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In the context of electromagnetism, Maxwell’s equations are the equations of motion for the electromagnetic field strength electric current and magnetic current.
$E$ 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: $\mu_0 \epsilon_0 = 1/c^2$).
Gauss’ law for electric fields
where $\partial V$ is a closed surface which is a boundary of a 3d domain $V$ (physicists say “volume”) and $Q = \int_V \rho d V$ 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)
where $\Sigma$ is any closed surface.
Faraday’s law of induction
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).
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).
Here we put units with $c = 1$. By $\rho$ we denote the density of the charge.
no magnetic charges (magnetic Gauss law): $div B = 0$
Faraday’s law: $\frac{d}{d t} B + rot E = 0$
Gauss’ law: $div D = \rho$
generalized Ampère’s law $- \frac{d}{d t} D + rot H = j_{el}$
This is adapted from electromagnetic field – Maxwell’s equations.
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
in $\Omega^2(U)$ and the electric charge density and current density combine to a differential 3-form
in $\Omega^3(U)$ such that the following two equations of differential forms are satisfied
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
then in terms of these components the field equations – called Maxwell’s equations – read as follows.
$d F = 0$ gives the magnetic Gauss law and Faraday’s law
$d \star F = 0$ gives Gauss’s law and Ampère-Maxwell law
Maxwell's equations originate in
Discussion in terms of differential forms is for instance in
Theodore Frankel, Maxwell’s equations, The American Mathematical Monthly, Vol 81, No 4 (1974) (pdf, JSTOR)
Theodore Frankel, section 3.5 in The Geometry of Physics - An Introduction
Gregory L. Naber, Topology, geometry and gauge fields, Appl. Math. Sciences vol. 141, Springer 2000
Some history and reflection is in
Last revised on January 5, 2019 at 12:52:21. See the history of this page for a list of all contributions to it.