Electromagnetic field

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Classically, the electromagnetic field is a region of space where electric charges experience a physical influence. The electromagnetic field arises from accelerating electric charges and describes one of the four fundamental forces of nature - electromagnetism. It can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz Force Law. From a quantum mechanical point of view, the electromagnetic field can be regarded as being composed of photons.

Nature of the electromagnetic field

As with many physical concepts, there are various ways of thinking about the electromagnetic field. The field may be viewed in two distinct ways.

Continuous structure

Classically, electric and magnetic fields are thought of as being produced by smooth motions of charged objects. For example, oscillating charges produce electric and magnetic fields that may be viewed in a 'smooth', continuous, wavelike manner. In this case, energy is viewed as being transferred continuously through the electromagnetic field between any two locations. For instance, the metal atoms in a radio transmitter appear to transfer energy continuously. This view is useful to a certain extent (radiation of low frequency), but problems are found at high frequencies (see ultraviolet catastrophe). This problem leads to another view.

Discrete structure

The electromagnetic field may be thought of in a more 'coarse' way. Experiments reveal that electromagnetic energy transfer is better described as being carried away in 'packets' or 'chunks' called photons with a fixed frequency. Planck's relation links the energy $E$ of a photon to its frequency $f$ through the equation:

$E=\,h\,\nu$

where $h$ is Planck's constant, named in honour of Max Planck, and $\nu$ is the frequency of the photon . For example, in the photoelectric effect - the emission of electrons from metallic surfaces by electromagnetic radiation - it is found that increasing the intensity of the incident radiation has no effect and only the frequency of the radiation is relevant in ejecting electrons.

This quantum picture of the electromagnetic field has proved very successful, giving rise to quantum electrodynamics, a quantum field theory which describes the interaction of electromagnetic radiation with charged matter.

Dynamics

In the past, electrically charged objects were thought to produce two types of field associated with their charge property. An electric field is produced when the charge is stationary with respect to an observer measuring the properties of the charge and a magnetic field (as well as an electric field) is produced when the charge moves (creating an electric current) with respect to this observer. Over time, it was realised that the electric and magnetic fields are better thought of as two parts of a greater whole - the electromagnetic field.

Once this electromagnetic field has been produced from a given charge distribution, other charged objects in this field will experience a force (in a similar way that planets experience a force in the gravitational field of the Sun). If these other charges and currents are comparable in size to the sources producing the above electromagnetic field, then a new net electromagnetic field will be produced. Thus, the electromagnetic field may be viewed as a dynamic entity that causes other charges and currents to move and which is also affected by them. These interactions are described by Maxwell's equations and the Lorentz force law.

Mathematical description

There are different mathematical ways of representing the electromagnetic field.

Vector field approach

The electric and magnetic fields are usually described by the use of three-dimensional vector fields. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as $\mathbf {E} (x,y,z,t)$ (electric field) and $\mathbf {B} (x,y,z,t)$ (magnetic field).

If only the electric field ($\mathbf {E}$) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field ($\mathbf {B}$) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed in a vacuum by Maxwell's equations:

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$ (Gauss' Law - electrostatics)

$\nabla \cdot \mathbf {B} =0$ (Gauss' Law - magnetostatics)

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ (Faraday's Law)

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$ (Ampère-Maxwell Law)

where $\rho$ is the charge density, which can (and often does) depend on time and position, $\epsilon _{0}$ is the permittivity of free space, $\mu _{0}$ is the permeability of free space, and $\mathbf {J}$ is the current density vector, also a function of time and position. The units used above are the standard SI units. Inside a linear material, Maxwell's equations change by switching the permeability and permitivity of free space with the permeability and permitivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these terms are often represented by complex numbers, or tensors.

Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity. Moreover, the apparent coincidences in which the same effect was observed due to different physical phenomena by two different observers would be shown to be not coincidental in the least by special relativity. In fact, half of Einstein's first paper on special relativity, On the Electrodynamics of Moving Bodies, is taken up by explanations of the transformation of Maxwell's equations.

