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Question #42745

Difficulty: 8/10 | 🟢 Free

The Question

Two identical coils, each having self-inductance $L$ and resistance $R$ , are placed near each other such that their mutual inductance is $M$ ( $M > 0$ ). At $t=0$ , a constant DC voltage source of EMF $V_0$ is connected across the first coil, while the second coil is maintained as a closed short-circuited loop. Which of the following expressions gives the time $t_m$ at which the induced current in the second coil reaches its maximum magnitude?

Options

A.
$\frac{L^2-M^2}{2RM} \ln\left(\frac{L+M}{L-M}\right)$
B.
$\frac{L}{R}$
C.
$\frac{L}{R} \ln\left(\frac{L+M}{L-M}\right)$
D.
$\frac{L^2+M^2}{2RM} \ln\left(\frac{L+M}{L-M}\right)$

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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

Kirchhoff’s Loop Rule|Mutual Inductance Reciprocity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle transient coupling in mutually inductive circuits.

A common trap is to treat the primary circuit as independent, ignoring the back-EMF induced by the secondary circuit's current ( $M \frac{di_2}{dt}$ ).

If a student makes this error, they will incorrectly assume the primary current follows an uncoupled $i_1(t) = \frac{V_0}{R}(1 - e^{-Rt/L})$ curve.

Substituting this into the secondary circuit yields a first-order differential equation that incorrectly predicts the maximum current occurs at $t = L/R$ (Option B).

The correct approach requires applying Kirchhoff's Loop Rule to both circuits simultaneously and solving the coupled differential equations to account for the continuous feedback loop between the coils.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Kirchhoff’s Loop Rule
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free