Name: Hysteresis: Property of Ferromagnetic Substances
Uploaded: 2019-07-19
Duration: 5 min 6 s
Description: The behaviour of ferromagnets is fascinating. The magnetic field due to a ferromagnet exhibits an interesting phenomenon known as hysteresis. Let’s explore.

Question

Derive an expression for the torque experienced by a magnetic dipole in a uniform magnetic field. Hence obtain the expression for the potential energy of the dipole.

Accepted Answer

Consider a bar magnet of length $2 l$ is placed in a uniform magnetic field $\vec{B}$ . Let m be the pole strength of its each pole and the angle made by the magnetic axis of a bar magnet with the magnetic field is $θ$ .

The force experienced by $N$ -pole, $= m B$ along $\vec{B}$ .

The force experienced by $S$ -pole, $= m B$ opposite to $\vec{B}$ .

The force on two poles are equal and opposite. They form a couple and the moment of a couple (or torques) is given by

$τ = Force \times perpendicular distance$

$= m B \times 2 l \sin θ = (m \times 2 l) B \sin θ = M B \sin θ$

Where $M = m \times 2 l$ , is called magnetic dipole moment of a bar magnet.

If the dipole is rotated against the torque of the magnetic field, the work has to be done. This work is stored as the potential energy of the dipole.

The work done in rotating the dipole by small angle $d θ$ is

$d W = M B \sin θ d θ$

If the dipole was rotated from the initial position $θ_{1}$ to final position $θ_{2}$ .

$W = \int_{θ_{1}}^{θ_{2}} d W = \int_{θ_{1}}^{θ_{2}} M B \sin θ d θ = - M B {[\cos θ]}_{θ_{1}}^{θ_{2}} = - M B (\cos θ_{2} - \cos θ_{1})$

The potential energy of the dipole is zero when $\vec{M} ⊥ \vec{B}$ . So potential energy of dipole in any orientation θ can be obtained by putting $θ_{1} = 90 °$ and $θ_{2} = θ$ .

$U = - M B (\cos θ - \cos 90) = - M B \cos θ$ .

Derive an expression for the torque experienced by a magnetic dipole in a uniform magnetic field. Hence obtain the expression for the potential energy of the dipole.

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