Before You Begin

Question #43038

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical shunt resistor of radius $a$ (in $\text{m}$ ), length $l$ (in $\text{m}$ ), electrical conductivity $\sigma$ (in $\text{S/m}$ ), and electrical permittivity $\epsilon$ (in $\text{F/m}$ ) is connected across an AC voltage source $V(t) = V_0 \sin(\omega t)$ , where $V_0$ is the voltage amplitude (in $\text{V}$ ) and $\omega$ is the angular frequency (in $\text{rad/s}$ ). Assuming the electric field is uniform throughout the volume of the resistor and edge effects are negligible, what is the maximum instantaneous magnitude of the Poynting vector, $S_{max}$ (in $\text{W/m}^2$ ), evaluated at the curved surface of the cylinder?

Options

A.
$\frac{a V_0^2}{4 l^2} \left( \sigma + \sqrt{\sigma^2 + (\epsilon \omega)^2} \right)$
B.
$\frac{a V_0^2}{2 l^2} \sqrt{\sigma^2 + (\epsilon \omega)^2}$
C.
$\frac{a \sigma V_0^2}{2 l^2}$
D.
$\frac{a V_0^2}{4 l^2} \left( \sigma + \epsilon \omega \right)$

Check Your Answer

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

Shunt Resistance Conversion|Ampere’s Law Displacement Current|Poynting Vector and Intensity

Why This Question Is Tricky

Common Mistakes

This question tests the transition from lumped-element circuit theory to Maxwell's field theory in high-frequency regimes.

Students commonly fall into the 'Ohmic Fallacy' by ignoring the displacement current ( $\epsilon \partial E/\partial t$ ) inside the resistor, leading to Option C.

Another severe trap is assuming that the maximum of the Poynting vector is simply the product of the maximum electric and magnetic fields ( $S_{max} = E_{max}B_{max}/\mu_0$ ), which leads to Option B.

This ignores the crucial phase difference between the conduction current (in phase with $E$ ) and the displacement current (in quadrature with $E$ ).

The correct approach requires evaluating the instantaneous cross product $\vec{E} \times \vec{B}$ and maximizing the resulting time-dependent function.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Shunt Resistance Conversion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free