5.1.2 Equations for the Electrodynamic Potentials

Equation (4.2) is satisfied by the expression

$$\vec{B} = \vec{\nabla}\times\vec{A}.$$ (5.3)

Thus (4.1) becomes

$$\vec{\nabla}\times\left(\vec{E} + \partial_t\vec{A}\right) = \vec{0} \Leftrightarrow \vec{E} + \partial_t\vec{A} = -\vec{\nabla}\varphi$$ (5.4)

and the electric field $\vec{E}$ can be given by the electrodynamic potentials -- the magnetic vector potential $\vec{A}$ and the electric scalar potential $\varphi$

$$\vec{E} = - \partial_t\vec{A} -\vec{\nabla}\varphi.$$ (5.5)

Using (4.7) and (5.3) the left hand side of (4.3) is expressed as

$$\vec{\nabla}\times\vec{H} = \vec{\nabla}\times\frac{1}{\mu}\vec{B} = \vec{\nabla}\times(\frac{1}{\mu}\vec{\nabla}\times\vec{A}).$$ (5.6)

With (4.6), (4.8), and (5.5) the right hand side of (4.3) is given by

$$\vec{J} + \partial_t\vec{D} = \gamma\vec{E} + \epsilon\partial_t\... ...\varphi - \epsilon\partial_{tt}\vec{A} - \epsilon\partial_t\vec{\nabla}\varphi.$$ (5.7)

Substituting (5.6) and (5.7) in (4.3) the following equation for $\vec{A}$ and $\varphi$ in the frequency domain is obtained

$$\vec{\nabla}\times(\frac{1}{\mu}\vec{\nabla}\times\vec{A}) = (\om... ...jmath\omega\gamma)\vec{A} - (\gamma + \jmath\omega\epsilon)\vec{\nabla}\varphi.$$ (5.8)

Analogously after partial differentiation of (4.4) with respect to time and using (4.5), (4.6), and (4.8) it can be written

$$\vec{\nabla}\cdot\epsilon\partial_t\vec{E} = \partial_t\rho = -\vec{\nabla}\cdot\left(\gamma\vec{E}\right),$$ (5.9)

which leads with (5.5) to

$$\vec{\nabla}\cdot\left\{ (\omega^2\epsilon - \jmath\omega\gamma)\vec{A} - (\gamma + \jmath\omega\epsilon)\vec{\nabla}\varphi \right\} = 0.$$ (5.10)

Equation (5.10) can also be obtained by applying the divergence operator to (5.8). The unknown fields $\vec{A}$ and $\varphi$ are obtained from the boundary value problem given by the partial differential equation system (5.8) and (5.10). A similar strategy can be found in [67,68], where $\vec{B} = \vec{\nabla}\times(\vec{A} + \vec{\nabla}\chi)$ and the auxiliary arbitrary scalar field $\chi$ is conveniently termed as a ghost field.