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Question #42989

Difficulty: 8/10 | 🟢 Free

The Question

A long elastic solenoid has length $l$ , $N$ total turns, un-stretched radius $r_0$ , and total mass $M$ . The solenoid is mounted on a platform and rotated about its longitudinal axis with a time-varying angular velocity $\omega(t) = \beta t$ , where $\beta$ is a positive constant. A constant direct current $I$ is maintained in the solenoid by an external source. The wire has an effective elastic force constant $F_0$ such that the tension $T$ in the wire produces a fractional change in its local radius given by $\frac{\Delta r}{r_0} = \frac{T}{F_0}$ . Assuming the radial expansion is small ( $\Delta r \ll r_0$ ) and ignoring any tangential deformation or self-inductance changes due to the wire's finite thickness, what is the magnitude of the induced electromotive force (EMF) across the solenoid at time $t$ ?

Options

A.
$\frac{2 \mu_0 N M r_0^3 \beta^2 I t}{l F_0}$
B.
$\frac{4 \pi \mu_0 N M r_0^3 \beta^2 I t}{l F_0}$
C.
$\frac{\mu_0 N M r_0^3 \beta^2 I t}{l F_0}$
D.
$\frac{2 \mu_0 N M r_0^3 \beta^2 I t}{\pi l F_0}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Self-Inductance of Solenoid|Frame of Reference|System of Particles and Rotational Motion

Why This Question Is Tricky

Common Mistakes

This problem tests the coupling of non-inertial mechanics with electromagnetic induction.

The primary trap lies in treating the solenoid as a rigid body, which would incorrectly yield zero induced EMF.

A secondary, more lethal trap involves miscalculating the tension in the wire by equating it directly to the total centrifugal force without accounting for the $2\pi$ geometric factor derived from integrating the radial force components over the circular loop.

Furthermore, students often forget that inductance scales with the square of the radius ( $A = \pi r^2$ ), missing a factor of 2 when applying the binomial approximation for small radial expansions.

The constant current ensures that the radial magnetic pressure is static, meaning the dynamic flux linkage is driven entirely by the time-varying centrifugal expansion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Self-Inductance of Solenoid
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free