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Question #42986

Difficulty: 8/10 | 🟢 Free

The Question

A uniform conducting rod of initial length $L_0$ , mass $m$ , and specific heat capacity $c$ is pivoted at one end and rotates with a constant angular velocity $\omega$ in a horizontal plane. A uniform vertical magnetic field $B$ exists throughout the region. The free end of the rod makes sliding contact with a fixed, perfectly conducting circular ring. A stationary circuit is completed by connecting a wire of negligible resistance between the pivot and the ring. Assume the rod is thermally insulated from its surroundings, so all the electrical power dissipated in the rod is entirely used to heat it. The coefficient of linear thermal expansion of the rod is $\alpha$ . To isolate the effect of length change, assume the cross-sectional area and electrical resistivity of the rod remain strictly constant as it heats up. If the initial electrical power dissipated in the rod at $t=0$ is $P_0$ , what is the initial rate of change of the power dissipated, $\left. \frac{dP}{dt} \right|_{t=0}$ ?

Options

A.
$\frac{3 \alpha P_0^2}{m c}$
B.
$\frac{4 \alpha P_0^2}{m c}$
C.
$\frac{\alpha P_0^2}{m c}$
D.
$\frac{2 \alpha P_0^2}{m c}$

Check Your Answer

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

Motional EMF in Rotating Rods|Power Dissipation Ratio|Thermal Properties of Matter

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of electromagnetic induction, circuit theory, and thermodynamics, specifically targeting the trap of treating resistance as a constant during thermal expansion.

Students often correctly identify the motional EMF dependence on length ( $\varepsilon \propto L^2$ ) but incorrectly assume the resistance $R$ remains constant, leading to $P \propto L^4$ and an erroneous factor of 4 in the derivative (Option B).

By explicitly constraining the cross-sectional area and resistivity to be constant, the problem forces the realization that $R \propto L$ , making the power dissipation proportional to $L^3$ .

The circular dependency—where power causes heating, which changes length, which in turn changes power—requires setting up a coupled rate equation evaluated precisely at $t=0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Motional EMF in Rotating Rods
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free