24. Maxwell’s equations

describe the interaction of magnetic and electric fields. It is a coupled set of differential equations

\[\begin{align*} \operatorname{curl} E &= -\frac{\partial B}{\partial t} \\ \operatorname{curl} H &= \frac{\partial D}{\partial t} + j \end{align*}\]

coupled with material laws

\[ B = \mu H, \qquad D = \varepsilon E, \qquad j = \sigma E \]

The fields are

symbol	unit	name
\(E\)	\(\frac{V}{m}\)	electric field
\(H\)	\(\frac{A}{m}\)	magnetic field
\(B\)	\(\frac{Vs}{m^2}\)	magnetic flux density
\(D\)	\(\frac{As}{m^2}\)	displacement current density
\(j\)	\(\frac{A}{m^2}\)	current density

material parameters are

symbol	unit	name
\(\mu\)	\(\frac{Vs}{Am}\)	permeability
\(\varepsilon\)	\(\frac{As}{Vm}\)	permittivity
\(\sigma\)	\(\frac{A}{Vm}\)	conductivity

often one writes

\[ \mu = \mu_r \mu_0, \qquad \varepsilon = \varepsilon_r \varepsilon_0 \]

with \(\mu_0 = 4 \pi 10^{-6} \frac{Vs}{Am} \) and \(\varepsilon_0 \approx 8.854 \cdot 10^{-12} \frac{As}{Vm}\) permeability and permittivity in vacuuum, and \(\mu_r\) and \(\varepsilon_r\) dimensionless factors depending on the material.

Integrating the differential equation over a surface \(S\), and applying Stokes’ theorem, we obtain the integral form of Maxwell’s equations:

\[\begin{align*} \int_{\partial S} E \cdot d\tau &= -\frac{d}{dt} \int_S B \cdot n \, ds & \text{(Faraday's induction law)}\\ \int_{\partial S} H \cdot d\tau &= \frac{d}{dt} \int_S D \cdot n \, ds + \int_S j \cdot n \, ds & \text{(Ampère-Maxwell law)} \end{align*}\]

If the vector field \(A\) solves the equation

\[ \varepsilon \frac{\partial^2 A}{\partial t^2} + \sigma \frac{\partial A}{\partial t} + \operatorname{curl} \frac{1}{\mu} \operatorname{curl} A = 0, \]

then \(E := -\frac{\partial A}{\partial t}\), \(B := \operatorname{curl} A\) (and the other fields defined via the material laws) solve Maxwell’s equations.

That equation is second order in time, and describes electromagnetic wave propagation effects. The wavespeed is \(1/ \sqrt{\varepsilon \mu}\). For low frequency applications (like electric motors or transformers) wave propagation is not relevant, and the second time derivative is just skipped. This leads to the so called Eddy-current approximation:

\[ \sigma \frac{\partial A}{\partial t} + \operatorname{curl} \frac{1}{\mu} \operatorname{curl} A = 0, \]