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Question #43473

Difficulty: 8/10 | 🟢 Free

The Question

An ideal inductor of inductance $L$ carrying an initial steady current $I_0$ is connected across a time-varying resistor at time $t=0$ . The resistance varies with time as $R(t) = \alpha t$ , where $\alpha$ is a positive constant. There is no other voltage source in the circuit for $t > 0$ . Which of the following correctly identifies the dimensional formula of $\alpha$ and the exact expression for the current $i(t)$ in the circuit at any time $t > 0$ ?

Options

A.
$[\mathrm{M L^2 T^{-4} A^{-2}}] \text{ and } i(t) = I_0 e^{-\frac{\alpha t^2}{2L}}$
B.
$[\mathrm{M L^2 T^{-4} A^{-2}}] \text{ and } i(t) = I_0 e^{-\frac{\alpha t^2}{L}}$
C.
$[\mathrm{M L^2 T^{-3} A^{-2}}] \text{ and } i(t) = I_0 e^{-\frac{\alpha t^2}{2L}}$
D.
$[\mathrm{M L^2 T^{-2} A^{-2}}] \text{ and } i(t) = I_0 e^{-\frac{\alpha t}{L}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Principle of Homogeneity Exceptions|Dynamic Resistance (AC Resistance)|Transient Current in RL Circuit

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from algebraic memorization to rigorous calculus in transient circuits, anchored by dimensional analysis.

The primary trap lies in the misapplication of the standard RL circuit time constant $\tau = L/R$ .

Students often incorrectly assume that for a time-varying resistance, they can simply substitute $R(t)$ into the standard decay formula $i(t) = I_0 e^{-t/\tau}$ , yielding the erroneous result $I_0 e^{-\alpha t^2 / L}$ .

By forcing them to write and integrate the fundamental differential equation $L(di/dt) + R(t)i = 0$ , the correct factor of $1/2$ emerges from the integration of $t dt$ .

The dimensional analysis of $\alpha$ serves as a secondary check on the Principle of Homogeneity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Principle of Homogeneity Exceptions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free