There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.
Vector-field approach
The most common description of the electromagnetic field uses 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 the vector-field approach
The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell–Heaviside's equations:
where ρ is the charge density, 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 equations take this form with the International System of Quantities. When dealing with only nondispersive isotropic linear materials, Maxwell's equations are often modified to ignore bound charges by replacing the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. For some materials that have 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 rapid 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).
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, φ {\displaystyle \varphi } , for the electric field, and the magnetic vector potential, A, 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:
E = − ∇ φ − ∂ A ∂ t {\displaystyle \mathbf {E} =-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {A} }{\partial t}}}
B = ∇ × A {\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} }
Maxwell's equations in potential formulation These relations can be substituted into Maxwell's equations to express the latter in terms of the potentials. Faraday's law and Gauss's law for magnetism (the homogeneous equations) turn out to be identically true for any potentials. This is because of the way the fields are expressed as gradients and curls of the scalar and vector potentials. The homogeneous equations in terms of these potentials involve the divergence of the curl ∇ ⋅ ∇ × A {\displaystyle \nabla \cdot \nabla \times \mathbf {A} } and the curl of the gradient ∇ × ∇ φ {\displaystyle \nabla \times \nabla \varphi } , which are always zero. The other two of Maxwell's equations (the inhomogeneous equations) are the ones that describe the dynamics in the potential formulation.
These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as the electric and magnetic fields together had six components to solve for. In the potential formulation, there are only four components: the electric potential and the three components of the vector potential. However, the equations are messier than Maxwell's equations using the electric and magnetic fields.
Gauge freedom These equations can be simplified by taking advantage of the fact that the electric and magnetic fields are physically meaningful quantities that can be measured; the potentials are not. There is a freedom to constrain the form of the potentials provided that this does not affect the resultant electric and magnetic fields, called gauge freedom. Specifically for these equations, for any choice of a twice-differentiable scalar function of position and time λ, if (φ, A) is a solution for a given system, then so is another potential (φ′, A′) given by:
φ ′ = φ − ∂ λ ∂ t {\displaystyle \varphi '=\varphi -{\frac {\partial \lambda }{\partial t}}}
A ′ = A + ∇ λ {\displaystyle \mathbf {A} '=\mathbf {A} +\mathbf {\nabla } \lambda }
This freedom can be used to simplify the potential formulation. Either of two such scalar functions is typically chosen: the Coulomb gauge and the Lorenz gauge.
Coulomb gauge
The Coulomb gauge is chosen in such a way that ∇ ⋅ A ′ = 0 {\displaystyle \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
∇ 2 λ = − ∇ ⋅ A . {\displaystyle \nabla ^{2}\lambda =-\mathbf {\nabla } \cdot \mathbf {A} .}
This choice of function results in the following formulation of Maxwell's equations:
∇ 2 φ ′ = − ρ ε 0 {\displaystyle \nabla ^{2}\varphi '=-{\frac {\rho }{\varepsilon _{0}}}}
∇ 2 A ′ − μ 0 ε 0 ∂ 2 A ′ ∂ t 2 = − μ 0 J + μ 0 ε 0 ∇ ( ∂ φ ′ ∂ t ) {\displaystyle \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 \varphi '}{\partial t}}\!\right)}
Several features about Maxwell's equations in the Coulomb gauge are as follows. 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 difficult. This is the big disadvantage of this gauge. The third thing to note, and something that 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 violates causality in special relativity, i.e. the impossibility of information, signals, or anything travelling faster than the speed of light. The resolution to this apparent problem lies in the fact that, as previously stated, no observers can measure the potentials; they measure the electric and magnetic fields. So, the combination of ∇φ 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.
