A,
q2 is the net charge on particle B,
d is the distance between the particles,
k is a proportionality constant which is 9.0 * 109 Nm2/C2.
The direction of the force is on the line from one particle to the other.
Example problem 3.
Object X has a positive charge of 5.0 * 10-6 C. Object Y, carrying a positive charge of 2.5 * 10-6 C, is 0.03 m away.
A. Calculate the force on X.
(e.g. "+1.0*10^10 N", or "-1.0*10^10 N")
B. What would the force be if the charge on Y were negative.
Solution
Example problem 4.
An object, A, with +8.25 * 10-6 C charge, has two other charges nearby. Object B, -3.5 * 10-6 C, is 0.030 m to the right. Object C, +2.50 * 10-6 C, is 0.050 m below. What is the net force and the angle on A?
(e.g. "1.0*10^10 N, 10 deg.")
The electric force is vector quantity. Try to solve the problem graphically.
Solution
By Keiji Oenoki and Hector Judez [physics@amersol.edu.pe]
We often experience the power of nature. One of the most fearsome natural phenomena is lightning. Some people might be curious why and how it occurs. Today, we know that lightning is a flow of electricity. In this chapter we are going to study about characteristics of electricity.
1. Net Electrical Charge
2. Conductors and Insulators
3. Charged Objects
1. Method of Charging
4. Unit of Charge: The Coulomb
5. Coulomb's Law
6. Chapter Quiz
Section
Section 1. Net Electrical Charge
Electrons around an atom Matters are made of atoms. An atom is basically composed of three different components -- electrons, protons, and neutrons. An electron can be removed easily from an atom.
Normally, an atom is electrically neutral, which means that there are equal numbers of protons and electrons. Positive charge of protons is balanced by negative charge of electrons. It has no net electrical charge.
When atoms gain or lose electrons, they are called "ions."
* A positive ion is a cation that misses electron(s).
* A negative ion is an anion that gains extra electron(s).
Important!
repell and attract What is charge? Objects that exert electric forces are said to have charge. Charge is the source of electrical force. There are two kinds of electrical charges, positive and negative. Same charges (+ and +, or - and -) repel and opposite charges (+ and -) attract each other.
The law of consevation of charge says that charge is neither created nor destroyed. This state ment brings us new question. What do you think about this statement?
Section
Section 2. Conductors and Insulators
Important!
Substances can be classified into three types -- insulators, conductors, and semiconductors.
Insulators are materials which allow very little electrical charges and heat energy to flow. Plastics, glass, dry air and wood are examples of insulators.
Conductors are materials which electrical charges and heat energy can be transmitted very easily. Almost all metals such as gold, silver, copper, iron, and lead are good conductors.
Semiconductors are materials which allow the electrical charges to flow better than insulators, but less than conductors. Examples are silicon and germanium.
Example Problem 1.
Which one of these is a conductor?
(a) dry air (b) lead (c) silicon (d) glass
(e.g. "a" )
Section
Section 3. Charged Objects
Net charges
When two objects are rubbed together, some electrons from one object move to another object. For example, when a plastic bar is rubbed with fur, electrons will move from the fur to the plastic stick. Therefore, plastic bar will be negatively charged and the fur will be positively charged.
a. Methods of Charging
Charging by Induction
(Electroscope) When you bring a negatively charged object close to another object, electrons in the second object will be repelled from the first object. Therefore, that end will have a negative charge. This process is called charging by induction.
When a negatively charged object touches a neutral body, electrons will spread on both objects and make both objects negatively charged. This process is called charging by conduction. The other case, positively charged object touching the neutral body, is just the same in principle. Charge by Conduction
(Electroscope)
How does lightning start and end?
Section
Section 4. Unit of Electrical Charge: The Coulomb " C "
Important!
The symbol for electric charge is written q, -q or Q. The unit of electric charge is coulomb "C". The charge of one electron is equal to the charge of one proton, which is 1.6 * 10-19C. This number is given a symbol "e".
Example Problem 2.
How many electrons are there in 1 C of charge?
(e.g. "1.00*10^10")
Section
Section 5. Coulomb's Law
Important!
The magnitude of force that a particle exerts on another particle is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Direction of the force between particles
where:
F is the force between the two particles,
q1 is the net charge on particle A,
q2 is the net charge on particle B,
d is the distance between the particles,
k is a proportionality constant which is 9.0 * 109 Nm2/C2.
