nLab Maxwell's equations

Idea

In the context of electromagnetism, Maxwell’s equations are the equations of motion for the electromagnetic field strength electric current and magnetic current.

(This is the specialization of the Yang-Mills equations for the case that the structure group is abelian, see there).

Three dimensional formulation

$E$ is here the (vector of) strength of electric field and $B$ the strength of magnetic field; $Q$ is the charge and $j_{el}$ the density of the electrical current; $\epsilon_0$, $c$, $\mu_0$ are constants (electrical permeability, speed of the light, and magnetic permeability; all of/in vacuum: $\mu_0 \epsilon_0 = 1/c^2$).

Integral formulation in vacuum

Gauss' law for electric fields

$$\textstyle{\int}_{\partial V} E\cdot d A = \frac{Q}{\epsilon_0}$$

where $\partial V$ is a closed surface which is a boundary of a 3d domain $V$ (physicists say “volume”) and $Q = \textstyle{\int}_V \rho d V$ the charge in the domain $V$; $\cdot$ denotes the scalar (dot) product. Surface element $d A$ is $\vec{n} d |A|$, i.e. it is the scalar surface measure times the unit vector of normal outwards.

No magnetic monopoles (Gauss’ law for magnetic fields)

$$\textstyle{\int}_\Sigma B\cdot d A = 0$$

where $\Sigma$ is any closed surface.

Faraday’s law of induction

$$\textstyle{\oint}_{\partial \Sigma} E\cdot d s = - \frac{d}{d t} \textstyle{\int}_\Sigma B\cdot d A$$

The line element $d s$ is the differential (or 1-d measure on the boundary) of the length times the unit vector in counter-circle direction (or parametrize the curve with $s$ being a vector in 3d space, express magnetic field in the same parameter and calculate the integral as a function of parameter: $\cdot$ is a scalar (“dot”) product).

Ampère-Maxwell law (or generalized Ampère’s law; Maxwell added the second term involving derivative of the flux of electric field to the Ampère’s law which described the magnetic field due electric current).

$$\textstyle{\oint}_{\partial \Sigma} B\cdot d s = \mu_0 I + \mu_0 \epsilon_0 \frac{d}{d t} \textstyle{\int}_\Sigma E\cdot d A$$

where $\Sigma$ is a surface and $\partial \Sigma$ its boundary; $I$ is the total current through $\Sigma$ (integral of the component of $j_{el}$ normal to the surface).

Differential equations

Here we put units with $c = 1$. By $\rho$ we denote the density of the charge.

In pregeometric form, Maxwell’s equations are differential equations for four fields (vector fields) called

• electric field $\vec E$

• magnetic flux density $\vec B$

• magnetic field $\vec H$

• displacement field $\vec D$ (or similar)

and state:

• (magnetic Gauss law)

  $div B = 0$

  (“there is no source for magnetic flux, hence no magnetic monopoles”)

• Faraday’s law:

  $\frac{d}{d t} B + rot E = 0$

• (electric Gauss law):

  $div D = \rho$

  (“the source of electric flux is electric current”)

• generalized Ampère’s law

  $- \frac{d}{d t} D + rot H = j_{el}$

In order to complete this pregeometric form to the actual equations of motion of the electromagnetic field, these four fields are to be subjected to a constraint called the constitutive equation which expresses $(\vec D, \vec H)$ as a function of $(\vec E, \vec B)$.

In vacuum and in the absence of background gravity, this constitutive relation:

• equates the displacement field $\vec D$ with the electric field $\vec E$ times a constant $\epsilon_0$

  called the “permitivity of the vacuum”,

• equates the magnetic flux $\vec B$ with the magnetic field $\vec H$ times a constant $\mu_0$

  called the “permeability of the vacuum”.

But for electromagnetic fields inside dielectric media other constitutive relations appear. For small field strengths these are typically linear functions $C$ (e.g. de Lange & Raab 2006 (19))

$$(\vec D, \vec H) \;\overset{!}{=}\; C\big((\vec E, \vec B)\big) \,,$$

but in general the constitutive relation can be a non-linear or even be a “multi-valued function” (namely when there are hysteresis effects in the dielectric medium).

