Question

Two coils X and Y having the same number of turns, carrying the same current and in the same sense are arranged coaxially so that they subtend the same solid angle at point O. If the smaller coil is midway between the larger coil and the point o, then the ratio of the magnetic fields at O due to the two coils is

• A. 1
• B. 4
• C. 2
• D. 8

Answer 1

The correct option is C 2

It is obvious from the diagram that
$tan\theta =\frac{R_1}{x}=\frac{R}{\frac{x}{2}}\Rightarrow R_1=2R$
The magnetic field at O due to coil X is
$B_1=\frac{{\mu }_{0}I{R}_1^2}{2[R_1^2+x^2{]}^{\frac{3}{2}}}=\frac{{\mu }_{0}I\times 4R^2}{2[4R^2+x^2{]}^{\frac{3}{2}}}$
The magnetic field at O due to coil Y is
$B_2=\frac{{\mu }_{0}I{R}^2}{2[R^2+\frac{x^2}{4}{]}^{\frac{3}{2}}}=\frac{{\mu }_{0}I{R}^2\times 4^{\frac{3}{2}}}{2[4R^2+x^2{]}^{\frac{3}{2}}}\therefore \frac{B_1}{B_2}=\frac{4}{4^{\frac{3}{2}}}=\frac{4}{2^3}=\frac{4}{8}=\frac{1}{2}\Rightarrow \frac{B_2}{B_1}=2$