Time period of oscillation of bar magnet in a uniform magnetic field derivation

In this article, we are going to derive an expression for the time period of oscillation of a bar magnet in a uniform magnetic field, so without wasting time, let’s get started…

Derivation for the time period of oscillation of a bar magnet

What is a magnetic dipole?

A magnetic dipole is the arrangement of two equal and opposite magnetic poles separated by a small distance of 2l. It is described by the magnetic dipole moment (M).

What is a magnetic dipole moment?

The magnetic dipole moment of a magnetic dipole is defined as the product of the magnetic pole strength and the magnetic length. It is a vector quantity directed from S-pole to N-pole. M=m×2l where m is the magnetic pole strength and 2l is the length between their magnetic poles.

Torque, and relationship with the time period

Fig. 1, magnetic dipole in a uniform magnetic field, source: toppr.com

When a magnetic dipole (bar magnet) is placed in a uniform magnetic field, then it experiences a torque. This torque is given by the product of the force applied and the perpendicular distance between the forces. The value of Torque can be given as τ=2lsinθ×F Now, the force applied on each magnetic pole can be given as the product of the pole strength and magnetic field. F=mB

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Now substitute the value of force F to the above formula of torque. After substitution we get: τ=(mB)×2lsinθ It can be rewritten as follows: τ=(m×2l)Bsinθ We know that (m×2l) is nothing but the expression of magnetic dipole moment, and we can say simply write (M) in place of this expression. Now, new expression of torque can be given as: τ=MBsinθ In vector form, it can be given as: τ=M×B Which is cross product of magnetic moment (M) and magnetic field (B). Now, since the magnetic dipole experiences a torque, then it can also experience angular acceleration as the value of θ keeps changing. This is due to the moment of inertia of the magnetic dipole and hence it keeps oscillating. Considering these factors, the equation of torque can be given as τ=Iα where I is the moment of inertia of the dipole and α is the angular acceleration of the dipole. Now we can combine both the equations of torque and we get: Iα=MBsinθ For small angular displacement θ, we get: Iα=MBθ α=MBIθ Since the magnetic dipole undergoes oscillations which are the type of angular simple harmonic motion. We can compare the equations of simple harmonic motion with α. On comparing, we see that: α=ω2θ Putting value of α in above equation, we get: ω2θ=MBIθ or ω2=MBI Where ω is the angular frequency. And in simple harmonic motion, the time period of the oscillations is given by the formula: T=2πω Therefore, on substituting the value of ω, we get the time period of the dipole as follows: T=2πIMB From this equation, expression of the magnetic field can be given in the terms of the time period as follows: B=4π2IMT2

Note: Equations of Magnetism are mostly analogs to the equation of electrostatic, viz. Force on magnetic dipole (FB=mB) is analogous to the force on electric dipole (FE=qE) and many more.

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