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 0 . The net magnetic field at O is:

Magnetic field on the axis of a circular current-carrying loop, $$B = \frac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}}$$

Taking direction from P to Q as positive $$\hat{i}$$: $$x = r$$

For loop P:

$$I_P = I$$

$$\vec{B}_P = -\frac{\mu_0 I r^2}{2(r^2 + r^2)^{3/2}} \hat{i}$$

$$\vec{B}_P = -\frac{\mu_0 I r^2}{2(2r^2)^{3/2}} \hat{i}$$

$$\vec{B}_P = -\frac{\mu_0 I}{2 \cdot 2\sqrt{2} r} \hat{i}$$

$$\vec{B}_P = -\frac{\mu_0 I}{4\sqrt{2}r} \hat{i}$$

For loop Q:

$$I_Q = 4I$$

$$\vec{B}_Q = \frac{\mu_0 (4I) r^2}{2(r^2 + r^2)^{3/2}} \hat{i}$$

$$\vec{B}_Q = \frac{4\mu_0 I}{4\sqrt{2}r} \hat{i}$$

Net magnetic field at O: $$\vec{B}_{\text{net}} = \vec{B}_P + \vec{B}_Q$$

$$\vec{B}_{\text{net}} = \left(-\frac{\mu_0 I}{4\sqrt{2}r} + \frac{4\mu_0 I}{4\sqrt{2}r}\right) \hat{i}$$

$$\vec{B}_{\text{net}} = \frac{3\mu_0 I}{4\sqrt{2}r} \hat{i}$$