Skin effect -- Calculate the electric field

[Lambda96]

Homework Statement

Calculate the electric field

Relevant Equations

none

Hi,

unfortunately, I can't get any further with the following task, or rather I don't know how to start:

I'm also a bit confused, the task says that you have to calculate the electric field in the metal. Since it is a metal, i.e. a conductor, there is no electric field in the conductor, is it rather the electromagnetic wave that penetrates the metal that is meant to be calculated? Because $\delta$, i.e. the penetration depth, appears in the formula.

Would I then have to calculate the wave vector $k$ when the wave penetrates the metal? Unfortunately, I don't know where to start to solve the problem.

 

[Lambda96]

I have now proceeded as follows, $E$, $B$ and $k$ are all perpendicular to each other and it is a complex electromagnetic wave, then you can represent Faraday's law of induction and the Ampère-Maxwell law with the wave vector $k$ as follows in the CGS unit system:

$$\begin{align*}
ik \times E&= i \omega \frac{B}{c}\\
ik \times \frac{B}{c}&= \frac{4 \pi}{c^2} \sigma(\omega)E - i \frac{4 \pi}{c^2} \epsilon(\omega) \omega E
\end{align*}$$

With the Maxwell-Faraday equation, you can now form a cross product with the wave vector $k$ again and divide it by $i$ beforehand, i.e.

$$k \times (k \times E)= \omega k \times \frac{B}{c}$$

The expression $k \times \frac{B}{c}$ is the Ampère-Maxwell law, which you can now insert there and you get the following:

$$-k^2 E=(-i \frac{4 \pi}{c^2} \omega \sigma(\omega)-\frac{4 \pi}{c^2} \epsilon(\omega) \omega^2)E$$

The expression E can now be ignored and the equation can be rearranged as follows:

$$k^2 +-i \frac{4 \pi}{c^2} \omega \sigma(\omega)-\frac{4 \pi}{c^2} \epsilon(\omega) \omega^2=0$$

Now you can use $\sigma(\omega)=\frac{\sigma_0}{1-i \omega \tau}$ and $\epsilon(\omega)=\frac{i4 \pi \sigma_0}{\omega - i\omega^2 \tau}$ from the task and get

$$k^2 +-i \frac{4 \pi}{c^2} \omega \frac{\sigma_0}{1-i \omega \tau}-\frac{4 \pi}{c^2} \frac{i4 \pi \sigma_0}{\omega - i\omega^2 \tau} \omega^2=0$$

You can now solve the expression for $k$ and get:

$$k=\frac{2\sqrt{\pi(1+4 \pi) \sigma \omega}}{\sqrt{-c^2(\omega \tau + i)}}$$

I would now have to multiply the expression by $-i z$, but I can see directly that my result does not match the one on the task sheet, not even with the $\delta$. Have I done something wrong?

 

[renormalize]

I'm also a bit confused, the task says that you have to calculate the electric field in the metal. Since it is a metal, i.e. a conductor, there is no electric field in the conductor...

Based on your second post, you've realized that a metal is an imperfect conductor and hence the electric field can indeed be non-zero inside.

$$\begin{align*}
ik \times E&= i \omega \frac{B}{c}\\
ik \times \frac{B}{c}&= \frac{4 \pi}{c^2} \sigma(\omega)E - i \frac{4 \pi}{c^2} \epsilon(\omega) \omega E
\end{align*}$$

Note that Faraday's and Ampere's laws in Gaussian units are:
\begin{align*}
\nabla\times\vec{E}&=-\frac{1}{c}\frac{\partial\vec{B}}{\partial t}\\
\nabla\times\vec{B}&=\frac{1}{c}\left(4\pi\sigma\vec{E}+\frac{\partial\vec{E}}{\partial t}\right)
\end{align*}From these, it's not clear to me how the factor $4\pi\epsilon(\omega)$ gets into the last term of your second equation above.

 

[Lambda96]

Thank you renormalize for your help

We had defined Maxwell's laws in the lecture as follows

$$\begin{align*}
ik \times E&= i \omega B\\
ik \times B&= \mu_0 \sigma(\omega)E - i \mu_0 \epsilon(\omega) \omega E
\end{align*}$$

Then we have the following relationship for the conversion from SI to CGS $\mu_0=\frac{4 \pi}{c^2}$

 

[renormalize]

Then we have the following relationship for the conversion from SI to CGS $\mu_0=\frac{4 \pi}{c^2}$

That's not right. The vacuum-permeability $\mu_0$ is related to the speed-of-light $c$ and the vacuum-permittivity $\varepsilon_0$ according to:$$\mu_0=\frac{1}{c^2\,\varepsilon_0}$$

 

[Lambda96]

Thanks for your help renormalize

I have asked my lecturer again if it was unfortunately a typing error in the script

 

[hutchphd]

The expression E can now be ignored and the equation can be rearranged

What does this mean? ( Please justify this step more explicitly )

 

[Lambda96]

$$\begin{align*}
-k^2 E&=(-i \frac{4 \pi}{c^2} \omega \sigma(\omega)-\frac{4 \pi}{c^2} \epsilon(\omega) \omega^2)E\\
0&=k^2 E+(-i \frac{4 \pi}{c^2} \omega \sigma(\omega)-\frac{4 \pi}{c^2} \epsilon(\omega) \omega^2)E\\
0&=E(k^2-i \frac{4 \pi}{c^2} \omega \sigma(\omega)-\frac{4 \pi}{c^2} \epsilon(\omega) \omega^2)
\end{align*}$$

For the expression to be zero, either $E$ must be zero or the expression in the parenthesis must be zero, which is why I get the following equation, which I can solve for $k$ and I meant that I can ignore $E$.

$$0=k^2-i \frac{4 \pi}{c^2} \omega \sigma(\omega)-\frac{4 \pi}{c^2} \epsilon(\omega) \omega^2$$

 

[hutchphd]

To get this equation you have already made the ansatz that E is a plane wave with possibly complex wavenumber, I believe, but you sounded a little bit unsure. This solution is important to know well.