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Question #43102

Difficulty: 8/10 | 🟢 Free

The Question

A long straight solid cylindrical conductor of radius $a$ carries a slowly varying current $I(t)$ . The material of the conductor exhibits non-Ohmic behavior such that the local electric field $E$ required to drive a current density $J$ is given by $E = \beta J^2$ , where $\beta$ is a positive constant. Assume the rate of change of current is small enough that the current density $J$ can be considered uniform across the cross-section at any instant. By considering the macroscopic energy balance (Poynting's theorem), which of the following expressions gives the magnitude of the Poynting vector at the surface of the conductor?

Options

A.
$\frac{\beta I^3}{2\pi^3 a^5} + \frac{\mu_0 I}{16\pi^2 a} \frac{dI}{dt}$
B.
$\frac{\beta I^3}{2\pi^3 a^5} + \frac{\mu_0 I}{8\pi^2 a} \frac{dI}{dt}$
C.
$\frac{\beta I^3}{\pi^3 a^5} + \frac{\mu_0 I}{16\pi^2 a} \frac{dI}{dt}$
D.
$\frac{\beta I^3}{2\pi^3 a^5}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Poynting Vector and Intensity|Current Density Vector|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the deep application of Poynting's theorem in a non-idealized macroscopic conductor.

Students commonly fall into the trap of assuming that the Poynting vector only accounts for Ohmic dissipation (analogous to $P = I^2 R$ ), leading them to select Option D.

By introducing a time-varying current, an induced longitudinal electric field is created, meaning energy must also flow radially inward to build up the internal magnetic field.

Furthermore, the non-Ohmic resistance ( $E = \beta J^2$ ) forces students to derive the dissipation from first principles rather than relying on standard formulas.

The correct approach requires reconciling the macroscopic energy balance ( $P_{\text{in}} = P_{\text{diss}} + dU_B/dt$ ) with the surface flux of the Poynting vector.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Poynting Vector and Intensity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free