Question

Two circular coils $X$ and $Y$, having equal number of turns and carrying currents in the same sense, subtend same solid angle at point $O$. If the smaller coil $X$ is midway between $O$ and $Y$, and if we represent the magnetic induction due to the bigger coil $Y$ at $O$ as $B_y$ and that due to the smaller coil $X$ at $O$ as $B_x$, then find the ratio $B_x/B_y$.

Answer 1

The magnetic induction or field due to circular loop at a distance $x$ on its axis from the center is $B=\frac{{\mu }_{0}I{R}^2}{2(R^2+x^2{)}^{3/2}}$
For $coilX:B_x=\frac{{\mu }_{0}I{r}^2}{2(r^2+d^2{)}^{3/2}}$
For $coilY:B_y=\frac{{\mu }_{0}I(2r{)}^2}{2\left[\right(2r{)}^2+(2d{)}^2{]}^{3/2}}$
Thus, $\frac{B_x}{B_y}=\frac{2^3}{4}=2$