Answer to Question #89209 in Electricity and Magnetism for Sridhar

Question

In which of one of the following materials is the skin depth is high
1)aluminium
2)gold
3)silver
4)graphite

Answer 1

Skin depth is a measure of the depth at which the current density falls to 1/e of its value near the surface. The general formula for the skin depth is

$$\delta = \sqrt { {2 \rho } / {\omega \mu } }*\sqrt { \sqrt{ 1 + ( \rho \omega \epsilon )^2} + \rho \omega \epsilon }$$

where

$\rho$  = resistivity of the conductor

$\omega$ = angular frequency of current = 2π × frequency

$\mu = \mu_r \mu_0$

$\mu_r$ = relative magnetic permeability of the conductor

$\mu_0$ = the permeability of free space

$\epsilon = \epsilon_r \epsilon_0$

$\epsilon_r$ = relative permittivity of the material

$\epsilon_0$ = the permittivity of free space

We can see that this formula can be simplified to

$$\delta = \sqrt { {2 \rho } / { \mu } }$$

in case of good conductors and

$$\delta = 2 \rho \sqrt { {\epsilon } / { \mu } }$$

for poor conductors ( graphite ) in a wide frequency range.

From this formulas one can see that the skin depth is proportional to the resistivity in the first approximation, meaning it would be high in graphite and small in the metals