Question

L is a circular ring made of a uniform wire. Current enters and leaves the rind through straight conductors which, if produced, would have passed through the centre C of the ring. The magnetic field at C :

• A. due to the straight conductors is zero
• B. due to the loop is zero
• C. due to the loop is proportional to $\theta$
• D. due to the loop is proportional to $(\pi -\theta )$

Answer 1

Answer: (A), (B)

The correct options are
A due to the straight conductors is zero
B due to the loop is zero

Magnetic field at center of the coil due to incoming current and outgoing current is zero since center O lies on the path of the currents.

Let the current along path be $i_1$ and current along path ADC be $i_2$.

Given $i=i_1+i_2$

Length of arc ABC $l_{ABC}=2\pi R\left(\frac{2\pi -\theta }{2\pi }\right)$

Length of arc ADC $l_{ADC}=2\pi R\left(\frac{\theta }{2\pi }\right)$

$\frac{i_1}{i_2}=\frac{\theta }{2\pi -\theta }$ since the current will be inversely proportional to resistance and resistance is directly proportional to length.

$\therefore i_1=i(\frac{\theta }{2\pi })$

$\therefore i_2=i(\frac{2\pi -\theta }{2\pi })$

$\overrightarrow{B_{ABC}}=\frac{{\mu }_0{i}_1}{2R}\times (\frac{l_{ABC}}{2\pi R})(-\hat{k})$

$\overrightarrow{B_{ADC}}=\frac{{\mu }_0{i}_2}{2R}=\frac{{\mu }_0{i}_2}{2R}\times (\frac{l_{ADC}}{2\pi R})\left(\hat{k}\right)$

$\overrightarrow{B}=\overrightarrow{B_{ABC}}+\overrightarrow{B_{ADC}}=\frac{{\mu }_0}{4\pi R^2}\times (i_1{l}_{ABC}-i_2{l}_{ADC})\hat{k}=0$ since $i_1{l}_{ABC}=i_2{l}_{ADC}$