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.

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 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]

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)