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Question #42687

Difficulty: 8/10 | 🟢 Free

The Question

A long cylindrical region of radius $R$ contains a time-varying magnetic field directed parallel to its central axis. The magnitude of the magnetic field is given by $B(r,t) = \beta r t$ , where $\beta$ is a positive constant, $r$ is the radial distance from the axis, and $t$ is time. The magnetic field outside this region is zero. Calculate the total electrical energy stored in the induced electric field per unit length of the cylinder, within the region $0 \le r \le R$ . (Assume $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{\pi \epsilon_0 \beta^2 R^6}{54}$
B.
$\frac{\pi \epsilon_0 \beta^2 R^6}{24}$
C.
$\frac{\pi \epsilon_0 \beta^2 R^6}{18}$
D.
$\frac{\pi \epsilon_0 \beta^2 R^6}{27}$

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

Energy Density Locality|Induced Electric Fields|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Faraday's Law with localized field energy density in a non-conservative setup.

The primary trap (Option B) catches students who incorrectly calculate magnetic flux by simply multiplying the local magnetic field $B(r)$ by the area $\pi r^2$ , ignoring the spatial variation of the field inside the integral.

Another conceptual trap is attempting to define a scalar potential or use $W = q\Delta V$ shortcuts, which fail because the induced electric field is non-conservative ( $\nabla \times \vec{E} \neq 0$ ).

Students must rely on the fundamental energy density volume integral $U = \int \frac{1}{2}\epsilon_0 E^2 d\tau$ .

Option C traps students who assume the electric field is uniform across the cross-section, and Option D catches those who forget the $1/2$ factor in the energy density formula.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Energy Density Locality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free