Question

A straight magnetised wire of magnetic moment '$M$' is bent so that it subtends an angle '$\theta$' at the centre of the curvature. Find resultant magnetic moment.

• A. $2\left(\frac{M}{\theta }\right)\sin \frac{\theta }{4}$
• B. $2\left(\frac{M}{\theta }\right)\sin \frac{\theta }{2}$
• C. $3\left(\frac{M}{\theta }\right)\sin \frac{\theta }{2}$
• D. $4\left(\frac{M}{\theta }\right)\sin \frac{\theta }{2}$

Answer 1

The correct option is B $2\left(\frac{M}{\theta }\right)\sin \frac{\theta }{2}$

For a straight wire,

Magnetic moment $=M=m\times l$ (1)

Where m= pole strength and

r= length of the wire = l

For new shape,$\theta$ angle

$r=\frac{l}{\theta }$

New magnetic moment, $M^{{}^{\prime }}=m\times r^{{}^{\prime }}$ (a)

$m=\frac{M}{l}=\frac{m}{r\theta }$ from (1) and

$r^{{}^{\prime }}=2\left(R\theta \right)=2\times r\sin \frac{\theta }{2}$

$\therefore M^{{}^{\prime }}=m\times r^{{}^{\prime }}$

$=\frac{M}{r\theta }\times (2\times r\sin \frac{\theta }{2})$

$=2\left(\frac{M}{\theta }\right)\sin \left(\frac{\theta }{2}\right)$