Question

An infinitely long conductor is bent into a circle as shown in figure. It carries a current $i$ ampere and the radius of loop is $R$ metre. The magnetic induction at the centre of loop is :

• A. $\frac{{\mu }_0}{4\pi }\frac{2I}{R}(\pi +1)$ tesla
• B. $\frac{{\mu }_0}{4\pi }\frac{2I}{R}(\pi -1)$ tesla
• C. $\frac{{\mu }_0}{4\pi }\frac{I}{2R}(\pi +1)$ tesla
• D. zero

Answer 1

Answer: (A)

The given shape is equivalent to the following diagram
The field at o due to straight part of conductor is
$B_1=\frac{{\mu }_0}{4{\pi }^{\ast }}\frac{2i}{r}\odot$.

The field at o due to circular coil is $B_2=\frac{{\mu }_0}{4\pi }\frac{2\pi i}{r}\otimes$.
Both fields will act in the opposite direction,
hence the total field at $O$.
i.e. $B=B_2-B_1$
$=\left(\frac{{\mu }_0}{4\pi }\right)\times (\pi -1)\frac{2i}{r}$
$=\frac{{\mu }_0}{4\pi }\cdot \frac{2i}{r}(\pi -1)$