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Question #43296

Difficulty: 8/10 | 🟢 Free

The Question

A conducting rod of mass $m$ and length $L$ rests on two long, frictionless, horizontal parallel rails. The rod is attached to a fixed wall by a light spring of force constant $k$ , parallel to the rails. A uniform vertical magnetic field $B$ exists throughout the region. The rails are connected at the end near the wall by a stationary circuit consisting of an uncharged capacitor of capacitance $C$ , an ideal diode, and an open switch $S$ in series. The diode is oriented to allow current to flow when the rod moves away from the wall. At $t=0$ , the rod is at its natural equilibrium position, moving away from the wall with velocity $v_0$ , and the switch $S$ is instantaneously closed. The resistance of the entire circuit is negligible. What is the maximum extension of the spring?

Options

A.
$\frac{m v_0}{m + C B^2 L^2} \sqrt{\frac{m}{k}}$
B.
$\frac{m v_0}{\sqrt{k(m + C B^2 L^2)}}$
C.
$v_0 \sqrt{\frac{m}{k}}$
D.
$v_0 \sqrt{\frac{m + C B^2 L^2}{k}}$

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

Magnetic Flux Scalar Sign|Dynamic Resistance (AC Resistance)|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the boundary between electromagnetic induction, impulsive forces, and non-linear circuit elements.

Students often fall into two traps.

First, they may ignore the impulsive magnetic force caused by the instantaneous charging of the capacitor when the switch is closed, incorrectly assuming energy is conserved from the start.

Second, even if they calculate the correct post-impulse velocity, they may fail to recognize that as the rod slows down, the induced EMF drops below the capacitor's voltage.

The diode then blocks the reverse discharge current, effectively decoupling the circuit.

Students who miss this will incorrectly use the effective mass $m_{eff} = m + CB^2L^2$ for the entire motion, leading to an erroneous energy conservation equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Magnetic Flux Scalar Sign
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free