Correct Answer - Option 1 :

\(\delta = {\left[ {\frac{2}{{\omega \sigma \mu }}} \right]^{\frac{1}{2}}}\)
__Derivation__:

The depth of penetration δ of a plane electromagnetic wave incident normally on a good conductor is mathematically defined as:

\(\delta=\frac{1}{α}\)

α is the attenuation constant given by:

\(α=\omega\sqrt{\frac{\mu\epsilon}{2}\left[\sqrt{1+\left(\frac{σ}{\omega\epsilon}\right)^2}-1\right]}\)

For a conducting medium σ >>1. The above expression for the attenuation constant can, therefore, be approximated as:

\(\alpha=\left[\sqrt{1+\left(\frac{σ}{\omega\epsilon}\right)^2}-1\right]≈\frac{σ}{\omega\epsilon}\)

∴ The attenuation constant becomes:

\(α=\omega \sqrt{\left(\frac{\muσ}{2\omega}\right)}\)

\(α=\sqrt{\left(\frac{\omega\muσ} {2}\right)}\) or \(\therefore~\alpha =\sqrt{\pi f\muσ}\)

We know that,

\(\delta=\frac{1}{α}\)

Thus, the skin depth becomes:

\(\delta= [\frac{2}{\omegaσ\mu }]^\frac{1}{2}\) or \(\delta=\frac{1}{\sqrt{\pi f \muσ}}\)

- The skin depth is inversely proportional to the square root of frequency.
- It is inversely proportional to the square root of the conductivity of the medium.
- It is inversely proportional to the square root of the permeability of the medium.