A plane monochromatic electromagnetic wave of angular frequency $\omega =3\pi \times 10^9{\text{rad s}}^{-1}$ propagates in a non‑magnetic conductor with conductivity $\sigma =1.0\times 10^6{\text{S m}}^{-1}$ and relative permeability ${\mu }_r=1$. Determine (i) the skin depth $\delta$, (ii) the factor by which the field amplitude is reduced after travelling a distance of $2\delta$, and (iii) the phase difference between the electric and magnetic fields inside the conductor.

Step-by-step solution

1. For a good conductor, the skin depth is $\delta =\sqrt{\frac{2}{\mu \sigma \omega }}$, where $\mu ={\mu }_0{\mu }_r$ and ${\mu }_0=4\pi \times 10^{-7}{\text{H m}}^{-1}$.
2. Calculate $\mu =4\pi \times 10^{-7}{\text{H m}}^{-1}$ (since ${\mu }_r=1$).
3. Compute the product $\mu \sigma \omega =(4\pi \times 10^{-7})(1.0\times 10^6)(3\pi \times 10^9)=12{\pi }^2\times 10^8$.
4. Evaluate $12{\pi }^2\approx 12\times 9.87=118.4$, so $\mu \sigma \omega \approx 1.184\times 10^{10}{\text{s}}^{-1}$.
5. Find $\delta =\sqrt{\frac{2}{1.184\times 10^{10}}}=\sqrt{1.688\times 10^{-10}}\approx 1.30\times 10^{-5}\text{m}$, i.e., $13\mu \text{m}$.
6. In a conductor the field varies as $e^{-x/\delta }$; therefore after travelling $2\delta$ the amplitude factor is $e^{-2\delta /\delta }=e^{-2}=0.135$.
7. For a good conductor the electric and magnetic fields are out of phase by $45^{\circ }$ (or $\pi /4$ rad) because the complex propagation constant has equal real and imaginary parts.
8. Thus the required quantities are $\delta =1.3\times 10^{-5}\text{m}$, attenuation factor $0.135$, and phase difference $45^{\circ }$.

Concept tested: Skin depth and field behaviour in conductors

Published 9 September 2026 · How this solution was prepared and checked