Q. A wire of length $I$ is bent into a circular coil of one turn of radius $R_{1}.$ Another wire of the same material and same area of cross-section and same lengths is bent into a circular coil of two turns of radius $R_{2}.$ When the same current flows, through the two coils, the ratio of magnetic induction at the centres of the two coils is
A) 1 : 2
B) 1 : 1
C) 1 : 4
D) 3 : 1
Explanation
(c) Magnetic induction at the centre of a circular coil
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B = \frac{\mu_{0}}{2}\ .\frac{ni}{R}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ B \propto \frac{n}{R}\ $
Here, $\ \ \ n_{1} = 1,\ n_{2} = 2,$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ l = 2\pi R_{1} = 2 \times 2\pi R_{2}$
$\Rightarrow \ \ \ \ \ \ \ \ R_{2} = \frac{R_{1}}{2}$
$\therefore\ \ \ \ \ \ \ \ \ \frac{B_{1}}{B_{2}} = \frac{n_{1}}{n_{2}} \times \frac{R_{2}}{R_{1}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{1}{2} \times \frac{R_{1}/2}{r} = \frac{1}{4}$
$\ \ \ \ \ \ B_{1}:B_{2} = 1\ :4$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.