Question

The wave number of an electromagnetic wave incident on a metal surface is \( (20 \pi + 750) \, \text{m<}^{-1} \) inside the metal, where \( i = \sqrt{-1} \). The skin depth of the wave in the metal is:

Hint:

The skin depth is inversely proportional to the imaginary part of the wave number, which determines the attenuation of the wave in a material.

Answer 1

Step 1: Understanding the skin depth.
The skin depth \( \delta \) is related to the wave number \( k \) in the metal by the formula: \[ \delta = \frac{1}{\text{Im}(k)} \] where \( k = k_0 + i \beta \) is the complex wave number, and \( \beta \) is the imaginary part representing the attenuation of the wave in the metal.
Step 2: Finding the imaginary part.
Given that the wave number is \( k = 20 \pi + 750 \), the imaginary part \( \beta \) is 750. The skin depth is: \[ \delta = \frac{1}{\beta} = \frac{1}{750} \, \text{m} = 1.33 \, \text{mm} \] Step 3: Conclusion.
Thus, the skin depth of the wave in the metal is 1.33 mm.