# For a good conductor, the depth of penetration of electromagnetic wave is given by:

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For a good conductor, the depth of penetration of electromagnetic wave is given by:
1. $\delta = {\left[ {\frac{2}{{\omega \sigma \mu }}} \right]^{\frac{1}{2}}}$
2. $\delta = {\left[ {\frac{1}{{\omega \sigma \mu }}} \right]^{\frac{1}{2}}}$
3. $\delta = {\left[ {\frac{2}{{\omega \sigma^2 \mu }}} \right]^{\frac{1}{2}}}$
4. $\delta = {\left[ {\frac{2}{{\omega \sigma }}} \right]^{\frac{1}{2}}}$

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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.