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Question #43441

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The Question

A long ideal solenoid of radius $R$ is aligned along the $z$ -axis. For $t \ge 0$ , the current in the solenoid is varied such that the uniform magnetic field inside is given by $\mathbf{B} = B_0 e^{-t/\tau} \hat{k}$ , where $B_0$ and $\tau$ are positive constants. The magnetic field outside the solenoid is negligible. Let $\mathbf{S}(r,t)$ be the Poynting vector at a radial distance $r$ ( $r < R$ ) from the axis. Let $U_{\text{cross}}$ be the total electromagnetic energy that crosses a coaxial cylindrical surface of radius $r$ and length $L$ during the time interval $t=0$ to $t=\infty$ . Which of the following correctly describes $\mathbf{S}(r,t)$ and $U_{\text{cross}}$ ? (Assume $\hat{r}$ is the outward radial unit vector in cylindrical coordinates).

Options

A.
$\mathbf{S}(r,t) = \frac{r B_0^2}{2\mu_0 \tau} e^{-2t/\tau} \hat{r}$ , $U_{\text{cross}} = \frac{\pi r^2 L B_0^2}{2\mu_0}$ (outward)
B.
$\mathbf{S}(r,t) = -\frac{r B_0^2}{2\mu_0 \tau} e^{-2t/\tau} \hat{r}$ , $U_{\text{cross}} = \frac{\pi r^2 L B_0^2}{2\mu_0}$ (inward)
C.
$\mathbf{S}(r,t) = \frac{r B_0^2}{2\mu_0 \tau} e^{-t/\tau} \hat{r}$ , $U_{\text{cross}} = \frac{\pi r^2 L B_0^2}{\mu_0}$ (outward)
D.
$\mathbf{S}(r,t) = -\frac{r B_0^2}{\mu_0 \tau} e^{-2t/\tau} \hat{r}$ , $U_{\text{cross}} = \frac{\pi r^2 L B_0^2}{\mu_0}$ (inward)

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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

Gauss’s Law for Magnetism|Energy Density Locality|Poynting Vector and Intensity

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of energy transport in time-varying electromagnetic fields via the Poynting vector.

A common misconception is that when a magnetic field collapses, its energy simply 'vanishes' locally or flows inward toward the central axis.

By applying Faraday's Law, students must determine that the induced electric field is in the $+\hat{\theta}$ direction (opposing the decrease of $\mathbf{B}$ ).

The cross product $\mathbf{E} \times \mathbf{B}$ then yields a Poynting vector directed radially outward ( $+\hat{r}$ ).

This demonstrates that the stored magnetic energy flows outward to the solenoid windings, where it does work on the current.

Furthermore, students often fall into the mathematical trap of forgetting to square the time-dependent exponential factor when calculating intensity, leading to an incorrect integration for the total energy crossed.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Gauss’s Law for Magnetism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free