First slide

Magnetism

Question

A magnet of length L and Moment M is cut into two halves perpendicular to its axis. One piece A is bent into a semicircle of radius R and is joined to the other piece at the poles as shown. The moment of the resulting magnet is,

Moderate

A

$\frac{M}{2 π}$
B

$\frac{M}{π}$
C

$\frac{M (2 + π)}{2 π}$
D

$\frac{M π}{2 + π}$

Solution

$Magnetic Moment is proportional to Magnetic Length$
$\begin{array}{l} ∴ M_{B} = M / 2 \\ For A, Magnetic length = 2 R \\ where π R = \frac{L}{2} \\ \Rightarrow R = \frac{L}{2 π} \\ \Rightarrow Magnetic length = 2 R = L / π \\ ∵ Magnetic Moment is proportional to Magnetic Length \\ ∴ M_{A} = \frac{\frac{L}{π}}{L} \times M = \frac{M}{π} \end{array}$
$H e n c e, M^{1} = M_{A} + M_{B} = \frac{M}{π} + \frac{M}{2} = \frac{M (2 + π)}{2 π}$

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