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Question #43413

Difficulty: 8/10 | 🟢 Free

The Question

A moving coil galvanometer consists of a coil with $N$ turns, each of area $A$ , and total resistance $R_c$ . The coil is wound on a continuous metallic rectangular frame of effective resistance $R_f$ . The moment of inertia of the entire rotating assembly (coil and frame) about its axis of rotation is $I$ . The assembly is placed in a perfectly radial magnetic field whose magnitude varies with time as $B(t) = B_0 e^{-\lambda t}$ , where $B_0$ and $\lambda$ are positive constants. A shunt resistance $R_s$ is connected across the terminals of the coil. The torsional constant of the suspension is negligible. At $t = 0$ , the assembly is given an initial angular velocity $\omega_0$ . Assuming air drag is negligible and all quantities are in standard SI units, which of the following expressions correctly represents the angular velocity $\omega(t)$ of the assembly as a function of time $t$ ?

Options

A.
$\omega_0 \exp\left[ - \frac{A^2 B_0^2}{2\lambda I} \left( \frac{N^2}{R_c + R_s} + \frac{1}{R_f} \right) (1 - e^{-2\lambda t}) \right]$
B.
$\omega_0 \exp\left[ - \frac{N^2 A^2 B_0^2}{2\lambda I (R_c + R_s)} (1 - e^{-2\lambda t}) \right]$
C.
$\omega_0 \exp\left[ - \frac{A^2 B_0^2}{\lambda I} \left( \frac{N^2}{R_c + R_s} + \frac{1}{R_f} \right) (1 - e^{-\lambda t}) \right]$
D.
$\omega_0 \exp\left[ - \frac{A^2 B_0^2}{I} \left( \frac{N^2}{R_c + R_s} + \frac{1}{R_f} \right) t \right]$

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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

Shunt Resistance Conversion|Eddy Currents Dissipation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests multiple advanced concepts in electromagnetic induction and rotational dynamics.

The primary trap is neglecting the eddy current dissipation in the metallic frame, which acts as a parallel energy sink to the shunt resistor.

Students often only calculate the damping from the coil.

A secondary, more subtle trap involves the time-varying magnetic field in a radial geometry: students might incorrectly attempt to add a $dB/dt$ transformer EMF term, failing to realize that the net magnetic flux through the coil in a perfectly radial field is perpetually zero.

Finally, the integration of the damping torque requires squaring the magnetic field term, leading to a $e^{-2\lambda t}$ decay factor rather than $e^{-\lambda t}$ , catching mathematical oversights.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Shunt Resistance Conversion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free