The electric and magnetic fields transform under a Lorentz boost, a relativistic transformation of coordinates, in the direction $\mathbf {v}$ as:

$\mathbf {E} '=\gamma \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)-\left({\frac {\gamma -1}{v^{2}}}\right)(\mathbf {E} \cdot \mathbf {v} )\mathbf {v}$

$\mathbf {B} '=\gamma \left(\mathbf {B} -{\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}\right)-\left({\frac {\gamma -1}{v^{2}}}\right)(\mathbf {B} \cdot \mathbf {v} )\mathbf {v}$

Component by component, for relative motion along the x-axis, this works out to be the following:

$\displaystyle E'_{x}=E_{x}$

$E'_{y}=\gamma \left(E_{y}-vB_{z}\right)$

$E'_{z}=\gamma \left(E_{z}+vB_{y}\right)$

$\displaystyle B'_{x}=B_{x}$

$B'_{y}=\gamma \left(B_{y}+{\frac {v}{c^{2}}}E_{z}\right)$

$B'_{z}=\gamma \left(B_{z}-{\frac {v}{c^{2}}}E_{y}\right)$

Finally, one thing worth noting is that if one of the fields is zero in one frame of reference, that doesn't necessarily mean it is zero in all other frames of reference. This can be seen by, for instance, making the unprimed electric field zero in the transformation to the primed electric field. In this case, depending on the orientation of the magnetic field, the primed system could see an electric field, even though there is none in the unprimed system.

It should be stressed when stating this that this does not mean two completely different sets of events are seen in the two frames, but that the same sequence of events is described in two different ways. The classic example, and the one cited by Einstein in his paper the Electrodynamics of Moving Bodies, is that of a magnet and a conductor. If the conductor is held at rest, but the magnet moves, then there is a magnetic field which changes with time, which according to Faraday's Law produces an electric field, which in turn causes a current to flow in the conductor. However, if the magnet is held stationary but the conductor moves, the charges in the conductor that are moving with the conductor as a whole form a kind of current, which produces a magnetic field which then causes current to flow. Assuming that in these cases, the object in motion in one of these cases has a velocity that is identical in speed but opposite in direction to the velocity of the object in motion in the other case, then the results are identical. A current, with the same strength, direction and electromotive force, is induced in the conductor.

Potential field approach

Many times in the use and calculation of electric and magentic fields, the approach used first computes an associated potential: the electric potential for the electric field, and the magnetic potential for the magnetic field. The electric potential is a scalar field, while the magnetic potential is a vector field. This is why sometimes the electric potential is called the scalar potential and the magnetic potential is called the vector potential. These potentials can be used to find their associated fields as follows:

$\mathbf {E} =-\mathbf {\nabla } V-{\frac {\partial \mathbf {A} }{\partial t}}$

$\mathbf {B} =\mathbf {\nabla } \times \mathbf {A}$

These relations can be plugged into Maxwell's equations to find them in terms of the potentials. Faraday's Law and Gauss's Law for magnetostatics reduce to identities (i.e. in the case of Gauss's Law for magnetostatics, 0 = 0). The other two of Maxwell's equations don't turn out so simply.

$\nabla ^{2}V+{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon _{0}}}$ (Gauss's Law for electrostatics)

$\left(\nabla ^{2}\mathbf {A} -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)-\mathbf {\nabla } \left(\mathbf {\nabla } \cdot \mathbf {A} +\mu _{0}\varepsilon _{0}{\frac {\partial V}{\partial t}}\right)=-\mu _{0}\mathbf {J}$ (Ampère-Maxwell Law)

These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as between the electric and magnetic fields, each had three components which needed to be solved for, meaning it was necessary to solve for six quantities. In the potential formulation, there are only four quantities, the electric potential and the three components of the scalar potential. However, this improvement is contrasted with the equations being much messier than Maxwell's equations using just the electric and magnetic fields.

