Question

A circular coil of one turn carries a current $I.$ The same wire is then bent to form a smaller circular coil of $2$ turns and the same current is passed through it. What is the relation between the fields at the centre of the coils in the second and first case?

• A. $B_2=B_1$
• B. $B_2=2B_1$
• C. $B_2=\frac{B_1}{2}$
• D. $B_2=4B_1$

Answer 1

The correct option is D $B_2=4B_1$

Let the length of the wire is $l$.

When the only single turn is formed then it can be written as,

$l=2\pi R$

The magnetic field is given as,

$B_1=\frac{{\mu }_{0}I}{2R}$

$B_1=\frac{\pi {\mu }_{0}I}{l}$

For the $2$ turns the length is given as,

$l=2\times 2\pi r$

$r=\frac{l}{4\pi }$

The magnetic field is given as,

$B_2=\frac{2{\mu }_{0}I}{2r}$

$B_2=\frac{2{\mu }_{0}I\times 4\pi }{2l}$

$B_2=4B_1$