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Question #43377

Difficulty: 8/10 | 🟢 Free

The Question

A rigid circular loop of area $A$ (in $\text{m}^2$ ) rotates with a constant angular velocity $\omega$ (in $\text{rad/s}$ ) about its diameter in a uniform magnetic field of magnitude $B$ (in $\text{T}$ ). The axis of rotation is perpendicular to the magnetic field. The loop is connected in series via slip rings to an ideal chemical cell of electromotive force $E$ (in $\text{V}$ ), a resistor of resistance $R$ (in $\Omega$ , representing the total resistance of the circuit), and a parallel-plate capacitor of capacitance $C$ (in $\text{F}$ ) with plate area $S$ (in $\text{m}^2$ ). After the circuit reaches a steady state, what is the maximum displacement current density $J_d$ (in $\text{A/m}^2$ ) between the plates of the capacitor?

Options

A.
$\frac{B A \omega^2 C}{S \sqrt{1 + (\omega R C)^2}} \, \text{A/m}^2$
B.
$\frac{(E + B A \omega) \omega C}{S \sqrt{1 + (\omega R C)^2}} \, \text{A/m}^2$
C.
$\frac{B A \omega}{S R} \, \text{A/m}^2$
D.
$\frac{E \omega C}{S \sqrt{1 + (\omega R C)^2}} \, \text{A/m}^2$

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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

Equivalent EMF for Cells|Magnetic Moment of Orbiting Electron|Motional EMF for Arbitrary Shapes

Why This Question Is Tricky

Common Mistakes

This question tests the superposition of static and time-varying EMFs and the application of the Maxwell-Ampere law in a steady-state AC/DC hybrid circuit.

The primary trap lies in the concept of Equivalent EMF.

Students often incorrectly assume that the maximum current is driven by the maximum total EMF ( $E + BA\omega$ ), failing to recognize that the capacitor acts as an infinite impedance block to the DC chemical cell in the steady state.

Consequently, they erroneously include the chemical EMF $E$ in their calculation of the displacement current.

By requiring the displacement current density, the problem forces students to correctly apply the Maxwell-Ampere law ( $I_d = I_c$ ) and realize that only the time-derivative of the motional EMF contributes to the steady-state oscillating current.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Equivalent EMF for Cells
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free