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Question #43933

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The Question

An inductor of inductance $L = 4 \text{ H}$ and a resistor of resistance $R = 1 \text{ }\Omega$ are connected in series to a battery of electromotive force $E = 3 \text{ V}$ . The battery has a non-constant internal resistance $r$ that depends on the current $I$ drawn from it as $r = r_0 + \alpha I$ , where $r_0 = 1 \text{ }\Omega$ and $\alpha = 1 \text{ }\Omega\text{A}^{-1}$ . If the circuit is closed at $t = 0$ , which of the following expressions correctly represents the current $I(t)$ in the circuit as a function of time $t$ (in seconds)?

Options

A.
$\frac{3(1 - e^{-t})}{3 + e^{-t}} \text{ A}$
B.
$\frac{3(1 - e^{-t/2})}{1 + 3e^{-t/2}} \text{ A}$
C.
$\frac{3}{2} \left(1 - e^{-t/2}\right) \text{ A}$
D.
$\frac{3(1 - e^{-t})}{1 + 3e^{-t}} \text{ A}$

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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

Transient Current in RL Circuit|Cell EMF vs Terminal Voltage|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Kirchhoff's Voltage Law in a transient circuit where the DC source is non-ideal and possesses a current-dependent internal resistance.

Students commonly fall into the trap of blindly applying the standard exponential growth formula $I(t) = I_0(1 - e^{-t/\tau})$ (Option C), failing to recognize that the dynamic resistance $r = r_0 + \alpha I$ introduces a non-linear $\alpha I^2$ term.

This transforms the standard linear first-order ODE into a Riccati-type separable differential equation, requiring partial fractions to integrate and correctly model the transient state.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Transient Current in RL Circuit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free