Question

Two circular coils $C$ and $D$ have equal number of turns and carry equal currents in the same direction in the same sense and subtend same solid angle at point $O$ as shown in figure. The smaller coil $C$ is midway between $O$ and $D$. If we represent magnetic field induction due to bigger coil and smaller coil $C$ as $B_D$ and $B_C$ respectively, then $B_D/B_C$ is

• A. $1:4$
• B. $1:2$
• C. $2:1$
• D. $1:1$

Answer 1

The correct option is B $1:2$

Since two coils subtend the same solid angle at $O$, hence area of coil $D=4x$ area of coil $C$
Therefore, radius of coil $D=2\times$ radius of coil $C$
$\therefore B_D=\frac{{\mu }_0}{4\pi }\frac{2\pi /(2r{)}^2}{\left[\right(2r{)}^2+a^2{]}^{3/2}}$
and $B_C=\frac{{\mu }_0}{4\pi }\frac{2\pi l(r{)}^2}{[r^2+(a/2{)}^2{]}^{3/2}}$
$\therefore \frac{B_D}{B_C}=\frac{4}{(4r^2+a^2{)}^{3/2}}\times {\left[\frac{4r^2+a^2}{4}\right]}^{3/2}$
$=\frac{4}{8}=\frac{1}{2}$.