3.2. Wave Equations in Electromagnetism
Wave equations for physical quantities in electromagnetism are derived from the governing equations for linear local isotropic homogeneous (\(\varepsilon\), \(\mu\) uniform, not function of space) media with constitutive equations
\[\vec{d} = \varepsilon \vec{e} \qquad , \qquad \vec{b} = \mu \vec{h} \ ,\]
using vector identity
\[\Delta \vec{v} = \nabla ( \nabla \cdot \vec{v} ) - \nabla \times \nabla \times \vec{v} \ .\]
If some of the assumptions made above is not true, slight modifications and extra terms in the equations are likely to appear during the manipulation of the equations done below.
3.2.1. Electromagnetic Potentials
3.2.1.1. Vector potential
Wave equation for the vector potential,
\[\vec{b} = \nabla \times \vec{a} \ ,\]
is derived taking the curl of its definition,
\[\begin{split}\begin{aligned}
\vec{0} & = \nabla \times \nabla \times \vec{a} - \nabla \times \vec{b} = && (1.a) \\
& = - \Delta \vec{a} + \nabla(\nabla \cdot \vec{a}) - \mu \nabla \times \vec{h} = && (2) \\
& = - \Delta \vec{a} + \nabla(\nabla \cdot \vec{a}) - \mu ( \partial_t \vec{d} + \vec{j}_f ) = && (1.b) \\
& = - \Delta \vec{a} + \nabla(\nabla \cdot \vec{a}) - \mu ( \varepsilon \partial_t \vec{e} + \vec{j}_f ) = && (3) \\
& = - \Delta \vec{a} + \nabla(\nabla \cdot \vec{a}) - \mu \varepsilon ( - \partial_t \nabla \varphi - \partial_{tt} \vec{a} ) + \mu \vec{j}_f = \\
& = - \Delta \vec{a} + \nabla(\nabla \cdot \vec{a}) + \frac{1}{c^2} \partial_t \nabla \varphi + \dfrac{1}{c^2} \partial_{tt} \vec{a} - \mu \vec{j}_f \\
\end{aligned}\end{split}\]
and using (1) the constitutive law for homogeneous isotropic linear media, (2) Ampére-Maxwell’s equation, (3), and (4) the definition of the electric field (3.1)(a) in terms of the potentials. Using the Lorentz gauge condition (3.2)
\[\nabla \cdot \vec{a} + \frac{1}{c^2} \partial_t \varphi = 0 \ ,\]
wave equation for the vector potential reads,
(3.3)\[ \dfrac{1}{c^2} \partial_{tt} \vec{a} - \Delta \vec{a} = \mu \vec{j} \ .\]
3.2.1.2. Scalar potential
Wave equation for the the scalar potential, \(\varphi(\vec{r},t)\), can be derived taking the time derivative of Lorentz’s gauge condition,
\[\begin{split}\begin{aligned}
0 & = \partial_t \left(\frac{1}{c^2} \partial_t \varphi + \nabla \cdot \vec{a} \right) = \\
& = \frac{1}{c^2} \partial_{tt} \varphi + \nabla \cdot \partial_t \vec{a} = && (1) \\
& = \frac{1}{c^2} \partial_{tt} \varphi - \nabla \cdot \nabla \varphi - \nabla \cdot \vec{e} = && (2) \\
& = \frac{1}{c^2} \partial_{tt} \varphi - \Delta \varphi - \frac{\rho_f}{\varepsilon} \ ,
\end{aligned}\end{split}\]
using (1) the definition (3.1)(a) of the electric field as a function of the potentials, and (2) Gauss’ law for the electric field,
(3.4)\[ \frac{1}{c^2} \partial_{tt} \varphi - \Delta \varphi = \frac{\rho_f}{\varepsilon} \ .\]
3.2.2. Electric Field and Magnetic Field
Exploiting the linearity - obviously, if the problem is linear - wave equations for the electric and the magnetic field can be readily derived from applying the wave operator
\[\square := \frac{1}{c^2} \partial_{tt} - \Delta \ ,\]
to the (1) definitions (3.1) of the electric and the magnetic fields as functions of the potentials, (2) swapping the order of the operator \(\square\) with \(\partial_t\) and \(\nabla\), and (3) using the expressions of the wave equations for the vector potential (3.3) and the scalar potential (3.4).
3.2.2.1. Electric field
\[\begin{split}\begin{aligned}
\square \vec{e} & = && (1) \\
& =\square ( -\nabla \varphi - \partial_t \vec{a}) = && (2) \\
& = - \nabla \square \varphi - \partial_t \square \vec{a} = && (3) \\
& = - \nabla \dfrac{\rho}{\varepsilon} - \mu \partial_t \vec{j} \ .
\end{aligned}\end{split}\]
3.2.2.2. Magnetic field
\[\begin{split}\begin{aligned}
\square \vec{b} & = && (1) \\
& = \square \nabla \times \vec{a} = && (2) \\
& = \nabla \times \square \vec{a} = && (3) \\
& = \mu \nabla \times \vec{j}
\end{aligned}\end{split}\]