Fortunately, there is a way to simplify these equations that takes advantage of the fact that the potential fields are not what is observed, the electric and magnetic fields are. Thus there is a freedom to impose conditions on the potentials so long as whatever condition we choose to impose does not affect the resultant electric and magnetic fields. This freedom is called gauge freedom. Specifically for these equations, for any choice of a scalar function of position and time $\lambda$, we can change the potentials as follows:

$\mathbf {A} '=\mathbf {A} +\mathbf {\nabla } \lambda$

$V'=V-{\frac {\partial \lambda }{\partial t}}$

This freedom can be used to greatly simplify the potential formulation. Generally, two such scalar functions are chosen. The first is chosen in such a way that $\mathbf {\nabla } \cdot \mathbf {A} =0$, which corresponds to the case of magnetostatics. In terms of $\lambda$, this means that it must satisfy the equation $\nabla ^{2}\lambda =-\mathbf {\nabla } \cdot \mathbf {A}$. This choice of function is generally called the Coloumb gauge, and results in the following formulation of Maxwell's equations:

$\nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}$

$\nabla ^{2}\mathbf {A} -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}=-\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}\nabla \left({\frac {\partial V}{\partial t}}\right)$

There are several things worth noting about Maxwell's equations in the Coloumb gauge. Firstly, solving for the electric potential is very easy, as the equation is a version of Poisson's equation. Secondly, solving for the magnetic vector potential is particularly hard to calculate. This is the big disadvantage of this gauge. The third thing to note, and something which is not immediately obvious, is that the electric potential changes instantly everywhere in response to a change in conditions in one locality.

For instance, if a charge is moved in New York at 1pm local time, then a hypothetical observer in Australia who could measure the electric potential directly would measure a change in the potential at 1pm New York time. This seemingly goes against the prohibition in special relativity of sending information, signals, or anything faster than the speed of light. The solution to this apparent problem lays in the fact that, as previously stated, no observer measures the potentials, they measure the electric and magnetic fields. So, the combination of $\nabla V$ and ${\frac {\partial \mathbf {A} }{\partial t}}$ used in determining the electric field restores the speed limit imposed by special relativity for the electric field, making all observable quantities consistent with relativity.

The second scalar function that is used very often is called the Lorenz gauge. This gauge chooses the scalar function $\lambda$ such that $\mathbf {\nabla } \cdot \mathbf {A} =-\mu _{0}\varepsilon _{0}{\frac {\partial V}{\partial t}}$. This means $\lambda$ must satisfy the equation $\nabla ^{2}\lambda =-\mathbf {\nabla } \cdot \mathbf {A} -\mu _{0}\varepsilon _{0}{\frac {\partial V}{\partial t}}$. The Lorenz gauge results in the following form of Maxwell's equations:

$\nabla ^{2}\mathbf {A} -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}=\Box ^{2}\mathbf {A} =-\mu _{0}\mathbf {J}$

$\nabla ^{2}V-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}V}{\partial t^{2}}}=\Box ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}$

The operator $\Box ^{2}$ is called the d'Alembertian. These equations are inhomogenous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave. These equations lead to two solutions: advanced potentials (which depend on the configuration of the sources at future points in time), and retarded potentials (which depend on the past configurations of the sources); the former are usually (and sensibly) dismissed as 'non-physical' in favor of the latter, which preserve causality.

It must be strongly emphasized that, as pointed out above, the Lorentz gauge is no more valid than any other gauge, as the potentials themselves are unobservable (with only a few loopholes, such as the Aharonov-Bohm effect, that still leave gauge invariance intact); any acausality exhibited by the potentials will vanish for the observable fields, which are the physically meaningful quantities.

Tensor field approach

The electric and magnetic fields can be combined together mathematically to form an antisymmetric, second-rank tensor, or a bivector, usually written as $F^{\mu \nu }$. This is called the electromagnetic field tensor, and it puts the electric and magnetic forces on the same footing. In matrix form, the tensor is as below.

$F^{\mu \nu }={\begin{vmatrix}0&{\frac {E_{x}}{c}}&{\frac {E_{y}}{c}}&{\frac {E_{z}}{c}}\\-{\frac {E_{x}}{c}}&0&B_{z}&-B_{y}\\-{\frac {E_{y}}{c}}&-B_{z}&0&B_{x}\\-{\frac {E_{z}}{c}}&B_{y}&-B_{x}&0\end{vmatrix}}$

where

   E is the electric field
   B the magnetic field and
   c the speed of light. When using natural units, the speed of light is taken to equal 1.

