Mathematical descriptions of the electromagnetic field

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental forces of nature. In this article four approaches are discussed.

Contents

Vector field approach

The most common description of the electromagnetic field to use two three-dimensional vector fields called the electric field and the magnetic field. 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 E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field).

If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field (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.

Maxwell's equations in vector field approach

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)
\nabla \cdot \mathbf{B} = 0 (Gauss's law for magnetism)
\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 ρ is the charge density per unit volume, which can (and often does) depend on time and position, ε0 is the electric constant, μ0 is the magnetic constant, and J is the current per unit area, 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 permittivity of free space with the permeability and permittivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these properties can be represented by tensors, with time-dependence related to the material's ability to respond to fast field changes (dispersion (optics), Green–Kubo relations), and possibly also field dependencies representing nonlinear and/or nonlocal material responses to large amplitude fields (nonlinear optics).

Relativistic transformation of fields in vector field approach

Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity.[1] 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 1905 first paper on special relativity, "On the Electrodynamics of Moving Bodies," explains how to transform 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 - v B_z \right )
E'_z = \gamma \left ( E_z + v B_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 in Einstein (1905), 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 magnetic 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

Maxwell's equation in potential formulation

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 magnetism reduce to identities (i.e. in the case of Gauss's Law for magnetism, 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 vector 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 is chosen 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 λ, the potentials can be changed 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 λ, this means that it must satisfy the equation \nabla^2 \lambda = - \mathbf \nabla \cdot \mathbf A. This choice of function is generally called the Coulomb 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[says who?] about Maxwell's equations in the Coulomb 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 1 pm local time, then a hypothetical observer in Australia who could measure the electric potential directly would measure a change in the potential at 1 pm 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 lies in the fact that, as previously stated, no observers measure the potentials, they measure the electric and magnetic fields. So, the combination of V and A/∂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 λ such that \mathbf \nabla \cdot \mathbf A' = - \mu_0 \varepsilon_0 \frac{\partial V'}{\partial t}. This means λ must satisfy the equation \nabla^2 \lambda - \mu_0 \varepsilon_0 \frac{\partial^2 \lambda }{\partial t^2}= - \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[says who?] 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 causality exhibited by the potentials will vanish for the observable fields, which are the physically meaningful quantities.

Extension to quantum electrodynamics

The previous Lorentz Gauge equations, can also be written:

\nabla^2 \mathbf A - \frac 1 {c^2} \frac{\partial^2 \mathbf A}{\partial t^2} = - \mu_0 \mathbf J
\nabla^2 V - \frac 1 {c^2} \frac{\partial^2 V}{\partial t^2} = - \frac{\rho}{\varepsilon_0}

This is the basis for the extension of Maxwell equations to quantum electrodynamics, which yields the equations[2]:


\nabla^2 \mathbf A - \frac 1 {c^2} \frac{\partial^2 \mathbf A}{\partial t^2} = 4 \pi e \psi^+ a \psi
\nabla^2 V - \frac 1 {c^2} \frac{\partial^2 V}{\partial t^2} = 4 \pi e \psi^+ \psi

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 Fμν. 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 Gμν.

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^\alpha_\mu \Lambda^\beta_\nu F^{\mu \nu}, where the \Lambda^\alpha_\nu 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.

Maxwell's equations in tensor notation

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

F^{\alpha \beta},_{\alpha} = \frac{\partial F^{\alpha \beta}}{\partial x^\alpha} = \mu_0 J^\beta
G^{\alpha \beta},_{\alpha} = \frac{\partial G^{\alpha \beta}}{\partial x^\alpha} = 0

In the above, the tensor notation f is used to denote partial derivatives, \frac{\partial f}{\partial x^\alpha}. The four-vector Jα 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 α = 0 ) and the Ampère-Maxwell Law ( for α = 1,2,3 ). The second equation corresponds to the two remaining equations, Gauss's law for magnetism ( for α = 0 ) and Faraday's Law ( for α = 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.

Geometric algebra (GA) approach

This is the additional detail referred to in Maxwell's equations.

The most straightforward method of demonstrating the recovery of Maxwell's equations via the GA formalism is to descend from the Spacetime algebra by selecting a timelike direction γ0 and then deal simply with the 3D spatial algebra (equivalent to the Pauli algebra). So we need to expand


\begin{align}\gamma_0 \left( \nabla F - c\mu_0J \right) = 0 \end{align}

To do so first note that


\begin{align}\gamma_0 \nabla &= \partial_t + \boldsymbol{\nabla} \end{align}

where a bold font is used for the spatial gradient.


\begin{align}\boldsymbol{\nabla} &= \sigma^k \partial_k\end{align}

Similarly, multiplication of the four vector current density also has scalar and spatial vector components. With \mathbf{J} = J^k \sigma_k,\ c\rho = J^0 this is


\begin{align}\gamma_0 J &= c \rho - \mathbf{J}\end{align}

One obtains


\begin{align}\left( \frac{1}{{c}} \partial_t + \boldsymbol{\nabla} \right) (\mathbf{E} + I c \mathbf{B}) - \frac{\rho}{\epsilon_0} - c\mu_0\mathbf{J} = 0\end{align}

Noting that the pseudoscalar I commutes with all spatial vectors, that I^2 = -1,\ \epsilon_0 \mu_0 c^2 = 1 and \boldsymbol{\nabla} \mathbf{X} = \boldsymbol{\nabla} \cdot \mathbf{X} + I \boldsymbol{\nabla} \times \mathbf{X} for spatial vectors \mathbf{X}, one can expand and regroup this yielding


\left( \boldsymbol{\nabla} \cdot \mathbf{E} - \frac{\rho}{\epsilon_0} \right)- c \left( \boldsymbol{\nabla} \times \mathbf{B} - \mu_0 \epsilon_0 \frac{\partial {\mathbf{E}}}{\partial {t}} - \mu_0 \mathbf{J} \right)+ I \left( \boldsymbol{\nabla} \times \mathbf{E} + \frac{\partial {\mathbf{B}}}{\partial {t}} \right)+ I c \left( \boldsymbol{\nabla} \cdot \mathbf{B} \right)= 0

We have scalar, vector, bivector, and trivector grades. Equating each to zero recovers all of Maxwell's equations in their traditional vector form.

References and notes

  1. ^ Questions remain about the treatment of accelerating charges: Haskell, "Special relativity and Maxwell's equations."
  2. ^ Quantum Electrodynamics, Mathworld [1]

See also


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