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
$r_1=\frac{L}{2\pi }$
$B_L=\frac{{\mu }_{0}I}{2r_1}=\frac{{\mu }_{0}I}{2L}\times 2\pi$ ...(1)
For a circular coil, radius
$r_2=\frac{L}{2\pi N}$
$B_c=\frac{{\mu }_{0}IN}{2r_2}=\frac{{\mu }_{0}IN}{2L}2\pi N$ ...(2)
$\frac{B_L}{B_C}=\frac{1}{N^2}$ (from eqns (1) and (2))