1.3. Electromagnetic waves

Similar to Section 1.1 and Section 1.2 the geometry is a given open domain \(\Omega\subset\mathbb R^3\), and time interval \([0,T], T>0\).

The constituents are the electric field \(E:[0,T]\times \Omega\to\mathbb R^3\) (unit \(m\,kg/(s^3A)\)) and the magnetic field intensity \(H:[0,T]\times \Omega\to\mathbb R^3\) (unit \(A/m\)), as well as the electric flux density \(D:[0,T]\times \Omega\to\mathbb R^3\) (also called electric displacement field, unit \(As/m^2\)) and the magnetic induction \(B:[0,T]\times \Omega\to\mathbb R^3\) (also called magnetic flux density unit \(kg/(s^2 A)\)). Further quantities, which act as forcings are the electric current density \(J:[0,T]\times \Omega\to\mathbb R^3\) (unit \(A/m^2\)) and the electric charge density \(\rho:[0,T]\times\Omega\to\mathbb R\) (unit \(As/m^3\)).

As balance relations we use Faraday’s law

\[-\partial_t B = \nabla\times E,\]

Ampère’s law with Maxwell’s correction

\[\partial_t D = \nabla\times H -J\]

and Gauß’ laws

\[\begin{aligned} \nabla\cdot D& = \rho,&\nabla\cdot B &= 0. \end{aligned}\]

Note that conservation of charge

\[\partial_t\rho+\nabla\cdot J = 0\]

follows from Ampere’s law and Gauß’ law for the electric field. One possible viewpoint is to take \(E,H\) as the primary quantities leading to the so called \(EH\)-formulation of Maxwell’s equations. To this end we also need the linear material laws

\[\begin{aligned} D&=\varepsilon E,&B&=\mu H, \end{aligned}\]

where the material parameters \(\varepsilon,\mu\) are the permittivity (unit \(A^2s^4/(kg\,m^3)\)) and permeability (unit \(kg\,m/(s^2A^2)\)) of the materials in question. The above leads to

\[\begin{split}\partial_t \mu H + \nabla\times E &=0,\\ \partial_t \varepsilon E-\nabla\times H&=-J,\\ E(0)&=E_0,\\ H(0)&= H_0,\\ E\times n = 0\end{split}\]

Similar to acoustic and elastic waves for constant \(\mu,\varepsilon\) we also may obtain the second order equation for \(E\)

(1.14)\[\partial_t^2 E+c^2\nabla\times\nabla\times E = 0,\]

where we assumed the absence of electric currents and charges and defined the speed of light by \(c^2=\frac{1}{\varepsilon\mu}\).