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Question #42722

Difficulty: 8/10 | 🟢 Free

The Question

A long coaxial cable consists of a solid inner cylindrical conductor of radius $a$ and a thick outer coaxial conducting cylindrical shell of inner radius $b$ and outer radius $c$ . Both conductors are made of a non-magnetic metal with resistivity $\rho$ and permeability $\mu_0$ . A high-frequency alternating current $I(t) = I_0 \cos(\omega t)$ flows down the inner conductor and returns via the outer shell. The angular frequency $\omega$ is sufficiently high such that the skin depth $\delta = \sqrt{\frac{2\rho}{\omega \mu_0}}$ is much smaller than $a$ , $b$ , and $(c-b)$ . What is the time-averaged power dissipated as heat per unit length of the cable?

Options

A.
$\frac{I_0^2}{4\pi} \sqrt{\frac{\rho \omega \mu_0}{2}} \left( \frac{a+b}{ab} \right)$
B.
$\frac{I_0^2}{2\pi} \sqrt{\frac{\rho \omega \mu_0}{2}} \left( \frac{a+b}{ab} \right)$
C.
$\frac{I_0^2}{4\pi} \sqrt{\frac{\rho \omega \mu_0}{2}} \left( \frac{a+c}{ac} \right)$
D.
$\frac{I_0^2}{4\pi} \sqrt{\frac{\rho \omega \mu_0}{2}} \left( \frac{1}{a} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Ohm's Law Validity|Frequency Sensitivity of Z|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the transition from lumped circuit theory to distributed electromagnetic field theory via the high-frequency skin effect.

The primary trap lies in identifying the correct surfaces where the current flows.

Students often incorrectly assume the return current in the outer shell distributes uniformly or flows on the outer surface (radius $c$ ).

However, to minimize the system's inductance (magnetic field energy), the high-frequency return current strongly crowds to the inner surface of the outer shell (radius $b$ ).

Additionally, students may forget the factor of $1/2$ when calculating time-averaged power from peak current $I_0$ , or fail to correctly approximate the effective cross-sectional area $2\pi r \delta$ for a thin cylindrical shell.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Ohm's Law Validity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free