Question

Two identical circular loops each of radius $r$ and carrying a current $i$ are arranged concentric with each other and in perpendicular planes as shown in the given figure. The magnetic field $\overrightarrow{B}$ at the common center is :

• A. $\frac{{\mu }_{0}i}{2r}$
• B. $\frac{{\mu }_{0}i}{\sqrt{2}r}$
• C. $zero$
• D. $\frac{{\mu }_{0}i}{r}$

Answer 1

The correct option is B $\frac{{\mu }_{0}i}{\sqrt{2}r}$

Given,

i) two coils (circular loop) arranged in perpendicular planes.

ii) The radius of each loop $=r$

iii) The magnetic field at common centre =?

We know that,

Induced magnetic field due to current flowing in a plane are mutually perpendicular. Thus,

$B_1$ is perpendicular to loop 1 plane

$B_2$ is perpendicular to loop 2 plane

Resultant at centre, $B_r=\sqrt{B_1^2+B_2^2}$

and $B_1=\frac{{\mu }_{0}I}{2r}=B_2$

$\therefore B_r=\sqrt{2}{B}_1$

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

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