There is actually another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing ${\frac {\mathbf {E} }{c}}\to \mathbf {B}$ and $\mathbf {B} \to -{\frac {\mathbf {E} }{c}}$, to get the dual tensor $G^{\mu \nu }$.

$G^{\mu \nu }={\begin{vmatrix}0&B_{x}&B_{y}&B_{z}\\-B_{x}&0&-{\frac {E_{z}}{c}}&{\frac {E_{y}}{c}}\\-B_{y}&{\frac {E_{z}}{c}}&0&-{\frac {E_{x}}{c}}\\-B_{z}&-{\frac {E_{y}}{c}}&{\frac {E_{x}}{c}}&0\end{vmatrix}}$

In the context of special relativity, both of these transform according to the Lorentz transformation like $F'^{\alpha \beta }=\Lambda _{\mu }^{\alpha }\Lambda _{\nu }^{\beta }F^{\mu \nu }$, where the $\Lambda _{\nu }^{\alpha }$ are the Lorentz transformation tensors for a given change in reference frame. Though there are two such tensors in the equation, they are the same tensor, just used in the summation differently.

Examples

Here are two examples of transformations of the field tensor. Both are transformations due to observers moving with repect to each other on the x-axis. The first transformation shows how the unprimed observer can see an electric field, designated $E$, only in the positive z-axis direction, transform such that the primed observer, moving with velocity $\beta ={\frac {v}{c}}$ along the x-axis with respect to the unprimed observer, sees both electric and magnetic fields.

$F^{\mu \nu }={\begin{vmatrix}0&0&0&{\frac {E}{c}}\\0&0&0&0\\0&0&0&0\\-{\frac {E}{c}}&0&0&0\end{vmatrix}}$

$\Lambda _{\mu }^{\sigma }=\Lambda _{\nu }^{\tau }={\begin{vmatrix}\gamma &-\gamma \beta &0&0\\-\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{vmatrix}}$

$F'^{\sigma \tau }=\Lambda _{\mu }^{\sigma }\Lambda _{\nu }^{\tau }F^{\mu \nu }$

So, in the above, it's clear that the field tensor term is zero everywhere, except where $\mu =0,\nu =3$ or where $\mu =3,\nu =0$. The results are as below.

$F'^{00}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{0}F^{\mu \nu }=0$

$F'^{01}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{1}F^{\mu \nu }=0$

$F'^{02}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{03}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{3}F^{\mu \nu }=\gamma \left({\frac {E}{c}}\right)$

$F'^{10}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{0}F^{\mu \nu }=0$

$F'^{11}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{1}F^{\mu \nu }=0$

$F'^{12}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{13}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{3}F^{\mu \nu }=-\gamma \beta \left({\frac {E}{c}}\right)$

$F'^{20}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{0}F^{\mu \nu }=0$

$F'^{21}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{1}F^{\mu \nu }=0$

$F'^{22}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{23}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{3}F^{\mu \nu }=0$

$F'^{30}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{0}F^{\mu \nu }=-\gamma \left({\frac {E}{c}}\right)$

$F'^{31}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{1}F^{\mu \nu }=\gamma \beta \left({\frac {E}{c}}\right)$

$F'^{32}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{33}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{3}F^{\mu \nu }=0$

The result, in matrix form, looks like this:

$F'^{\sigma \tau }={\begin{vmatrix}0&0&0&\gamma {\frac {E}{c}}\\0&0&0&-\gamma \beta {\frac {E}{c}}\\0&0&0&0\\-\gamma {\frac {E}{c}}&\gamma \beta {\frac {E}{c}}&0&0\end{vmatrix}}$

As can be seen, if one compares this result with the general form of the field tensor shown above, two things have occurred. Firstly, the primed observer sees the electrical field as being stronger than the unprimed observer. Secondly, the primed observer sees a magnetic field in the positive y-axis direction that the unprimed observer does not see. This hints at the reason that magnetism is sometimes called a relativistic phenomenon.

