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A magnetic dipole is free to rotate in a...

A magnetic dipole is free to rotate in a uniform magnetic field. For what orientation of the magnet with respect to the fiel (a) torque is maximum? (b) potential energy is maximum? (c) rate of change of torque with deflection is maximum?

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(a) As `vectau=vecMxxvecB` i.e. `tau=MB sin theta`,
so torque will be maximum when `sin theta=max.=1`.
`:. theta=90^@`. Figure.
(b) As P.E. `=-MB cos theta`,
therefore, P.E. will be maximum `(=MB)`, when
`cos theta=-1`, i.e., `theta=180^@` as shown in figure.
(c) From `tau=MB sin theta, (dtau)/(d theta)=MB cos theta`. Now `(dtau)/(d theta)` will be maximum when `cos theta=max.=1`, i.e., `theta=0^@`.
i.e., `vecM` is along `vecB`, as shown in figure.
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