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Question #42999

Difficulty: 8/10 | 🟢 Free

The Question

A conducting rod of mass $m$ and length $l$ is given an initial velocity $v_0$ at $x=0$ on two long, frictionless, horizontal parallel conducting rails. The rails are connected at $x=0$ by a resistor of resistance $R$ . A spatially varying magnetic field $\vec{B} = \alpha x \hat{k}$ exists in the region, where $\alpha$ is a positive constant and $\hat{k}$ is perpendicular to the plane of the rails. The self-inductance of the circuit is negligible. What fraction of the rod's initial kinetic energy is dissipated as heat in the resistor by the time the rod has traveled half of its maximum stopping distance?

Options

A.
$\frac{15}{64}$
B.
$\frac{7}{16}$
C.
$\frac{3}{4}$
D.
$\frac{1}{8}$

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

AC Generator Back-EMF|Variable Force Integration|Inductive Reactance Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the integration of power and the Work-Energy Theorem in a non-linear electro-mechanical system.

A common trap is assuming the magnetic force scales linearly with position or velocity independently, leading students to incorrectly apply constant-acceleration kinematics or assume velocity decreases linearly with distance ( $v = v_0(1 - x/x_f)$ ).

By introducing a spatial magnetic gradient $B(x) = \alpha x$ , the Lorentz force becomes proportional to $x^2 v$ .

Students must correctly set up and integrate the differential equation $m dv = -(\alpha^2 l^2 x^2 / R) dx$ to find the true non-linear velocity-position coupling $v(x) \propto x^3$ , before applying energy conservation to find the dissipated heat.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	AC Generator Back-EMF
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free