However, it is not true that all Lorentz transformations on a field tensor with only an electric component will produce a magnetic component. The following example illustrates this, with the same two observers as above, but with the electric field being in the positive x-axis direction instead of the positive z-axis direction. This direction is in the same direction of the relative velocity between the two observers.

$F^{\mu \nu }={\begin{vmatrix}0&{\frac {E}{c}}&0&0\\-{\frac {E}{c}}&0&0&0\\0&0&0&0\\0&0&0&0\end{vmatrix}}$

$\Lambda _{\mu }^{\sigma }=\Lambda _{\nu }^{\tau }={\begin{vmatrix}\gamma &-\gamma \beta &0&0\\-\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{vmatrix}}$

$F'^{\sigma \tau }=\Lambda _{\mu }^{\sigma }\Lambda _{\nu }^{\tau }F^{\mu \nu }$

So, in the above, it's clear that the field tensor term zero everywhere except where $\mu =0,\nu =1$ or where $\mu =1,\nu =0$. The results are as below.

$F'^{00}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{0}F^{\mu \nu }=\Lambda _{0}^{0}\Lambda _{1}^{0}F^{01}+\Lambda _{1}^{0}\Lambda _{0}^{0}F^{10}=-\gamma ^{2}\beta {\frac {E}{c}}+\gamma ^{2}\beta {\frac {E}{c}}=0$

$F'^{01}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{1}F^{\mu \nu }=\Lambda _{0}^{0}\Lambda _{1}^{1}F^{01}+\Lambda _{1}^{0}\Lambda _{0}^{1}F^{10}=\gamma ^{2}{\frac {E}{c}}-\gamma ^{2}\beta ^{2}{\frac {E}{c}}={\frac {E}{c}}$

$F'^{02}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{03}=\Lambda _{\mu }^{0}\Lambda _{\nu }^{3}F^{\mu \nu }=0$

$F'^{10}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{0}F^{\mu \nu }=\Lambda _{0}^{1}\Lambda _{1}^{0}F^{01}+\Lambda _{1}^{1}\Lambda _{0}^{0}F^{10}=\gamma ^{2}\beta ^{2}{\frac {E}{c}}-\gamma ^{2}{\frac {E}{c}}=-{\frac {E}{c}}$

$F'^{11}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{1}F^{\mu \nu }=\Lambda _{0}^{1}\Lambda _{1}^{1}F^{01}+\Lambda _{1}^{1}\Lambda _{0}^{1}F^{10}=-\gamma ^{2}\beta {\frac {E}{c}}+\gamma ^{2}\beta {\frac {E}{c}}=0$

$F'^{12}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{13}=\Lambda _{\mu }^{1}\Lambda _{\nu }^{3}F^{\mu \nu }=0$

$F'^{20}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{0}F^{\mu \nu }=0$

$F'^{21}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{1}F^{\mu \nu }=0$

$F'^{22}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{23}=\Lambda _{\mu }^{2}\Lambda _{\nu }^{3}F^{\mu \nu }=0$

$F'^{30}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{0}F^{\mu \nu }=0$

$F'^{31}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{1}F^{\mu \nu }=0$

$F'^{32}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{2}F^{\mu \nu }=0$

$F'^{33}=\Lambda _{\mu }^{3}\Lambda _{\nu }^{3}F^{\mu \nu }=0$

In the above, the following relation was used less explicitly.

$\gamma ^{2}-\gamma ^{2}\beta ^{2}={\frac {1}{{\sqrt {1-\beta ^{2}}}^{2}}}-{\frac {\beta ^{2}}{{\sqrt {1-\beta ^{2}}}^{2}}}={\frac {1-\beta ^{2}}{1-\beta ^{2}}}=1$

The result, in matrix form, looks like this:

$F'^{\mu \nu }={\begin{vmatrix}0&{\frac {E}{c}}&0&0\\-{\frac {E}{c}}&0&0&0\\0&0&0&0\\0&0&0&0\end{vmatrix}}$

Not only does no magnetic component show up, but the whole tensor is unchanged.

