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Question #43162

Difficulty: 8/10 | 🟢 Free

The Question

A time-varying electric field $\vec{E}(\rho, t)$ is confined to a cylindrical region of radius $R$ and is directed parallel to the axis of the cylinder. The magnitude of the electric field at a radial distance $\rho$ ( $\rho \le R$ ) from the axis is given by: $E(\rho, t) = E_0 \left( 1 - \frac{\rho}{R} \right) \sin(\omega t) + E_0 \left( \frac{\rho}{R} \right) \sin(2\omega t)$ where $E_0$ and $\omega$ are positive constants. What is the root-mean-square (RMS) value of the induced magnetic field, $B_{rms}$ , at a distance $r$ ( $r > R$ ) from the axis of the cylinder?

Options

A.
$\frac{\sqrt{34} \mu_0 \epsilon_0 R^2 E_0 \omega}{12r}$
B.
$\frac{\sqrt{10} \mu_0 \epsilon_0 R^2 E_0 \omega}{12r}$
C.
$\frac{5\sqrt{2} \mu_0 \epsilon_0 R^2 E_0 \omega}{12r}$
D.
$\frac{\sqrt{10} \mu_0 \epsilon_0 R^2 E_0 \omega}{8r}$

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

Displacement Current Source|Instantaneous vs RMS vs Peak|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Ampere-Maxwell law in the presence of a non-sinusoidal, spatially varying electric field.

A common trap for JEE Advanced students is to blindly apply the standard $1/\sqrt{2}$ peak-to-RMS conversion factor, which is only valid for pure sinusoidal signals.

Furthermore, students often fail to recognize that the displacement current density $J_d = \epsilon_0 \frac{\partial E}{\partial t}$ introduces a frequency-dependent amplitude multiplier (the chain rule brings out a factor of 2 for the harmonic).

Because the spatial distribution of the fundamental and harmonic components differ, the spatial integration alters their relative weights in the total displacement current.

The correct approach requires finding the exact time-domain expression for the total enclosed displacement current, and then applying Parseval's theorem (or the integral definition of RMS) to account for the orthogonal harmonic components.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Displacement Current Source
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free