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Question #42474

Difficulty: 8/10 | 🟢 Free

The Question

A conducting rod of mass $m$ and length $l$ is placed on two long, parallel, smooth horizontal rails separated by a distance $l$ . The rails are connected at one end by a resistor of resistance $R$ and an inductor of inductance $L$ in series. A uniform vertical magnetic field of magnetic induction $B$ exists in the region. At $t=0$ , a constant horizontal force $F$ is applied to the rod, pulling it along the rails. The rod starts from rest. Let $W(t)$ be the total work done by the external force $F$ and $H(t)$ be the total heat dissipated in the resistor from time $0$ to $t$ . Find the exact value of $\lim_{t \to \infty} [W(t) - H(t)]$ .

Options

A.
$\frac{F^2}{2 B^2 l^2} \left( \frac{m R^2}{B^2 l^2} + L \right)$
B.
$\frac{F^2}{2 B^2 l^2} \left( \frac{m R^2}{B^2 l^2} - L \right)$
C.
$\frac{m F^2 R^2}{2 B^4 l^4}$
D.
$\frac{L F^2}{2 B^2 l^2}$

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

Variable Force Integration|Transient Current in RL Circuit|Faraday’s Law Vector Form

Why This Question Is Tricky

Common Mistakes

This question tests the mastery of the Work-Energy theorem in coupled electro-mechanical systems.

The primary trap is attempting to solve the second-order differential equations for $v(t)$ and $i(t)$ to directly integrate $Fv$ and $i^2R$ , which is mathematically tedious and prone to errors.

Students often fail to recognize that the motional EMF term ( $iBvl$ ) acts as the power transfer mechanism between the mechanical and electrical domains.

By applying energy conservation, the difference between the work done by the external force and the Joule heating is simply the sum of the final kinetic energy of the rod and the magnetic energy stored in the inductor at steady state.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Variable Force Integration
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free