The direction of the force is on the line from one particle to the other.
Example problem 3.
Object X has a positive charge of 5.0 * 10-6 C. Object Y, carrying a positive charge of 2.5 * 10-6 C, is 0.03 m away.
A. Calculate the force on X.
(e.g. "+1.0*10^10 N", or "-1.0*10^10 N")
B. What would the force be if the charge on Y were negative.
Solution
Example problem 4.
An object, A, with +8.25 * 10-6 C charge, has two other charges nearby. Object B, -3.5 * 10-6 C, is 0.030 m to the right. Object C, +2.50 * 10-6 C, is 0.050 m below. What is the net force and the angle on A?
(e.g. "1.0*10^10 N, 10 deg.")
The electric force is vector quantity. Try to solve the problem graphically.
Solution
Non-contact force
From Wikipedia, the free encyclopedia
Jump to: navigation, search
A non-contact force is any force applied to an object by another body that is not in direct contact with it. The most common example of a non-contact force is gravity. A non-contact force is different from a contact force, which is a force applied to a body by another body that is in contact with it. However it is to be noted that the origin of all contact forces can be traced to non-contact forces.
Examples of non-contact forces include:
* Gravity, a non-contact force between two objects. The force exerted on each body by the other through gravity is proportional to the mass of the first body times the mass of the second body divided by the square of the distance between them. The direction of the force is from the body acted on towards the body applying the force. A human body's weight is a non-contact force exerted by the Earth on their mass.
* Magnetism, a unique non-contact force similar to gravity that generally acts to produce a stronger force over much shorter distances, and only affects specific materials. Magnetism can produce a force towards or away from the magnet.
* Strong nuclear force: Unlike Gravity and electromagnetism, the strong nuclear force is a short distance force that takes place between fundamental particles within a nucleus. It is charge independent and acts equally between a proton and a proton, a neutron and a neutron and a proton and a neutron. The strong nuclear force is the strongest force in nature; however, it's range is small (acting only over distances of the order of 10-15 m)
* Weak nuclear force: The weak nuclear force appears only in certain nuclear processes like β decay of a nucleus, in which the nucleus emits a β particle and an uncharged particle called a neutrino. Both the strong and weak forces form an important part of quantum mechanics.gooooo
From Wikipedia, the free encyclopedia
Jump to: navigation, search
A non-contact force is any force applied to an object by another body that is not in direct contact with it. The most common example of a non-contact force is gravity. A non-contact force is different from a contact force, which is a force applied to a body by another body that is in contact with it. However it is to be noted that the origin of all contact forces can be traced to non-contact forces.
Examples of non-contact forces include:
* Gravity, a non-contact force between two objects. The force exerted on each body by the other through gravity is proportional to the mass of the first body times the mass of the second body divided by the square of the distance between them. The direction of the force is from the body acted on towards the body applying the force. A human body's weight is a non-contact force exerted by the Earth on their mass.
* Magnetism, a unique non-contact force similar to gravity that generally acts to produce a stronger force over much shorter distances, and only affects specific materials. Magnetism can produce a force towards or away from the magnet.
* Strong nuclear force: Unlike Gravity and electromagnetism, the strong nuclear force is a short distance force that takes place between fundamental particles within a nucleus. It is charge independent and acts equally between a proton and a proton, a neutron and a neutron and a proton and a neutron. The strong nuclear force is the strongest force in nature; however, it's range is small (acting only over distances of the order of 10-15 m)
* Weak nuclear force: The weak nuclear force appears only in certain nuclear processes like β decay of a nucleus, in which the nucleus emits a β particle and an uncharged particle called a neutrino. Both the strong and weak forces form an important part of quantum mechanics.gooooo
♣r0lLing fricti0n♣
Rolling Friction
In mechanics, friction plays a major role both in the laboratory and industrial worlds. Friction is the resistance to the sliding, rolling, or flowing motion of an object due to its contact with another object. Rolling friction is caused primarily by the interference of small indentations formed as one surface rolls over another. This is the idea behind the frictional forces involved with wheels, cylinders, and spheres. In the ideal case of the wheel, we must first look at the forces acting on the wheel. In pure rolling motion friction is required to start, stop, change the motion of a wheel. Below we can see the frictional force necessary to begin motion, and get the wheel moving at a velocity v. In pure rolling motion, friction causes the wheel to catch and stops the sliding and slipping motion; for example when a car spins its tires, slipping is taking place, thus the frictional force works to stop the spinning out and causes the tires to catch and begin pure rolling motion.