Similarly (interestingly), in the presence of background gravity (such as for electromagnetic fields in and around a star) there is a linear such relation depending on the pseudo-Riemannian metric. This is most transparently expressed in terms of the Hodge star operator acting on the electromagnetic fields re-packaged as a Faraday tensor differential 2-form (see below).

In terms of Faraday tensor $F$

This is adapted from electromagnetic field – Maxwell’s equations, for more see the references below.

In modern language, the insight of (Maxwell, 1865) is that locally, when physical spacetime is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional Minkowski space $U \subset (\mathbb{R}^4, g = diag(-1,1,1,1))$, the electric field $\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right]$ and magnetic field $\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right]$ combine into a differential 2-form

$$\begin{aligned} F & \coloneqq E \wedge d t + B \\ & \coloneqq E_1 d x^1 \wedge d t + E_2 d x^2 \wedge d t + E_3 d x^3 \wedge d t \\ & \phantom{\coloneqq} + B_1 d x^2 \wedge d x^3 + B_2 d x^3 \wedge d x^1 + B_3 d x^1 \wedge d x^2 \end{aligned}$$

in $\Omega^2(U)$ and the electric charge density and current density combine to a differential 3-form

$$\begin{aligned} j_{el} & \coloneqq j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\ & \coloneqq j_1 d x^2 \wedge d x^3 \wedge d t + j_2 d x^3 \wedge d x^1 \wedge d t + j_3 d x^1 \wedge d x^2 \wedge d t - \rho \; d x^1 \wedge d x^2 \wedge d x^3 \end{aligned}$$

in $\Omega^3(U)$ such that the following two equations of differential forms are satisfied

$$\begin{aligned} d F = 0 \\ d \star F = j_{el} \end{aligned} \,,$$

where $d$ is the de Rham differential operator and $\star$ the Hodge star operator. If we decompose $\star F$ into its components as before as

$$\begin{aligned} \star F &= -D + H\wedge dt \\ &= -D_1 \; d x^2 \wedge d x^3 -D_2 \; d x^3 \wedge d x^1 -D_3 \; d x^1 \wedge d x^2 \\ & \phantom{=} + H_1 \; d x^1 \wedge d t + H_2 \; d x^2 \wedge d t + H_3 \; d x^3 \wedge d t \end{aligned}$$

then in terms of these components the Maxwell equations read as follows:

$d F = 0$ gives the magnetic Gauss law and Faraday’s law

$d \star F = j_{el}$ gives Gauss's law and Ampère-Maxwell law

In terms of the Faraday bivector $F$

In the geometric algebra formalism of electromagnetism, one works in the 4-dimensional Clifford algebra $\mathrm{Cl}^{1, 3}(\mathbb{R})$ representing spacetime, where the orthonormal basis vectors $\{\gamma_i\}$ have signature $(+, -, -, -)$ with $\gamma_0$ representing the time dimension and the other three basis vectors representing the spacial dimensions. The pseudoscalar of $\mathrm{Cl}^{1, 3}(\mathbb{R})$ is represented by the product of all the basis vectors

$$I = \prod_{i} \gamma_i = \gamma_0 \gamma_1 \gamma_2 \gamma_3$$

The basis bivectors of $\mathrm{Cl}^{1, 3}(\mathbb{R})$ come in two sets, the timelike bivectors $\sigma_1 = \gamma_1 \gamma_0$, $\sigma_2 = \gamma_2 \gamma_0$, $\sigma_3 = \gamma_3 \gamma_0$, and the spacelike bivectors $I \sigma_1 = -\gamma_2 \gamma_3$, $I \sigma_2 = \gamma_1 \gamma_3$, $I \sigma_3 = \gamma_1 \gamma_2$. The bivector subalgebra of $\mathrm{Cl}^{1, 3}(\mathbb{R})$ is equivalent to the three-dimensional Clifford algebra $\mathrm{Cl}^{3, 0}(\mathbb{R})$ corresponding to the relative space in the rest frame defined by $\gamma_0$, where the basis timelike bivectors correspond to the basis relative vectors and the basis spacelike bivectors correspond to the basis relative bivectors of the rest frame.

