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Question #42991

Difficulty: 8/10 | 🟢 Free

The Question

An ideal battery of electromotive force $V_0$ is connected in series with a resistor of resistance $R$ , a switch, and a variable inductor. At $t = 0$ , the switch is closed. A mechanical system alters the inductor such that its inductance increases linearly with time as $L(t) = L_0 + \alpha t$ , where $L_0$ and $\alpha$ are positive constants. Which of the following expressions correctly represents the current $i(t)$ in the circuit for $t > 0$ ?

Options

A.
$\frac{V_0}{R+\alpha} \left[ 1 - \left( \frac{L_0}{L_0 + \alpha t} \right)^{\frac{R+\alpha}{\alpha}} \right]$
B.
$\frac{V_0}{R} \left[ 1 - \left( \frac{L_0}{L_0 + \alpha t} \right)^{\frac{R}{\alpha}} \right]$
C.
$\frac{V_0}{R} \left[ 1 - \exp\left(-\frac{R t}{L_0 + \alpha t}\right) \right]$
D.
$\frac{V_0}{R+\alpha} \left[ 1 - \exp\left(-\frac{(R+\alpha) t}{L_0}\right) \right]$

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Correct Answer

Step-by-Step Solution

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

Energy Stored in Inductor|Kirchhoff’s Loop Rule|Average vs. Instantaneous Calculus

Why This Question Is Tricky

Common Mistakes

Students often memorize the inductive EMF as $\varepsilon = -L \frac{di}{dt}$ and the standard RL circuit current growth as an exponential function.

This question tests the fundamental definition of Faraday's law where flux $\Phi = L(t)i(t)$ .

When inductance is time-dependent, the product rule generates an additional motional EMF term, $-i \frac{dL}{dt}$ , which acts as an extra effective resistance ( $\alpha$ ) in the circuit.

Ignoring this term leads to Trap B, while blindly substituting $L(t)$ into the standard exponential decay formula leads to Trap C.

This highlights the breakdown of algebraic formulas when electromechanical coupling is introduced.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Energy Stored in Inductor
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free