A circular coil of 16 turns and radius 10 cm carrying a current of 0.75 A rests with its plane

step 1 In discussion, we have number of tons of coils n is equal to 16. Radius of circular coil, small

A circular coil of 16 turns and radius 10 cm carrying a...

A circular coil of 16 turns and radius $10 \mathrm{~cm}$ carrying a current of $0.75$ A rests with its plane normal to an external field of magnitude $5.0 \times 10^{-2} \mathrm{~T} .$ The coil is free to turn about an axis in its plane perpendicular to the field direction. When the coil is turned slightly and released, it oscillates about its stable equilibrium with a frequency of $2.0 / \mathrm{s}$. What is the moment of inertia of the coil about its axis of rotation? (a) $1.2 \times 10^{-4} \mathrm{~g}-\mathrm{cm}^{2}$ (b) $3 \times 10^{-4} \mathrm{~kg}-\mathrm{m}^{2}$ (c) $0.3 \times 10^{-4} \mathrm{~kg}-\mathrm{m}^{2}$ (d) $1.2 \times 10^{-4} \mathrm{~kg}-\mathrm{m}^{2}$

A circular coil of 16 turns and radius $10 \mathrm{~cm}$ carrying a current of $0.75$ A rests with its plane normal to an external field of magnitude $5.0 \times 10^{-2} \mathrm{~T} .$ The coil is free to turn about an axis in its plane perpendicular to the field direction. When the coil is turned slightly and released, it oscillates about its stable equilibrium with a frequency of $2.0 / \mathrm{s}$. What is the moment of inertia of the coil about its axis of rotation?
(a) $1.2 \times 10^{-4} \mathrm{~g}-\mathrm{cm}^{2}$
(b) $3 \times 10^{-4} \mathrm{~kg}-\mathrm{m}^{2}$
(c) $0.3 \times 10^{-4} \mathrm{~kg}-\mathrm{m}^{2}$
(d) $1.2 \times 10^{-4} \mathrm{~kg}-\mathrm{m}^{2}$

Key Concepts

Simple Harmonic Motion in a Magnetic Field

In the context of small angular displacements, the restoring torque acting on a magnetic dipole in an external field results in simple harmonic motion. The angular frequency of these oscillations depends on the ratio of the magnetic restoring torque (derived from the dipole moment and field strength) to the moment of inertia of the coil.

Moment of Inertia

The moment of inertia quantifies how an object’s mass is distributed relative to a chosen axis of rotation, determining its resistance to angular acceleration. In oscillatory systems, it plays a crucial role in setting the angular frequency of the motion, making it a key parameter in analyzing dynamics of rotational systems.

Magnetic Dipole Moment

The magnetic dipole moment is a fundamental property of current loops that characterizes their ability to interact with external magnetic fields. It is calculated as the product of the current, the number of turns in the loop, and the area of the loop. This quantity determines how the coil responds to the magnetic field, including the strength of the torque exerted on it.

Magnetic Torque on a Dipole

When placed in an external magnetic field, a magnetic dipole experiences a torque that tends to align it with the field direction. For small deviations from equilibrium, this torque is proportional to the sine of the angle between the dipole moment and the field, which, under the small angle approximation, behaves like a restoring force leading to oscillatory motion.

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