PX284 - S3a - conductors

no free charge, but there may be free current:

\begin{matrix} ρ_{f} = 0 \\ {\vec{J}}_{f} \neq 0 \end{matrix}

assuming linear and isotropic medium:

\begin{matrix} \vec{D} = ε \vec{E} \\ \vec{D} = μ \vec{H} \\ {\vec{J}}_{f} = g \vec{E} \end{matrix}

where, $g$ is conductivity

assuming that $ε$ , $μ$ and $g$ are scalars, ie. they don't depend on the field strength, but may depend on $ω$

\begin{aligned} \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) & = \vec{\nabla} (\vec{\nabla} \cdot \vec{E}) - \nabla^{2} \vec{E} \\ = - \frac{\partial}{\partial t} (\vec{\nabla} \times μ \vec{H}) \\ = - μ \frac{\partial}{\partial t} (g \vec{E}) - μ ε_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} \\ \nabla^{2} \vec{E} & = μ g \frac{\partial \vec{E}}{\partial t} + μ ε \frac{\partial^{2} \vec{E}}{\partial t^{2}} \end{aligned}

looking for plane wave solutions:

\vec{E} = {\vec{E}}_{0} \exp (i (\vec{k} \cdot \vec{r} - ω t))

\begin{matrix} \vec{\nabla} \to i \vec{k} \\ \frac{\partial}{\partial t} \to - i ω \end{matrix}

⟹ k^{2} = i μ ω g + μ ε ω^{2}

this is the dispersion relation
considering good conductors, at frequencies, $ω ≪ g / ε$ , can neglect the second term

\begin{matrix} k^{2} ≃ i μ g ω \\ k = \pm \frac{1 + i}{\sqrt{2}} \sqrt{μ g ω} = \pm \frac{1 + i}{δ} \end{matrix}

where, $δ = \sqrt{2 / μ g ω}$ , is called the 'skin depth'

for a wave propagating in $\vec{x}$ -direction:

\vec{E} = {\vec{E}}_{0} \exp (i (k x - ω t)) = {\vec{E}}_{0} \exp (- \frac{x}{δ}) \exp (i (\frac{x}{δ} - ω t))

this is a decaying solution

\begin{matrix} \vec{k} \cdot \vec{E} = 0 \\ \vec{k} \times \vec{E} = ω μ \vec{H} \end{matrix}

$\vec{k}$ , $\vec{E}$ and $\vec{H}$ are all perpendicular to each other
the impedance:

Z_{c} = \frac{E}{H} = \frac{ω μ}{k} = \frac{ω μ δ}{i + 1} = \frac{ω μ δ}{\sqrt{2}} \exp (- \frac{i π}{4})

$\vec{H}$ is out of phase with $\vec{E}$ by $π / 4$

\begin{matrix} \vec{E} = E_{0} \hat{z} \exp (i (- k r - ω t)) \\ \vec{H} = H_{0} \hat{θ} \exp (i (- k r - ω t)) \end{matrix}

$\vec{k}$ is the direction of $- \hat{r}$ , ie. into the wire

\begin{matrix} k = \frac{1 + i}{δ} \\ \vec{E} = E_{0} \hat{z} \exp (\frac{e}{δ}) \exp (i (- \frac{r}{δ} - ω t)) \end{matrix}

growing with $r$ up to $r = r_{0}$
current flows in a cylindrical shell of thickness $δ$ of edge of the wire
this is called the skin effect