Question

The figure shows a circular loop made of a wire of radius R. The resistivity of the material varies as a function of $\theta$ such that $\rho ={\rho }_{0}si{n}^2\theta .$ The positions of the jockey such that the magnetic field at the center (o) due to the current in the loop is zero, will be

• A. $\theta =\frac{\pi }{2}$
• B. $\theta =\pi$
• C. $\theta =\frac{3\pi }{2}$
• D. $\theta =\frac{\pi }{4}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A

$\theta =\frac{\pi }{2}$

B

$\theta =\pi$

C

$\theta =\frac{3\pi }{2}$

Consider two sectors, one of $\alpha$ and other of $(2\pi -\alpha )$.

As magnetic field $B\propto I\theta$ or $B\propto \frac{\theta }{\text{Resistance}}$

So $\frac{\alpha }{R_1}=\left(\frac{2\pi -\alpha }{R_2}\right)$-------(1) and $R_1={\int }_0^{\alpha }\frac{{\rho }_{0}si{n}^2\theta }{A}Rd\theta$ where A is the crossectional area of the wire.

Similarly we can get $R_2$

On solving we get $\alpha =\frac{\pi }{2},\pi ,\frac{3\pi }{2}$