The frictional force, f, the force required to slow the wheel produces a torque which tends to decrease the angular velocity, w. (The normal and gravitational forces produce no effect because their line of action is through the center of rotation.) However, the surface could not possibly have such an effect on the wheel once the wheel has achieved pure rolling motion and constant angular and linear velocity. Zero friction occurs only for horizontal motion at constant velocity, but it is non-zero for any case in which acceleration is occurring parallel to the direction of motion of the center of mass, as when the object is rolling-without-slipping up or down a sloped surface. If we consider the rotation as being about the center of mass of the object, then the frictional force must be in a direction to provide the torque necessary to decrease or increase the angular velocity, depending on whether the object is accelerating or decelerating, respectively. Note that the friction can be in the direction of motion (rolling downhill) or opposite to it (rolling uphill). In pure rolling motion there is no sliding or slipping, thus the contact points have no relative motion (no relative velocity). This results in a frictional force of zero. Therefore, the wheel will roll forward with constant velocity, v = Rw, where R is the radius of the wheel.
In the actual case of the rolling wheel, the free-body diagram is much different. Both the wheel and the surface will undergo deformations due to their particular elastic characteristics. At the contact points, the wheel flattens out while a small trench is formed in the surface. The normal force is now distributed over the actual contact area rather than the point just below the center of the wheel.
Furthermore, the wheel takes on a sort of plowing motion resulting in increased deformation at the front of the wheel, while the rear of the wheel undergoes little deformation which results in the majority of the normal force being located at the front. When the wheel and the surface deform there is a minute amount of slipping, but the majority of the force is due to static friction. The overall rolling friction results in a force at the center of the wheel and is parallel to the surface of contact, and is represented by the equation:
This resultant frictional force is still rearward acting (tending to contribute to the angular velocity), but the new distribution of the normal forces creates a net torque negating the rotational contribution of the friction and causing an overall deceleration of the wheel’s forward velocity. Also, the major source of energy losses during rolling motion is primarily due to the deformation that occurs when the two surfaces are compressed and relaxed. This continual process generates heat in both the wheel and the surface.
In mechanics, friction plays a major role both in the laboratory and industrial worlds. Friction is the resistance to the sliding, rolling, or flowing motion of an object due to its contact with another object. Rolling friction is caused primarily by the interference of small indentations formed as one surface rolls over another. This is the idea behind the frictional forces involved with wheels, cylinders, and spheres. In the ideal case of the wheel, we must first look at the forces acting on the wheel. In pure rolling motion friction is required to start, stop, change the motion of a wheel. Below we can see the frictional force necessary to begin motion, and get the wheel moving at a velocity v. In pure rolling motion, friction causes the wheel to catch and stops the sliding and slipping motion; for example when a car spins its tires, slipping is taking place, thus the frictional force works to stop the spinning out and causes the tires to catch and begin pure rolling motion.
The frictional force, f, the force required to slow the wheel produces a torque which tends to decrease the angular velocity, w. (The normal and gravitational forces produce no effect because their line of action is through the center of rotation.) However, the surface could not possibly have such an effect on the wheel once the wheel has achieved pure rolling motion and constant angular and linear velocity. Zero friction occurs only for horizontal motion at constant velocity, but it is non-zero for any case in which acceleration is occurring parallel to the direction of motion of the center of mass, as when the object is rolling-without-slipping up or down a sloped surface. If we consider the rotation as being about the center of mass of the object, then the frictional force must be in a direction to provide the torque necessary to decrease or increase the angular velocity, depending on whether the object is accelerating or decelerating, respectively. Note that the friction can be in the direction of motion (rolling downhill) or opposite to it (rolling uphill). In pure rolling motion there is no sliding or slipping, thus the contact points have no relative motion (no relative velocity). This results in a frictional force of zero. Therefore, the wheel will roll forward with constant velocity, v = Rw, where R is the radius of the wheel.
In the actual case of the rolling wheel, the free-body diagram is much different. Both the wheel and the surface will undergo deformations due to their particular elastic characteristics. At the contact points, the wheel flattens out while a small trench is formed in the surface. The normal force is now distributed over the actual contact area rather than the point just below the center of the wheel.