Lorenz gauge condition
A gauge that is often used is the Lorenz gauge condition. In this, the scalar function λ is chosen such that
∇ ⋅ A ′ = − μ 0 ε 0 ∂ φ ′ ∂ t , {\displaystyle \mathbf {\nabla } \cdot \mathbf {A} '=-\mu _{0}\varepsilon _{0}{\frac {\partial \varphi '}{\partial t}},}
meaning that λ must satisfy the equation
∇ 2 λ − μ 0 ε 0 ∂ 2 λ ∂ t 2 = − ∇ ⋅ A − μ 0 ε 0 ∂ φ ∂ t . {\displaystyle \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 \varphi }{\partial t}}.}
The Lorenz gauge results in the following form of Maxwell's equations:
∇ 2 φ ′ − μ 0 ε 0 ∂ 2 φ ′ ∂ t 2 = − ◻ 2 φ ′ = − ρ ε 0 {\displaystyle \nabla ^{2}\varphi '-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\varphi '}{\partial t^{2}}}=-\Box ^{2}\varphi '=-{\frac {\rho }{\varepsilon _{0}}}}
∇ 2 A ′ − μ 0 ε 0 ∂ 2 A ′ ∂ t 2 = − ◻ 2 A ′ = − μ 0 J {\displaystyle \nabla ^{2}\mathbf {A} '-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} '}{\partial t^{2}}}=-\Box ^{2}\mathbf {A} '=-\mu _{0}\mathbf {J} }
The operator ◻ 2 {\displaystyle \Box ^{2}} is called the d'Alembertian (some authors denote this by only the square ◻ {\displaystyle \Box } ). These equations are inhomogeneous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave. As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and retarded potentials (which are related to the past configurations of the sources); the former are usually disregarded where the field is to analyzed from a causality perspective. As pointed out above, the Lorenz gauge is no more valid than any other gauge since the potentials cannot be directly measured, however the Lorenz gauge has the advantage of the equations being Lorentz invariant.
Extension to quantum electrodynamics Canonical quantization of the electromagnetic fields proceeds by elevating the scalar and vector potentials; φ(x), A(x), from fields to field operators. Substituting 1/c2 = ε0μ0 into the previous Lorenz gauge equations gives:
∇ 2 A − 1 c 2 ∂ 2 A ∂ t 2 = − μ 0 J {\displaystyle \nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}=-\mu _{0}\mathbf {J} }
∇ 2 φ − 1 c 2 ∂ 2 φ ∂ t 2 = − ρ ε 0 {\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}
Here, J and ρ are the current and charge density of the matter field. If the matter field is taken so as to describe the interaction of electromagnetic fields with the Dirac electron given by the four-component Dirac spinor field ψ, the current and charge densities have form:
J = − e ψ † α ψ ρ = − e ψ † ψ , {\displaystyle \mathbf {J} =-e\psi ^{\dagger }{\boldsymbol {\alpha }}\psi \,\quad \rho =-e\psi ^{\dagger }\psi \,,}
where α are the first three Dirac matrices. Using this, we can re-write Maxwell's equations as:
which is the form used in quantum electrodynamics.
Geometric algebra formulations Analogous to the tensor formulation, two objects, one for the electromagnetic field and one for the current density, are introduced. In geometric algebra (GA) these are multivectors, which sometimes follow Ricci calculus.
