Question

A steady current is flowing in a circular coil of radius $R$, made up of a thin conducting wire. The magnetic field at the center of the loop is $B_L$. Now, a circular loop of radius $R/n$ is made from the same wire without changing its length, by unfolding and refolding the loop, and the same current is passed through it. If new magnetic field at the center of the coil is $B_C$, then the ratio $\frac{B_L}{B_C}$ is

• A. $1:n^2$
• B. $n^{1/2}$
• C. $n:1$
• D. none of these.

Answer 1

The correct option is A $1:n^2$

$B_L=\frac{{\mu }_{0}I}{2\pi R}$
If the radius is $R/n$, then number of turns will be $n$
$B_C=\frac{n{\mu }_{0}I}{2\pi \left(R/n\right)}=n^2\frac{{\mu }_{0}I}{2\pi R}$
Hence ,$\frac{B_L}{B_C}=\frac{1}{n^2}$