Furthermore, the wheel takes on a sort of plowing motion resulting in increased deformation at the front of the wheel, while the rear of the wheel undergoes little deformation which results in the majority of the normal force being located at the front. When the wheel and the surface deform there is a minute amount of slipping, but the majority of the force is due to static friction. The overall rolling friction results in a force at the center of the wheel and is parallel to the surface of contact, and is represented by the equation:
This resultant frictional force is still rearward acting (tending to contribute to the angular velocity), but the new distribution of the normal forces creates a net torque negating the rotational contribution of the friction and causing an overall deceleration of the wheel’s forward velocity. Also, the major source of energy losses during rolling motion is primarily due to the deformation that occurs when the two surfaces are compressed and relaxed. This continual process generates heat in both the wheel and the surface.
☺sl!ding friction☺
Force of Friction
* When the surface of one object slides across the surface of another object and some other force is pressing the surfaces together, a drag force is created between the two surfaces that is parallel to the surfaces.
For Objects at Rest: Static Friction
f ≤ μs n
fmax = μs n
f = Force of Friction. SI: N
μs = Coefficient of static friction. SI: Dimensionless: Table
n = The normal force pressing the surfaces together. SI: N
* The coefficient of static friction depends upon the roughness of the surfaces.
* The coefficient of static friction does not depend upon contact surface area, provided the two surfaces are made of dissimilar enough material so that any cohesive force between the two surfaces is small.
* The force of static friction (like the normal force) is a reaction force in that its value depends upon the magnitude of some external force trying to push the object along the surface.
* The force of friction is always oppsite to direction of the other forces imposed.
* The force of static friction increases up to a maximum value, after which the object "breaks loose" and begins to start moving.
For Objects in Motion: Kinetic Friction
f = μkn
f = Force of Friction. SI: N
μk = Coefficient of kinetic friction. SI: Dimensionless
n = The normal force pressing the surfaces together. SI: N
* The coefficient of kinetic friction (like the static coefficient) depends upon the roughness of the surfaces. However it is generally smaller in value than the static coefficient.
μk< μs
* For relative velocities in the range of cm/s to m/s (typical ranges we will encounter) the coefficient of kinetic friction is approximately constant.
Force of Friction on a Block
An horizontal force is applied to a block resting on a rough surface. Displayed are the various forces acting on the block as it move along the surface. The mass of the block, the applied force, and the coefficient of friction can be altered. Force of Friction on a Block QT Movie
Block on an Incline Plane
A block is pushed up an incline plane by an force parallel to the incline plane. The coefficient of friction between the block and the incline plane, the applied force, the angle of the incline plane, and the mass of the block can be varied. Displayed are the resulting forces acting on the block as it moves up or down the incline plane. Block on Incline Plane QT Movie
* When the surface of one object slides across the surface of another object and some other force is pressing the surfaces together, a drag force is created between the two surfaces that is parallel to the surfaces.
For Objects at Rest: Static Friction
f ≤ μs n
fmax = μs n
f = Force of Friction. SI: N
μs = Coefficient of static friction. SI: Dimensionless: Table
n = The normal force pressing the surfaces together. SI: N
* The coefficient of static friction depends upon the roughness of the surfaces.
* The coefficient of static friction does not depend upon contact surface area, provided the two surfaces are made of dissimilar enough material so that any cohesive force between the two surfaces is small.
* The force of static friction (like the normal force) is a reaction force in that its value depends upon the magnitude of some external force trying to push the object along the surface.
* The force of friction is always oppsite to direction of the other forces imposed.
* The force of static friction increases up to a maximum value, after which the object "breaks loose" and begins to start moving.
For Objects in Motion: Kinetic Friction
f = μkn
f = Force of Friction. SI: N
μk = Coefficient of kinetic friction. SI: Dimensionless
n = The normal force pressing the surfaces together. SI: N
* The coefficient of kinetic friction (like the static coefficient) depends upon the roughness of the surfaces. However it is generally smaller in value than the static coefficient.
μk< μs
* For relative velocities in the range of cm/s to m/s (typical ranges we will encounter) the coefficient of kinetic friction is approximately constant.
