Two similar coils of radius R are lying concentrically with their planes perpendicular to each other and carrying current I and 2 I respectively. What is the resultant magnetic field at the center?

$\frac{\sqrt{5} \mu_0 I}{2 R}$

The correct answer is Option (3) → $\frac{\sqrt{5} \mu_0 I}{2 R}$

Magnetic field at the center coil of radius R, carrying a current I.

$B=\frac{μ_0I}{2R}$

for first coil,

Current = I

$∴\vec B=\frac{μ_0I}{2R}$

for second coil,

Current = 2I

$B=\frac{μ_02I}{2R}=\frac{μ_0I}{R}$

$∴B_{resultant}=\sqrt{{B_1}^2+{B_2}^2}$ [coil are perpendicular to each other]

$=\sqrt{\left(\frac{μ_0I}{R}\right)^2\left(\frac{1}{4}+1\right)}$

$=\frac{μ_0I}{R}×\frac{\sqrt{5}}{2}=\frac{\sqrt{5} \mu_0 I}{2 R}$