Question

The skin depth in a conductor is:

• A. Inversely proportional to conductivity
• B. Directly proportional to square root of conductivity
• C. Inversely proportional to square root of conductivity
• D. Independent of material properties

Hint:

A higher electrical conductivity value ($\sigma \uparrow$) forces the currents to confine themselves tightly to the outer perimeter, making the skin depth much smaller ($\delta \downarrow$). This explains why high-frequency AC signals require silver or copper plating on structural waveguide walls.

Answer 1

The Correct Option is C

Concept: Skin depth ($\delta$), also referred to as penetration depth, is an electromagnetic measure describing the distance an electromagnetic wave can penetrate into a highly conducting medium before its electric field amplitude decays to $\frac{1}{e}$ (approximately $37\%$) of its initial boundary surface value. The mathematical formula to solve for skin depth inside a good conductor is given by: $$\delta = \frac{1}{\alpha} = \frac{1}{\sqrt{\pi f \mu \sigma}}$$ Where:
• $f$ = frequency of the alternating electromagnetic field ($\text{Hz}$).
• $\mu$ = magnetic permeability of the conductive material ($\text{H/m}$).
• $\sigma$ = electrical conductivity of the material ($\text{S/m}$ or $\Omega^{-1}\text{m}^{-1}$). Step-by-step Proportionality Isolation:
• From the mathematical expression above, let us isolate the variable representing conductivity ($\sigma$) while keeping the frequency and permeability variables constant: $$\delta = \frac{1}{\sqrt{\pi f \mu}} \cdot \frac{1}{\sqrt{\sigma}}$$
• This demonstrates that the skin depth depends on the inverse of the radical containing conductivity: $$\delta \propto \frac{1}{\sqrt{\sigma}}$$
• Therefore, skin depth is explicitly inversely proportional to the square root of conductivity.