# Magnetic field

File:Electromagnetism.png
Current flowing through a wire produces a magnetic field (M) around the wire. The field is oriented according to the right-hand rule.
For other senses of this term, see magnetic field (disambiguation).

In physics, a magnetic field is an entity produced by moving electric charges (electric currents) that exerts a force on other moving charges. (The quantum-mechanical spin of a particle produces magnetic fields and is acted on by them as though it were a current; this accounts for the fields produced by "permanent" ferromagnets.) A magnetic field is a vector field: it associates with every point in space a (pseudo-)vector that may vary in time. The direction of the field is the equilibrium direction of a compass needle placed in the field.

## Symbols and terminology

Magnetic field is usually denoted by the symbol ${\displaystyle \mathbf {B} \ }$. Historically, ${\displaystyle \mathbf {B} \ }$ was called the magnetic flux density, magnetic induction, or magnetic field strength. ${\displaystyle \mathbf {H} }$ was called the magnetic field (or magnetic field intensity), and this terminology is still often used to distinguish the two in the context of magnetic materials (non-trivial permeability μ). Otherwise, however, this distinction is often ignored, and both symbols are frequently referred to as the magnetic field. (Some authors call H the auxiliary field, instead.) In linear materials, such as air or free space, the two quantities are linearly related: ${\displaystyle \mathbf {H} =\mathbf {B} /\mu \ }$.

In SI units, ${\displaystyle \mathbf {B} \ }$ and ${\displaystyle \mathbf {H} \ }$ are measured in teslas (T) and amperes per meter (A/m), respectively; or, in cgs units, in gauss (G) and oersteds (Oe), respectively. Two parallel wires carrying an electric current in the same sense will generate a magnetic field which will cause a force of attraction to each other. This fact is used to generate the value of an ampere of electric current. Note that while like charges repel and unlike ones attract, the opposite holds for currents: if the current in one of the two parallel wires is reversed, the two will repel.

## Definition

Like the electric field, the magnetic field can be defined by the force it produces. In SI units, this is:

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$

where

F is the force produced, measured in newtons
${\displaystyle \times \ }$ indicates a vector cross product
${\displaystyle q\ }$ is electric charge, measured in coulombs
${\displaystyle \mathbf {v} \ }$ is velocity of the electric charge ${\displaystyle q\ }$, measured in metres per second
B is magnetic flux density, measured in teslas

This law is called the Lorentz force law. (More precisely, it is the special case of that law when there is no electric field. It holds in any reference frame, although the force due to the magnetic field may be different in different frames because magnetic fields transform into electric fields under Lorentz transformations. The total force due to the electric and magnetic fields is the same in any frame.)

### Current loop

A simpler form of the force equation in a wire current loop is: Force = BLi = (Tesla)x(meter length of wire)x(ampere current of wire).

A more complex explanation is that if the moving charge is part of a current in a wire, then an equivalent form of the law is

${\displaystyle {\frac {d\mathbf {F} }{dl}}=\mathbf {i} \times \mathbf {B} }$

In words, this equation says that the force per unit length of wire is the cross product of the current vector and the magnetic field. In the equation above, the current vector, ${\displaystyle \mathbf {i} }$, is a vector with magnitude equal to the usual scalar current, ${\displaystyle i}$, and direction pointing along the wire that the current is flowing.

### Vector calculus

The most compact and elegant mathematical statements describing how magnetic fields are produced makes use of vector calculus.

In free space:

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$
${\displaystyle \nabla \cdot \mathbf {B} =0}$

where

${\displaystyle \nabla \times }$ is the curl operator
${\displaystyle \nabla \cdot }$ is the divergence operator
${\displaystyle \mu _{0}\ }$ is permeability
${\displaystyle \mathbf {J} \ }$ is current density
${\displaystyle \partial \ }$ is the partial derivative
${\displaystyle \epsilon _{0}\ }$ is the free-space permittivity
${\displaystyle \mathbf {E} \ }$ is the electric field
${\displaystyle t\ }$ is time

The first equation is known as Ampère's law with James Clerk Maxwell's correction. The second term of this equation (Maxwell's correction) disappears in static or quasi-static systems. The second equation is a statement of the observed non-existence of magnetic monopoles. These are two of four Maxwell's equations; the notation is due to Oliver Heaviside.

## Energy in the magnetic field

The general relation for nonlinear materials, the differential energy is:

${\displaystyle dU_{H}=\int _{V}^{}H\cdot dB\,dV}$

Where V is the volume and dV is the differential volume.

For linear materials, H is proportional to B, so the above equation can be simplified:

${\displaystyle U_{H}={\frac {1}{2}}\int _{V}^{}B\cdot H\,dV}$

For linear materials and a constant volume:

${\displaystyle U_{H}={\frac {B^{2}V}{2\mu }}}$

Energy can produce a force, so

${\displaystyle F={\frac {dU_{H}}{dl}}}$
${\displaystyle F={\frac {B^{2}A}{2\mu }}}$

Where dl is differential distance and A is the surface area. Force per unit area (pressure) is

${\displaystyle P={\frac {B^{2}}{2\mu }}}$

In the case of free space (air), ${\displaystyle \mu _{o}=4\pi \cdot 10^{-7}{\frac {\mbox{H}}{\mbox{m}}}}$:

${\displaystyle P=57.7\,{\frac {\mbox{lb}}{{\mbox{in}}^{2}}}}$ @ B = 1 Tesla
${\displaystyle P=230.8\,{\frac {\mbox{lb}}{{\mbox{in}}^{2}}}}$ @ B = 2 Tesla

This is the force observed when a high permeability, ferromagnetic materials, such as iron and steel alloys, are in the proximity of magnetic fields.

