Question

Review. (a) Show that a magnetic dipole in a uniform magnetic field, displaced from its equilibrium orientation and released, can oscillate as a torsional pendulum (Section 15.5) in simple harmonic motion.

Answer 1

The answer is τ is proportional to θ

What is SHM?

A motion in which the restoring force is directly proportional to the body's displacement from its mean position is known as a simple harmonic motion, or SHM. This restoring force always moves in the direction of the mean position. The French mathematician Joseph Fourier made the discovery that any regularly recurring motion or wave, regardless of how complex its shape, may be thought of as the sum of a collection of straightforward harmonic motions or waves in 1822.

Consider a magnetic dipole that has been liberated from its equilibrium orientation in a homogeneous magnetic field.  Let the angle between the magnetic moment and the magnetic field be θ. The torque on the dipole due

to the external field is, τ=μXB

and magnitude of,     τ=μB sinθ

if the angle is small, then,    τ=μBθ

or,                                          τproportionalθ

which is the equation for a restoring torque, so the dipole tends to turn the dipole toward its equilibrium

orientation if displaced from its original position. So a magnetic dipole in a uniform magnetic field can oscillate as

a torsional pendulum in SHM if displaced from its equilibrium position.

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