Force of Friction on a Block
An horizontal force is applied to a block resting on a rough surface. Displayed are the various forces acting on the block as it move along the surface. The mass of the block, the applied force, and the coefficient of friction can be altered. Force of Friction on a Block QT Movie
Block on an Incline Plane
A block is pushed up an incline plane by an force parallel to the incline plane. The coefficient of friction between the block and the incline plane, the applied force, the angle of the incline plane, and the mass of the block can be varied. Displayed are the resulting forces acting on the block as it moves up or down the incline plane. Block on Incline Plane QT Movie
☻stat!c fR!ction 2
Static Friction
PIRA #: 1K20.35
setup time: 1 minute
Description:
An inclined plane is used to measure the coeffiecient of static friction between it and a wooden block.
Theory:
If an object is at rest on a surface and you push against it and it doesn't move, is there friction on the object?
The answer is "yes", but this kind of friction is different from kinetic friction. Whereas kinetic friction acts to resist the motion of an object sliding across a surface, static friction is the force which keeps a motionless object from being pushed or pulled across a surface.
If a wooden block is at rest on a horizontal wooden surface, it is acted upon only by the normal force and the gravitational force.
If the surface is inclined by a small angle, θ, a component of the gravitational force acts downward along the surface of the board. The magnitude of this component is mg sinθ. If the block doesn't slide, then it is acted on by the static frictional force, fs, which exactly balances the mg sinθ component of the object's weight.
If the inclination of the surface is increased further, the static friction reaches a maximum strength. Unlike kinetic friction, which is roughly constant at low speeds, static friction varies to resist other forces on the block. The formula for static friction, therefore, gives only the maximum possible value of the static frictional force. The formula is as follows:
If the surface is inclined so much that the mg sinθ component of the object's weight exceeds fs max, then static friction is overcome, and the block begins to slide. At the angle where the block is just on the verge of slipping, fs max is equal to mg sinθ., and Newton's laws give us that
So the coeffiecient of static friction is equal to the tangent of the angle of inclination where the block is just on the verge of slipping.
Apparatus:
* Wooden block
* Incline plane equipment
Procedure:
Place the wooden block on the plane surface and slowly incline the plane in small increments. As you are inclining the plane, gently hold the wooden block with your finger tips. This will ensure that the block will not accidentally be shaken and slide prematurely. At a certain inclination, the static friction will be overcome by the weight of the wooden block and the block will slide down the incline. The tangent of this angle is equal to the coefficient of static friction for this block and this plane.
Video:
high resolution video
3.6 MB
lower resolution video
2.0 MB
- Return to Top -
- Return to M
PIRA #: 1K20.35
setup time: 1 minute
Description:
An inclined plane is used to measure the coeffiecient of static friction between it and a wooden block.
Theory:
If an object is at rest on a surface and you push against it and it doesn't move, is there friction on the object?
The answer is "yes", but this kind of friction is different from kinetic friction. Whereas kinetic friction acts to resist the motion of an object sliding across a surface, static friction is the force which keeps a motionless object from being pushed or pulled across a surface.
If a wooden block is at rest on a horizontal wooden surface, it is acted upon only by the normal force and the gravitational force.
If the surface is inclined by a small angle, θ, a component of the gravitational force acts downward along the surface of the board. The magnitude of this component is mg sinθ. If the block doesn't slide, then it is acted on by the static frictional force, fs, which exactly balances the mg sinθ component of the object's weight.
If the inclination of the surface is increased further, the static friction reaches a maximum strength. Unlike kinetic friction, which is roughly constant at low speeds, static friction varies to resist other forces on the block. The formula for static friction, therefore, gives only the maximum possible value of the static frictional force. The formula is as follows:
If the surface is inclined so much that the mg sinθ component of the object's weight exceeds fs max, then static friction is overcome, and the block begins to slide. At the angle where the block is just on the verge of slipping, fs max is equal to mg sinθ., and Newton's laws give us that
So the coeffiecient of static friction is equal to the tangent of the angle of inclination where the block is just on the verge of slipping.
Apparatus:
* Wooden block
* Incline plane equipment
Procedure:
Place the wooden block on the plane surface and slowly incline the plane in small increments. As you are inclining the plane, gently hold the wooden block with your finger tips. This will ensure that the block will not accidentally be shaken and slide prematurely. At a certain inclination, the static friction will be overcome by the weight of the wooden block and the block will slide down the incline. The tangent of this angle is equal to the coefficient of static friction for this block and this plane.
Video:
high resolution video
3.6 MB
lower resolution video
2.0 MB
- Return to Top -
- Return to M
Subscribe to:
Posts (Atom)