Algebra of physical space In the Algebra of physical space (APS), also known as the Clifford algebra C ℓ 3 , 0 ( R ) {\displaystyle C\ell _{3,0}(\mathbb {R} )} , the field and current are represented by multivectors. The field multivector, known as the Riemann–Silberstein vector, is
F = E + I c B = E k σ k + I c B k σ k , {\displaystyle \mathbf {F} =\mathbf {E} +Ic\mathbf {B} =E^{k}\sigma _{k}+IcB^{k}\sigma _{k},}
and the four-current multivector is
c ρ − J = c ρ − J k σ k {\displaystyle c\rho -\mathbf {J} =c\rho -J^{k}\sigma _{k}}
using an orthonormal basis { σ k } {\displaystyle \{\sigma _{k}\}} . Similarly, the unit pseudoscalar is I = σ 1 σ 2 σ 3 {\displaystyle I=\sigma _{1}\sigma _{2}\sigma _{3}} , due to the fact that the basis used is orthonormal. These basis vectors share the algebra of the Pauli matrices, but are usually not equated with them, as they are different objects with different interpretations. After defining the derivative
∇ = σ k ∂ k , {\displaystyle {\boldsymbol {\nabla }}=\sigma ^{k}\partial _{k},}
Maxwell's equations are reduced to the single equation
In three dimensions, the derivative has a special structure allowing the introduction of a cross product:
∇ F = ∇ ⋅ F + ∇ ∧ F = ∇ ⋅ F + I ∇ × F {\displaystyle {\boldsymbol {\nabla }}\mathbf {F} ={\boldsymbol {\nabla }}\cdot \mathbf {F} +{\boldsymbol {\nabla }}\wedge \mathbf {F} ={\boldsymbol {\nabla }}\cdot \mathbf {F} +I{\boldsymbol {\nabla }}\times \mathbf {F} }
from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as
( ∇ ⋅ E − ρ ε 0 ) − c ( ∇ × B − μ 0 ε 0 ∂ E ∂ t − μ 0 J ) + I ( ∇ × E + ∂ B ∂ t ) + I c ( ∇ ⋅ B ) = 0 {\displaystyle \left({\boldsymbol {\nabla }}\cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)-c\left({\boldsymbol {\nabla }}\times \mathbf {B} -\mu _{0}\varepsilon _{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)+Ic\left({\boldsymbol {\nabla }}\cdot \mathbf {B} \right)=0}
Spacetime algebra
We can identify APS as a subalgebra of the spacetime algebra (STA) C ℓ 1 , 3 ( R ) {\displaystyle C\ell _{1,3}(\mathbb {R} )} , defining σ k = γ k γ 0 {\displaystyle \sigma _{k}=\gamma _{k}\gamma _{0}} and I = γ 0 γ 1 γ 2 γ 3 {\displaystyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}} . The γ μ {\displaystyle \gamma _{\mu }} s have the same algebraic properties of the gamma matrices but their matrix representation is not needed. The derivative is now
∇ = γ μ ∂ μ . {\displaystyle \nabla =\gamma ^{\mu }\partial _{\mu }.}
The Riemann–Silberstein becomes a bivector
F = E + I c B = E 1 γ 1 γ 0 + E 2 γ 2 γ 0 + E 3 γ 3 γ 0 − c ( B 1 γ 2 γ 3 + B 2 γ 3 γ 1 + B 3 γ 1 γ 2 ) , {\displaystyle F=\mathbf {E} +Ic\mathbf {B} =E^{1}\gamma _{1}\gamma _{0}+E^{2}\gamma _{2}\gamma _{0}+E^{3}\gamma _{3}\gamma _{0}-c(B^{1}\gamma _{2}\gamma _{3}+B^{2}\gamma _{3}\gamma _{1}+B^{3}\gamma _{1}\gamma _{2}),}
and the charge and current density become a vector
J = J μ γ μ = c ρ γ 0 + J k γ k = γ 0 ( c ρ − J k σ k ) . {\displaystyle J=J^{\mu }\gamma _{\mu }=c\rho \gamma _{0}+J^{k}\gamma _{k}=\gamma _{0}(c\rho -J^{k}\sigma _{k}).}
Owing to the identity
γ 0 ∇ = γ 0 γ 0 ∂ 0 + γ 0 γ k ∂ k = ∂ 0 + σ k ∂ k = 1 c ∂ ∂ t + ∇ , {\displaystyle \gamma _{0}\nabla =\gamma _{0}\gamma ^{0}\partial _{0}+\gamma _{0}\gamma ^{k}\partial _{k}=\partial _{0}+\sigma ^{k}\partial _{k}={\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }},}
Maxwell's equations reduce to the single equation
Differential-forms approach
In what follows, cgs-Gaussian units, not SI units are used. (To convert to SI, see here.) By Einstein notation, we implicitly take the sum over all values of the indices that can vary within the dimension.
