Question

One of the two identical conducting wires of length $L$ is bent in the form of a circular loop and the other one into a circular coil of $N$ identical turns. If the same current is passed in both, the ratio of the magnetic field at the centre of the loop $\left(B_{L}\right)$ to that at the centre of the coil $\left(B_{C}\right),$ i.e. $\frac{B_{L}}{B_{C}}$ will be

• A. $\frac{1}{N^{2}}$
• B. $\frac{1}{N}$
• C. $N$
• D. $N^{2}$

Answer 1

Answer: (A)

For a single loop, radius
$r_{1}=\frac{L}{2 \pi }$
$B_{L}=\frac{\mu _{0} I}{2 r_{1}}=\frac{\mu _{0} I}{2 L}\times 2\pi $ ...(1)
For a circular coil, radius
$r_{2}=\frac{L}{2 \pi N}$
$B_{c}=\frac{\mu _{0} I N}{2 r_{2}}=\frac{\mu _{0} I N}{2 L}2\pi N$ ...(2)
$\frac{B_{L}}{B_{C}}=\frac{1}{N^{2}}$ (from eqns (1) and (2))