Maxwell's Equations in Tensor Notation

Using this tensor notation, Maxwell's equations have the following form.

$F_{,\beta }^{\alpha \beta }={\frac {\partial F^{\alpha \beta }}{\partial x^{\beta }}}=\mu _{0}J^{\alpha }$

$G_{,\beta }^{\alpha \beta }={\frac {\partial G^{\alpha \beta }}{\partial x^{\beta }}}=0$

In the above, the tensor notation $f_{,\alpha }$ is used to denote partial derivatives, ${\frac {\partial f}{\partial x^{\alpha }}}$. The four-vector $J^{\alpha }$ is called the current density four-vector, which is the relativistic analogue to the charge density and current density. This four-vector is as follows.

$J^{\alpha }={\begin{pmatrix}c\rho &J_{x}&J_{y}&J_{z}\end{pmatrix}}$

The first equation listed above corresponds to both Gauss's Law ( for $\alpha =0$ ) and the Ampère-Maxwell Law ( for $\alpha =1,2,3$ ). The second equation corresponds to the two remaining equations, Gauss's Law for magnetism ( for $\alpha =0$ ) and Faraday's Law ( for $\alpha =1,2,3$ ).

This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using tensors.

Properties of the field

Reciprocal behaviour of electric and magnetic fields

The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field. Faraday's Law may be stated roughly as 'a changing magnetic field creates an electric field'. This is the principle behind the electric generator.

The Ampère-Maxwell Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field and run an electric motor.

Light as an electromagnetic disturbance

Maxwell's equations take the following, free space, form in an area that is very far away from any charges or currents - that is where $\rho$ and $\mathbf {J}$ are zero.

$\nabla \cdot \mathbf {E} =0$

$\nabla \cdot \mathbf {B} =0$

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

$\nabla \times \mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}$

In the above, the substitution $\mu _{0}\epsilon _{0}={\frac {1}{c^{2}}}$ has been made, where $c$ is the speed of light. Taking the curl of the last two equations, the result is as follows.

$\nabla \times \nabla \times \mathbf {E} =\nabla \left(\nabla \cdot \mathbf {E} \right)-\nabla ^{2}\mathbf {E} =\nabla \times \left(-{\frac {\partial \mathbf {B} }{\partial t}}\right)$

$\nabla \times \nabla \times \mathbf {B} =\nabla \left(\nabla \cdot \mathbf {B} \right)-\nabla ^{2}\mathbf {B} =\nabla \times \left({\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\right)$

However, the first two equations mean $\nabla \left(\nabla \cdot \mathbf {E} \right)=\nabla \left(\nabla \cdot \mathbf {B} \right)=0$. So plugging this in, and moving the curls within the time derivates and then plugging in for the resultant curls, the result is as follows.

$-\nabla ^{2}\mathbf {E} =-{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {B} \right)=-{\frac {\partial }{\partial t}}\left({\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\right)=-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}$

$-\nabla ^{2}\mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {E} \right)={\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}\left(-{\frac {\partial \mathbf {B} }{\partial t}}\right)=-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}$

Or:

$\nabla ^{2}\mathbf {E} ={\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}$

$\nabla ^{2}\mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}$

Or even:

$\Box ^{2}\mathbf {E} =0$

$\Box ^{2}\mathbf {B} =0$

In this last form, the $\Box ^{2}$ is the d'Alembertian, which is $\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}$, so the last two forms are the same thing written in two different ways. These can be identified as wave equations, that is, valid electric fields and magnetic fields have an oscillatory form, such as a sinusoid, which result in wave behaviors. Moreover, the first two of the free space Maxwell's equations imply that the waves are transverse waves. The last two of the free space Maxwell's equations imply that the wave of the electric field is in phase with and perpendicular to the magnetic field wave. Moreover, the $c^{2}$ term represents the speed of the wave. So these electromagnetic waves travel at the speed of light. James Clerk Maxwell, after whom Maxwell's equations are named, suggested when he made these calculations that as these waves travel at the same speed as light, that light would actually be such a wave. His suggestion proved correct, and light is indeed an electromagnetic wave.

