Two identical circular loops $P$ and $Q$ each of radius $r$ are lying in parallel planes such that they have common axis. The current through $P$ and $Q$ are $I$ and $4I$ respectively in clockwise direction as seen from $O$. The net magnetic field at $O$ is :
Answer: (B) $\dfrac{3\mu_oI}{4\sqrt2r}$ towards $Q$
$O$ is on the axis at distance $x = r$ from each loop. Field of a loop of current $i$:
$$B = \frac{\mu_0ir^2}{2(r^2 + x^2)^{3/2}} = \frac{\mu_0ir^2}{2(2r^2)^{3/2}} = \frac{\mu_0i}{4\sqrt2\,r}$$
A current that looks clockwise to an observer produces a field pointing **away** from the observer on the axis. Seen from $O$, both currents are clockwise, so the field of $P$ at $O$ points towards $P$, and the field of $Q$ points towards $Q$.
$$B_{\text{net}} = \frac{\mu_0(4I)}{4\sqrt2r} - \frac{\mu_0I}{4\sqrt2r} = \frac{3\mu_0I}{4\sqrt2r}\ \text{towards }Q$$
Solution by Sreeraj P, M.Sc Physics