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

To solve this problem, let's first understand the situation: We have two identical circular loops \( P \) and \( Q \) each with radius \( r \), and they are situated in parallel planes such that they share a common axis. Currents \( I \) and \( 4I \) flow through loops \( P \) and \( Q \) respectively, both in the clockwise direction as observed from the point \( O \). Our task is to find the net magnetic field at the point \( O \) which lies on the common axis of the loops.

The formula for the magnetic field at a point on the axis of a current-carrying loop of radius \( r \) and current \( I \) at a distance \( x \) from the loop is given by:

\(B = \dfrac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}}\)

Since the loops are identical and the axis is common, the magnetic field due to loop \( P \) at \( O \) is towards one direction, while the magnetic field due to loop \( Q \) at \( O \) is towards the opposite direction. Let us assume the magnetic field due to loop \( P \) at \( O \) points towards \( Q \), and vice versa. The magnetic field magnitudes are:

• Magnetic Field by Loop \( P \): \(B_P = \dfrac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}}\)
• Magnetic Field by Loop \( Q \): \(B_Q = \dfrac{\mu_0 (4I) r^2}{2(r^2 + x^2)^{3/2}} = 4 \times B_P\)

The net magnetic field at point \( O \) is given by the vector sum of these two fields. Assuming \( B_P \) is towards \( Q \) and \( B_Q \) is towards \( P \):

\(B_{\text{net}} = B_Q - B_P = 4B_P - B_P = 3B_P\) (towards \( P \))

Substituting the value of \( B_P \):

\(B_{\text{net}} = 3 \times \dfrac{\mu_0 I}{2x^3} (because x = \sqrt{2}r, according to geometry given) = \dfrac{3\mu_0 I}{4\sqrt{2}r}\)

This confirms that the magnitude of the net magnetic field at \( O \) comes out to be \( \dfrac{3\mu_0 I}{4\sqrt{2}r} \) towards \( P \), which matches the correct option provided in the question.

Therefore, the correct answer is:

Correct Option: \( \dfrac{3\mu_0 I}{4\sqrt{2}r} \) towards \( P \).