Relation to and comparison with other physical fields

Main article: Fundamental forces

Being one of the four fundamental forces of nature, it is useful to compare the electromagnetic field with the gravitational, strong and weak fields. The word 'force' is sometimes replaced by 'interaction'.

Electromagnetic and gravitational fields

Sources of electromagnetic fields consist of two types of charge - positive and negative. This contrasts with the sources of the gravitational field, which are masses. Masses are sometimes described as 'gravitational charges', the important feature of them being that there is only one type (no 'negative masses'), or, in more colloquial terms, 'gravity is always attractive'.

The relative strengths and ranges of the four interactions and other information are tabulated below:

Theory	Interaction	mediator	Relative Magnitude	Behavior	Range
Chromodynamics	Strong interaction	gluon	1038	1	infinite
Electrodynamics	Electromagnetic interaction	photon	1036	1/r2	infinite
Flavordynamics	Weak interaction	W and Z bosons	1025	1/r5 to 1/r7	10-18 m
Geometrodynamics	Gravitation	graviton	100	1/r2	infinite

Applications

Properties of the electromagnetic field are exploited in many areas of industry. The use of electromagnetic radiation is seen in various disciplines. For example, X-rays are high frequency electromagnetic radiation and are used in radio astronomy, radiography in medicine and radiometry in telecommunications. Other medical applications include laser therapy, which is an example of photomedicine. Applications of lasers are found in military devices such as laser-guided bombs, as well as more down to earth devices such as barcode readers and CD players. Something as simple as a relay in any electrical device uses an electromagnetic field to engage or to disengage the two different states of output (ie, when electricity is not applied, the metal strip will connect output A and B, but if electricity is applied, an electromagnetic field will be created and the metal strip will connect output A and C).

The electromagnetic field as a feedback loop

The behavior of the electromagnetic field can be resolved into four different parts of a loop: (1) the electric and magnetic fields are generated by electric charges, (2) the electric and magnetic fields interact only with each other, (3) the electric and magnetic fields produce forces on electric charges, (4) the electric charges move in space.

The feedback loop can be summarized in a list, including phenomena belonging to each part of the loop:

• charges generate fields
  • Gauss's law Coulomb's law: charges generate electric fields
  • Ampère's law: currents generate magnetic fields ($\star$)
• the fields interact with each other
  • displacement current: changing electric field acts like a current, generating 'vortex' (curl) of magnetic field
  • Faraday induction: changing magnetic field induces (negative) vortex of electric field
  • Lenz's law: negative feedback loop between electric and magnetic fields
  • Maxwell-Hertz equations: simplified version of Maxwell's equations
  • electromagnetic wave equation
• fields act upon charges
  • Lorentz force: force due to electromagnetic field
    • electric force: same direction as electric field
    • magnetic force: perpendicular both to magnetic field and to velocity of charge ($\star$)
• charges move
  • continuity equation: current is movement of charges

Phenomena in the list are marked with a star ($\star$) if they consist of magnetic fields and moving charges which can be reduced by suitable Lorentz transformations to electric fields and static charges. This means that the magnetic field ends up being (conceptually) reduced to an appendage of the electric field, i.e. something which interacts with reality only indirectly through the electric field.

See also

• Classification of electromagnetic fields
• Electric field
• Electromagnetism
• Electromagnetic tensor
• Fundamental interaction
• Electromagnetic radiation
• Electromagnetic spectrum
• Gravitational field
• List of environment topics
• Magnetic field
• Maxwell's equations
• Photoelectric effect
• Photon
• Quantum electrodynamics

External links

• On the Electrodynamics of Moving Bodies by Albert Einstein, June 30, 1905.
  • On the Electrodynamics of Moving Bodies (pdf)
• Non-Ionizing Radiation, Part 1: Static and Extremely Low-Frequency (ELF) Electric and Magnetic Fields (2002) by the IARC.
  • A summary of the previous report by GreenFacts.
• Environmental Health Criteria 232: Static Fields by the WHO (2006).
  • Health effects of static magnetic and electric fields - a summary of the above WHO report by GreenFacts.