Question

Three identical circular coils carrying a current 'i' each and having radius a and number of turn one are arranged such that their centers coincide and their planes are mutually perpendicular, Magnitude of the magnetic field at the centre is

• A. $\frac{{\mu }_{0}i}{2a}$
• B. $\frac{3{\mu }_{0}i}{2a}$
• C. $\frac{\sqrt{3{\mu }_{0}i}}{2a}$
• D. zero

Answer 1

The correct option is C $\frac{\sqrt{3{\mu }_{0}i}}{2a}$

Given

3 circular coil perpendicular to each othe and share common centre

$\underset{B_1}{\to }$ is magnetic field of coil in XY plane

$\underset{B_2}{\to }$ is magnetic field of coil in YZ plane

$\underset{B_3}{\to }$ is magnetic field of coil in ZX plane

Solution

$\underset{B_1}{\to }=\frac{{\mu }_0\ast i}{2a}\ast \hat{k}$

$\underset{B_2}{\to }=\frac{{\mu }_0\ast i}{2a}\ast \hat{i}$

$\underset{B_3}{\to }=\frac{{\mu }_0\ast i}{2a}\ast \hat{j}$

$B_{eff}=\underset{B_1}{\to }+\underset{B_2}{\to }+\underset{B_3}{\to }$

=$\frac{{\mu }_0\ast i}{2a}\ast ({\hat{i}}^2+{\hat{j}}^2+{\hat{k}}^2{)}^{\frac{1}{2}}$

=$3^{\frac{1}{2}}\ast \frac{{\mu }_0\ast i}{2a}$