Question

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

• A. $\frac{B_Y}{B_X}=1$
• B. $\frac{B_Y}{B_X}=2$
• C. $\frac{B_Y}{B_X}=\frac{1}{2}$
• D. $\frac{B_Y}{B_X}=\frac{1}{4}$

Answer 1

Answer: (C)

[ For same angle $\alpha$ ]

$B=\frac{{\left(\mu \right)}_0}{4\pi }\frac{2M}{(R^2+d^2{)}^{\frac{3}{2}}}$
$\propto \frac{R^2}{(R^2+d^2{)}^{\frac{3}{2}}}$
$\frac{B_y}{B_x}=\frac{\frac{R_y^2}{{\left([R{y}^2+(2d{)}^2]\right)}^{\frac{3}{2}}}}{\frac{R_x^2}{{\left([R_x^2+d^2]\right)}^{\frac{3}{2}}}}$
$R_y=2R_x$
$\Rightarrow \frac{B_y}{B_x}=\frac{1}{2}$