Let $x$ be a vector in $\mathrm{Cl}^{1, 3}(\mathbb{R})$. Then we define the coordinates of $x$ relative to the basis $\{\gamma_i\}$ to be $x^i = \gamma_i \cdot x$. $x^0$ is also denoted as $t$ since it represents the time coordinate. The spacetime vector derivative is defined as the operator

$$\nabla = \gamma_0 \frac{\partial}{\partial t} + \gamma_1 \frac{\partial}{\partial x^1} + \gamma_2 \frac{\partial}{\partial x^2} + \gamma_3 \frac{\partial}{\partial x^3}$$

The relative vector derivative is defined as the operator

$$\mathbf{\nabla} = \sigma_1 \frac{\partial}{\partial x^1} + \sigma_2 \frac{\partial}{\partial x^2} + \sigma_3 \frac{\partial}{\partial x^3}$$

Assuming the use of natural units where $c = \epsilon_0 = 1$ and ignoring the polarization and magnatization fields for the time being, Maxwell’s equations are written as:

• electric Gauss’s law:

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

• Faraday’s law:

  $$\mathbf{\nabla} \wedge E = -\frac{\partial}{\partial t}\left(I B\right)$$

• magnetic Gauss’s law:

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

• Ampère-Maxwell law:

  $$\mathbf{\nabla} \wedge B = I\left(\mathbf{J} + \frac{\partial}{\partial t}\left(E\right)\right)$$

where $E$ and $B$ are the relative electric and magnetic relative fields, $\rho$ is the density of the charge, and $\mathbf{J}$ is the relative current of the charge.

The Faraday bivector is the bivector $F = E + I B$. By multiplying the last two equations by the pseudoscalar, one gets

$$\mathbf{\nabla} \cdot (I B) = 0$$

$$\mathbf{\nabla} \wedge (I B) = -\mathbf{J} - \frac{\partial}{\partial t}\left(E\right)$$

and by adding the four equations together, one gets the equation

$$\mathbf{\nabla} \cdot (E + I B) + \mathbf{\nabla} \wedge (E + I B) = \rho -\mathbf{J} -\frac{\partial}{\partial t}\left(E + I B\right)$$

which then becomes

$$\mathbf{\nabla} \cdot F + \mathbf{\nabla} \wedge F + \frac{\partial}{\partial t}\left(F\right) = \rho - \mathbf{J}$$

Given a timelike bivector $v$ and a general bivector $M$ in $\mathrm{Cl}^{1, 3}(\mathbb{R})$, $v M = v \cdot M + v \wedge M$. Thus, the above equation could be simplified even further to

$$\mathbf{\nabla} F + \frac{\partial}{\partial t}\left(F\right) = \rho - \mathbf{J}$$

or

$$\left(\mathbf{\nabla} + \frac{\partial}{\partial t}\right)\left(F\right) = \rho - \mathbf{J}$$

Now, the spacetime current is given by $J = \gamma_0 (\rho - \mathbf{J})$, which is a vector in spacetime. The spacetime vector derivative is related to the relative vector derivative by the following equation:

$$\nabla = \gamma_0 (\mathbf{\nabla} + \frac{\partial}{\partial t})$$

Thus, by left multiplying each side by $\gamma_0$, one gets

$$\gamma_0 \left(\mathbf{\nabla} + \frac{\partial}{\partial t}\right)\left(F\right) = \gamma_0 (\rho - \mathbf{J})$$

or

$$\nabla F = J$$

where $\nabla$ is the spacetime vector derivative, $F$ is the Faraday bivector, and $J$ is the spacetime current.

Related concepts

• Hodge-Maxwell theorem

• Kirchhoff's laws

• Yang-Mills equations

References

For more see the references at electromagnetism.