## Properties

Maxwell did much to unify static electricity and magnetism, producing a set of four equations relating the two fields. However, under Maxwell's formulation, there were still two distinct fields describing different phenomena. It was Albert Einstein who showed, using special relativity, that electric and magnetic fields are two aspects of the same thing (a rank-2 tensor), and that one observer may perceive a magnetic force where a moving observer perceives only an electrostatic force. Thus, using special relativity, magnetic forces are a manifestation of electrostatic forces of charges in motion and may be predicted from knowledge of the electrostatic forces and the movement (relative to some observer) of the charges.

A thought experiment one can do to show this is with two identical infinite and parallel lines of charge having no motion relative to each other but moving together relative to an observer. Another observer is moving alongside the two lines of charge (at the same velocity) and observes only electrostatic repulsive force and acceleration. The first or "stationary" observer seeing the two lines (and second observer) moving past with some known velocity also observes that the "moving" observer's clock is ticking more slowly (due to time dilation) and thus observes the repulsive acceleration of the lines more slowly than that which the "moving" observer sees. The reduction of repulsive acceleration can be thought of as an attractive force, in a classical physics context, that reduces the electrostatic repulsive force and also that is increasing with increasing velocity. This pseudo-force is precisely the same as the electromagnetic force in a classical context.

Changing magnetic fields, according to Faraday's law of induction, can induce an electric field and thus an electric current; similar currents can be induced by conductors moving in a fixed magnetic field. These phenomena are the basis for many electric generators and electric motors.

### Magnetic field lines

File:Magnet0873.png
Magnetic field lines emanate primarily from the north pole of a magnet and curve around to the south pole

Technically, the magnetic field isn't a vector according to the formal definition, it is a pseudovector: it gains an extra sign flip under improper rotations of the coordinate system. (The distinction is important when using symmetry to analyze magnetic-field problems.) This is a consequence of the fact that B is related to two true vectors by a cross product (e.g. in the Lorentz force law). To simplify the study of magnets an arbitrary (but valid) description of magnetic field lines was created. 1 magnetic field line = 1 gauss line. 10,000 gauss lines per square meter is equal to 1 tesla. The total number of lines emanating from a magnet pole is the magnetic flux. Count only north or only south pole lines, i.e. monopole or one sided value.

Although the field line orientation is typically indicated in diagrams with an arrow, the arrow should not be interpreted to indicate any actual movement or flow of the field line.

### Pole labeling confusions

It is necessary to note that the labeling of north and south on a compass is in opposition to the labeling of the north and south pole of the Earth.

If you have two labeled magnets, it is clear that like poles repel, while opposing poles attract. However, this is clearly wrong when using a compass to find the North Pole of the Earth, because the "north" end of the compass points to the "North" Pole.

By convention, the pole of a magnet is labelled according to the direction it points, hence when we speak of the "north pole" of a magnet, we really mean the "north-seeking pole". Magnetic field lines point from north to south of a magnet, and hence the natural magnetic field lines run from south to north along the Earth's surface. This choice, along with the choice of sign convention in the Biot-Savart law, is equivalent to choosing a sign convention for charge.

## Rotating magnetic fields

A rotating magnetic field is a magnetic field which rotates in polarity at non-relativistic speeds. This is a key principle to the operation of alternating-current motor. A permanent magnet in such a field will rotate so as to maintain its alignment with the external field. This effect is utilised in alternating current electric motors. A good rotating magnetic field can be constructed using three phase alternating currents (or even with higher order polyphase systems). Synchronous motors and induction motors use a stator's rotating magnetic fields to turn rotors. In 1882, Nikola Tesla identified the concept of the rotary magnetic field. In 1885, Galileo Ferraris independently researched the concept. In 1888, Tesla gained Template:US patent for his work. Also in 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.

General

• Electric field - effect produced by an electric charge that exerts a force on charged objects in its vicinity.
• Electromagnetic field - a field composed of two related vectorial fields, the electric field and the magnetic field.
• Electromagnetism - the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field.
• Magnetism - phenomenon by which materials exert an attractive or repulsive force on other materials.
• Magnetohydrodynamics - the academic discipline which studies the dynamics of electrically conducting fluids.

Mathematics

• Ampere's law - magnetic equivalent of Gauss's law.
• Biot-Savart law - describes the magnetic field set up by a steadily flowing line current.
• Magnetic helicity - extent to which a magnetic field "wraps around itself".
• Maxwell's equations - four equations describing the behavior of the electric and magnetic fields, and their interaction with matter.

Applications

• Helmholtz coil - a device for producing a region of nearly uniform magnetic field.
• Maxwell coil - a device for producing a large volume of almost constant magnetic field.
• Earth's magnetic field - a discussion of the magnetic field of the Earth.
• Dynamo theory - a proposed mechanism for the creation of the Earth's magnetic field.
• Electric motor - AC motors used magnetic fields

## References

Books

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.), Prentice Hall. ISBN 013805326X.
• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.), Wiley. ISBN 047130932X.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.), W. H. Freeman. ISBN 0716708108.

## External articles

Information

Rotating magnetic fields

Diagrams

Journal Articles