Question

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: 

• A. \( \dfrac{\mu_0 I}{4\sqrt{2}r} \) towards \(Q\)
• B. \( \dfrac{\mu_0 I}{4\sqrt{2}r} \) towards \(P\)
• C. \( \dfrac{3\mu_0 I}{4\sqrt{2}r} \) towards \(P\)
• D. \( \dfrac{3\mu_0 I}{4\sqrt{2}r} \) towards \(Q\)

Hint:

Always determine magnetic field direction using the right-hand thumb rule before adding magnitudes.

Answer 1

The Correct Option is C

Step 1: Magnetic field on the axis of a circular loop.
Magnetic field at a point on the axis of a circular loop is given by \[ B = \frac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}} \] where \(x\) is the distance from the centre of the loop.
Step 2: Magnetic field due to loop \(P\).
For loop \(P\), current is \(I\) and distance from point \(O\) is \(r\).
\[ B_P = \frac{\mu_0 I r^2}{2(2r^2)^{3/2}} = \frac{\mu_0 I}{4\sqrt{2}r} \] Direction is towards \(P\) using right-hand thumb rule.
Step 3: Magnetic field due to loop \(Q\).
For loop \(Q\), current is \(4I\) and distance from point \(O\) is also \(r\).
\[ B_Q = \frac{4\mu_0 I r^2}{2(2r^2)^{3/2}} = \frac{\mu_0 I}{\sqrt{2}r} \] Direction is towards \(Q\).
Step 4: Find net magnetic field.
Since fields are in opposite directions, net field is \[ B_{\text{net}} = B_Q - B_P = \frac{\mu_0 I}{\sqrt{2}r} - \frac{\mu_0 I}{4\sqrt{2}r} = \frac{3\mu_0 I}{4\sqrt{2}r} \] Direction is towards \(P\).