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Question #42059

Difficulty: 8/10 | 🟢 Free

The Question

A conducting rod of mass $m$ and length $l$ is placed on two long, parallel, frictionless horizontal conducting rails. The rails are connected at one end by an ideal inductor of inductance $L$ . A uniform vertical magnetic field $B$ exists throughout the region. The resistance of the rod and the rails is negligible. At $t=0$ , the rod is at rest. An external horizontal force $F(t)$ is applied to the rod such that the acceleration it would produce in the absence of the magnetic field is a constant $a_0$ . What is the maximum energy stored in the inductor during the subsequent motion?

Options

A.
$\frac{2 L m^2 a_0^2}{B^2 l^2}$
B.
$\frac{L m^2 a_0^2}{2 B^2 l^2}$
C.
$\frac{L m^2 a_0^2}{B^2 l^2}$
D.
$\frac{4 L m^2 a_0^2}{B^2 l^2}$

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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

Area Under Acceleration-Time|Faraday’s Law Vector Form|Energy Stored in Inductor

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple Faraday's Law with mechanical equations of motion, leading to a non-homogeneous differential equation.

The lethal trap lies in incorrectly mapping the area under the given acceleration-time graph directly to the velocity profile ( $v = a_0 t$ ), ignoring the back-EMF's effect on the net acceleration.

Furthermore, even if students set up the forces correctly, they often assume the system reaches a steady-state 'terminal' condition where $F_{ext} = F_B$ , yielding $I = m a_0 / B l$ .

Because the circuit contains an inductor rather than a resistor, the system forms an LC-like mechanical oscillator.

The current overshoots the equilibrium value by a factor of 2, making the true maximum energy four times larger than the steady-state assumption.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Area Under Acceleration-Time
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free