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

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

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

Derivation for time period of oscillation of a bar magnet

What is magnetic dipole?

Magnetic dipole is the arrangement of two equal and opposite magnetic poles seperated by a small distance 2l. It is described by the magnetic dipole moment (\vec{M}.

What is magnetic dipole moment?

The magnetic dipole moment of a magnetic dipole is the 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.

    \[\vec{M}=\vec{m}\times 2\vec{l}\]

where m is the magnetic pole strength and 2l is the length between their magnetic poles.

Torque, and relationship with time period

Time period of oscillation of bar magnet in a uniform magnetic field derivation
Fig. 1, magnetic dipole in a uniform magnetic field, source: toppr.com

When a magnetic dipole (bar magnet) is placed in 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:

    \[\tau = 2l\sin\theta \times \vec{F}\]

Now, force applied on each magnetic pole can be given as the product of the pole strength and magnetic field.

    \[\vec{F}= mB\]

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Now substitute the value of force \vec{F} to the above formula of torque. After substitution we get:

    \[\tau=(mB)\times 2l\sin\theta\]

It can be rewritten as follows:

    \[\tau=(m\times 2l)B\sin\theta\]

We know that (m\times 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:

    \[\tau =MB\sin\theta\]

In vector form, it can be given as:

    \[\tau =\vec{M}\times \vec{B}\]

Which is cross product of magnetic moment (\vec{M}) and magnetic field (\vec{B}). Now, since the magnetic dipole experiences a torque, then it can also experiences angular acceleration as the value of θ keeps changing. This is due to the moment of inertia of magnetic dipole and hence it keeps oscillating. Considering these factors, the equation of torque can be given as:

    \[\tau =I\alpha\]

where, I is the moment of inertia of the dipole and \alpha is the angular acceleration of the dipole. Now we can combine both the equations of torque and we get:

    \[I\alpha =MB\sin\theta\]

For small angular displacement \theta, we get:

    \[I\alpha = MB\theta\]

    \[\implies \quad \alpha =\frac{MB}{I}\theta\]

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 \alpha. On comparing, we see that:

    \[\alpha =\omega^2\theta\]

Putting value of \alpha in above equation, we get:

    \[\omega^2\theta =\frac{MB}{I}\theta\]

or

    \[\omega^2 =\frac{MB}{I}\]

Where \omega is the angular frequency. And in simple harmonic motion, the time period of the oscillations is given by the formula:

    \[T=\frac{2\pi}{\omega}\]

Therefore, on substituting the value of \omega, we get the time period of the dipole as follows:

    \[T=2\pi\sqrt{\frac{I}{MB}}\]

From this equation, expression of magnetic field can be given in the terms of time period as follows:

    \[B=\frac{4\pi^{2}I}{MT^2}\]

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

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