Question

A wave travelling in conducting medium, if its amplitude decreases by a factor of _____ it is called penetration depth of the medium.

• A. 25 %
• B. 37 %
• C. 50 %
• D. 75 %

Hint:

Penetration depth (skin depth) \(\delta\) is the distance into a conductor at which the amplitude of an EM wave decays to \(1/e\) (approx. 36.8% or 37%) of its surface value.
\(\delta = 1/\alpha\), where \(\alpha\) is the attenuation constant.
The wave amplitude is \(A_0 e^{-z/\delta}\). At \(z=\delta\), amplitude is \(A_0 e^{-1}\).

Answer 1

The Correct Option is B

Penetration Depth (Skin Depth) Explanation

The penetration depth (or skin depth), denoted by \( \delta \), in a conducting medium is defined as the depth at which the amplitude of an electromagnetic wave decreases to \( \frac{1}{e} \) (approximately 36.78% or 37%) of its initial amplitude at the surface.

The amplitude of the wave attenuates as:

\[ A(z) = A_0 e^{-\alpha z} \]

where \( \alpha \) is the attenuation constant. The penetration depth is:

\[ \delta = \frac{1}{\alpha} \]

At \( z = \delta \), the amplitude becomes:

\[ A(\delta) = A_0 e^{-\alpha \delta} = A_0 e^{-1} \]

\[ e^{-1} \approx 0.36788 \approx 36.79\% \]

So, the amplitude decreases to approximately 37% of its initial value. This means the amplitude has decreased by \( 100\% - 37\% = 63\% \).

Key Interpretation:
If the question asks "amplitude decreases by a factor of ____", it can be ambiguous:

• If it means “amplitude becomes X% of initial”, then X = 37%.
• If it means “amplitude reduces by Y%”, then Y = 63%.
• If it means “amplitude is divided by a factor”, then that factor is \( e \approx 2.718 \).

The most standard interpretation in the context of skin depth is that the amplitude drops to 37% of its initial value.

Thus, the correct answer is:

\[ \boxed{37\% \text{ (meaning amplitude decreases TO 37% of its initial value)}} \]