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Question #43698

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor of vacuum capacitance $C_0$ is filled with a radioactive gas. The gas acts as a dielectric whose relative permittivity varies with time as $K(t) = 1 + \beta A(t)$ , where $A(t)$ is the instantaneous radioactive activity of the gas and $\beta$ is a positive constant. The initial activity of the gas at $t=0$ is $A_0$ and its decay constant is $\lambda$ . The capacitor is connected across an ideal DC voltage source $V_0$ . The charging current flows to the capacitor through a very long straight wire. A square conducting loop of side length $a$ is placed in the same plane as the long straight wire, with its nearest edge parallel to the wire at a distance $a$ . What is the magnitude of the induced EMF $\varepsilon$ in the square loop as a function of time $t$ ?

Options

A.
$\frac{\mu_0 a \ln 2}{2\pi} C_0 V_0 \beta A_0 \lambda^2 e^{-\lambda t}$
B.
$\frac{\mu_0 a \ln 2}{2\pi} C_0 V_0 \beta A_0 \lambda e^{-\lambda t}$
C.
$\frac{\mu_0 a}{2\pi} C_0 V_0 \beta A_0 \lambda^2 e^{-\lambda t}$
D.
$\frac{\mu_0 a}{3\pi} C_0 V_0 \beta A_0 \lambda^2 e^{-\lambda t}$

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Correct Answer

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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

Activity Units and Transitions|Energy Level Scaling|Induced Electric Fields|Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the radioactive decay law, electrostatics of dielectrics, and electromagnetic induction.

The primary trap lies in the misapplication of the capacitor current formula; students often memorize $I = C(dV/dt)$ and incorrectly assume $I=0$ for a DC source, ignoring the $V(dC/dt)$ term caused by the decaying dielectric.

A secondary trap is failing to differentiate the current a second time for Faraday's Law (missing a factor of $\lambda$ ), or incorrectly approximating the magnetic flux by multiplying the field at the loop's center by its area instead of performing the required logarithmic integration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Activity Units and Transitions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free