General

Maxwell’s equations originate in:

• James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the Royal Society of London 155 (1865) 459-512 [doi:10.1098/rstl.1865.0008, jstor:108892, Wikipedia entry]

• James Clerk Maxwell, A Treatise on Electricity and Magnetism, Clarendon Press Series, Macmillan & Co. (1873), Cambridge University Press (2010) [ark:/13960/t9s17v886, doi:10.1017/CBO9780511709333, pdf, Wikipedia entry]

Some history and reflection:

• A. C. T. Wu, Chen Ning Yang: Evolution of the concept of vector potential in the description of the fundamental interactions, International Journal of Modern Physics A 21 16 (2006) 3235-3277 [doi:10.1142/S0217751X06033143]

• Freeman J. Dyson, Why is Maxwell’s Theory so hard to understand?, Proceedings of The Second European Conference on Antennas and Propagation, EuCAP 2007 (doi: 10.1049/ic.2007.1146)

• Chen Ning Yang, The conceptual origins of Maxwell’s equations and gauge theory, Phyics Today 67 11 (2014) [doi:10.1063/PT.3.2585, pdf]

See also:

• Freeman J. Dyson: Feynman’s proof of Maxwell’s equations, American Journal of Physics 58 (1990) 209-211 [pdf, pdf]

  (recounting an observation by Richard Feynman)

  comment by N. Dombey: Am. J. Phys. 59 1 (1991) 85 [journal, scan]

  comment by I. E. Farquhar: Am. J. Phys. 59 1 (1991) 87 [journal, scan]

For Maxwell’s equations in the generality of dielectric media, see the references there, such as:

• G. Russakoff, A Derivation of the Macroscopic Maxwell Equations, American Journal of Physics 38 (1970) 1188–1195 [doi:10.1119/1.1976000]

• O. L. de Lange, R. E. Raab Surprises in the multipole description of macroscopic electrodynamics, American Journal of Physics 74 (2006) 301–312 [doi:10.1119/1.2151213]

On the Maxwell Green's function (propagator) and numerical solutions:

• Boris Lo, Victor Minden, Phillip Colella, A real-space Green’s function method for the numerical solution of Maxwell’s equations, 11 2 (2016) 143–170 [doi:10.2140/camcos.2016.11.143, pdf]

Maxwell’s equations via differential forms

On the expression of classical electromagnetism, and especially of Maxwell's equations, in terms of differential forms, the de Rham differential and Hodge star operators:

• Élie Cartan, §80 in: Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite), Annales scientifiques de l’É.N.S. 3e série, tome 41 (1924) 1-25 [numdam:ASENS_1924_3_41__1_]

  (already in pregeometric form)

• Charles Misner, Kip Thorne, John Wheeler, §3.4 and §4.3 in: Gravitation, W. H. Freeman, San Francisco (1973) [ISBN:9780716703440]

• Theodore Frankel, Maxwell’s equations, The American Mathematical Monthly 81 4 (1974) [doi:10.1080/00029890.1974.11993557, jstor:2318995, doi:10.2307/2318995]

• Walter Thirring, vol 2 §1.3 in: A Course in Mathematical Physics – 1 Classical Dynamical Systems and 2 Classical Field Theory, Springer (1978, 1992) [doi:10.1007/978-1-4684-0517-0]

• Theodore Frankel, §§9-10 in: Gravitational Curvature, Freeman, San Francisco (1979) [ark:13960/t58d7nn19]

• Dominic G. B. Edelen, §9.2 in: Applied exterior calculus, Wiley (1985) [GoogleBooks]

• Theodore Frankel, §3.5 & §7.2b in: The Geometry of Physics - An Introduction, Cambridge University Press (1997, 2004, 2012) [doi:10.1017/CBO9781139061377]

• Gregory L. Naber, §2.2 in: Topology, Geometry and Gauge fields – Interactions, Applied Mathematical Sciences 141 (2011) [doi:10.1007/978-1-4419-7895-0]

• Masao Kitano, Reformulation of Electromagnetism with Differential Forms, Chapter 2 in: Trends in Electromagnetism – From fundamentals to applications, InTech (2012) 21-44 [ISBN:978-953-51-0267-0, pdf]

• Sébastien Fumeron, Bertrand Berche, Fernando Moraes, Improving student understanding of electrodynamics: the case for differential forms, American Journal of Physics 88 (2020) 1083 [doi:10.1119/10.0001754, arXiv:2009.10356]

Via “geometric algebra: (Clifford algebra)

Formulation of Maxwell’s equations via “geometric algebra”:

• Chris Doran, Anthony Lasenby: Geometric algebra for physicists, Cambridge University Press (2003) (pdf)

• John W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory, John Wiley & Sons Inc. (2011). (ISBN:978-0470941638, doi:10.1002/9781118078549)