A non‑magnetic good conductor has conductivity $\sigma =4.0\times 10^7{\text{S m}}^{-1}$ and relative permeability ${\mu }_r=1$. For a sinusoidal field of frequency $f=10\text{MHz}$, (i) find the skin depth $\delta$, (ii) determine the factor by which the amplitude of the electric field is reduced after travelling a distance $3\delta$, and (iii) state the phase difference between the electric and magnetic fields inside the conductor.

Step-by-step solution

1. The skin depth for a good conductor is $\delta =\sqrt{\frac{2}{\mu \sigma \omega }}$, where $\mu ={\mu }_0{\mu }_r$ and $\omega =2\pi f$.
2. Calculate $\omega =2\pi \times 10\times 10^6=6.283\times 10^7{\text{rad s}}^{-1}$.
3. Use ${\mu }_0=4\pi \times 10^{-7}{\text{H m}}^{-1}$, so $\mu =4\pi \times 10^{-7}\times 1=1.257\times 10^{-6}{\text{H m}}^{-1}$.
4. Compute the denominator $\mu \sigma \omega =1.257\times 10^{-6}\times 4.0\times 10^7\times 6.283\times 10^7=3.15\times 10^9$.
5. Now $\delta =\sqrt{\frac{2}{3.15\times 10^9}}=\sqrt{6.35\times 10^{-10}}=2.52\times 10^{-5}\text{m}=25.2\mu \text{m}$ (rounded to $2.5\times 10^{-5}\text{m}$).
6. (ii) The field amplitude decays as $E=E_0{e}^{-x/\delta }$. For $x=3\delta$, the factor is $e^{-3}=0.0498\approx 5.0\times 10^{-2}$.
7. (iii) In a good conductor the electric and magnetic fields are out of phase by $\varphi =\frac{\pi }{4}=45^{\circ }$.

Concept tested: Skin depth and attenuation in good conductors

Published 17 September 2026 · How this solution was prepared and checked