Field 2-form In free space, where ε = ε0 and μ = μ0 are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. The Faraday tensor F μ ν {\displaystyle F_{\mu \nu }} (electromagnetic tensor) can be written as a 2-form in Minkowski space with metric signature (− + + +) as
F ≡ 1 2 F μ ν d x μ ∧ d x ν = B x d y ∧ d z + B y d z ∧ d x + B z d x ∧ d y + E x d x ∧ d t + E y d y ∧ d t + E z d z ∧ d t {\displaystyle {\begin{aligned}\mathbf {F} &\equiv {\frac {1}{2}}F_{\mu \nu }\mathrm {d} x^{\mu }\wedge \mathrm {d} x^{\nu }\\&=B_{x}\mathrm {d} y\wedge \mathrm {d} z+B_{y}\mathrm {d} z\wedge \mathrm {d} x+B_{z}\mathrm {d} x\wedge \mathrm {d} y+E_{x}\mathrm {d} x\wedge \mathrm {d} t+E_{y}\mathrm {d} y\wedge \mathrm {d} t+E_{z}\mathrm {d} z\wedge \mathrm {d} t\end{aligned}}}
which is the exterior derivative of the electromagnetic four-potential A : {\displaystyle \mathbf {A} :}
A = − ϕ d t + A x d x + A y d y + A z d z . {\displaystyle \mathbf {A} =-\phi \,\mathrm {d} t+A_{x}\mathrm {d} x+A_{y}\mathrm {d} y+A_{z}\mathrm {d} z.}
The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss's law and the Ampère–Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is the number of dimensions. Here, it takes the 2-form (F) and gives another 2-form (in four dimensions, n − p = 4 − 2 = 2). For the basis cotangent vectors, the Hodge dual is given as (see Hodge star operator § Four dimensions)
⋆ ( d x ∧ d y ) = − d z ∧ d t , ⋆ ( d x ∧ d t ) = d y ∧ d z , {\displaystyle {\star }(\mathrm {d} x\wedge \mathrm {d} y)=-\mathrm {d} z\wedge \mathrm {d} t,\quad {\star }(\mathrm {d} x\wedge \mathrm {d} t)=\mathrm {d} y\wedge \mathrm {d} z,}
and so on. Using these relations, the dual of the Faraday 2-form is the Maxwell tensor,
⋆ F = − B x d x ∧ d t − B y d y ∧ d t − B z d z ∧ d t + E x d y ∧ d z + E y d z ∧ d x + E z d x ∧ d y {\displaystyle {\star }\mathbf {F} =-B_{x}\mathrm {d} x\wedge \mathrm {d} t-B_{y}\mathrm {d} y\wedge \mathrm {d} t-B_{z}\mathrm {d} z\wedge \mathrm {d} t+E_{x}\mathrm {d} y\wedge \mathrm {d} z+E_{y}\mathrm {d} z\wedge \mathrm {d} x+E_{z}\mathrm {d} x\wedge \mathrm {d} y}
Current 3-form, dual current 1-form Here, the 3-form J is called the electric current form or current 3-form:
J = ρ d x ∧ d y ∧ d z − j x d t ∧ d y ∧ d z − j y d t ∧ d z ∧ d x − j z d t ∧ d x ∧ d y . {\displaystyle \mathbf {J} =\rho \,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z-j_{x}\mathrm {d} t\wedge \mathrm {d} y\wedge \mathrm {d} z-j_{y}\mathrm {d} t\wedge \mathrm {d} z\wedge \mathrm {d} x-j_{z}\mathrm {d} t\wedge \mathrm {d} x\wedge \mathrm {d} y.}
That F is a closed form, and the exterior derivative of its Hodge dual is the current 3-form, express Maxwell's equations:
Here d denotes the exterior derivative – a natural coordinate- and metric-independent differential operator acting on forms, and the (dual) Hodge star operator ⋆ {\displaystyle {\star }} is a linear transformation from the space of 2-forms to the space of (4 − 2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric). The fields are in natural units where 1/(4πε0) = 1. Since d2 = 0, the 3-form J satisfies the conservation of current (continuity equation):
d J = d 2 ⋆