Two coils $P$ and $Q$ each of radius $R$ carry currents I and $\sqrt{8} \mathrm{I}$ respectively in same…
Two coils $P$ and $Q$ each of radius $R$ carry currents I and $\sqrt{8} \mathrm{I}$ respectively in same direction. Those coils are lying in perpendicular planes such that they have a common centre. The magnitude of the magnetic field at the common centre of the two coils is ( $\mu_0=$ permeability of free space)
$\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}$
$\frac{3 \mu_0 I}{2 R}$
$\frac{5 \mu_0 I}{2 R}$
$\frac{7 \mu_0 \mathrm{I}}{2 \mathrm{R}}$
Solution
Magnetic field at the centre of P due to its current I is
$\overrightarrow{\mathrm{B}_{\mathrm{p}}}=\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}...(i)$
Magnetic field at the centre of Q due to its current $\sqrt{8} 1$ is
$\begin{aligned}
& \overrightarrow{\mathrm{B}_{\mathrm{Q}}} =\frac{\mu_0 \sqrt{8} \mathrm{I}}{2 \mathrm{R}}...(ii) \\
& \therefore \overline{\mathrm{~B}_{\text {net }}} \quad \ldots \text{[From(i) and (ii)]} \\
& =\sqrt{\mathrm{B}_{\mathrm{p}}^2+\mathrm{B}_{\mathrm{Q}}^2} \\
& =\sqrt{\left(\frac{\mu_0 I}{2 R}\right)^2+\left(\frac{\mu_0 \sqrt{8} I}{2 R}\right)^2} \\ & =\frac{3 \mu_0 I}{2 R}\end{aligned}$