First slide

Magnetism

Question

When a thin bar magnet of Magnetic moment is bent into an arc of a circle subtending at an angle ‘ $θ$ ’ radians at the centre of the circle. Then its new magnetic moment is given by

Moderate

A

$M^{1} = \frac{M \sin \frac{θ}{2}}{θ (redians)}$
B

$M^{1} = \frac{2 M \sin θ / 2}{θ (redians)}$
C

$M^{1} = \frac{M θ (radians)}{\sin \frac{θ}{2}}$
D

$M^{1} = \frac{M θ}{\sin θ / 2}$

Solution

We know
$\begin{array}{l} angle = \frac{arc length}{radius} \\ θ = \frac{2 l}{R}, R = \frac{2 l}{θ} \end{array}$
From fig. Effective length = 2y
$= 2 R \sin \frac{θ}{2} (∵ \sin \frac{θ}{2} = \frac{y}{R})$
$∴$ $2 y = 2 (\frac{2 l}{θ}) \sin \frac{θ}{2}$
New Magnetic moment = $M^{1} = m \times 2 y$
$\Rightarrow M^{1} = m 2 (\frac{2 l}{θ}) \sin \frac{θ}{2}$ $M^{1} = 2 M \frac{\sin \frac{θ}{2}}{θ} (∵ M=m \times 2l)$

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