Question

Find magnetic field at midpoint O. Rings have radius $R$ and direction of current is in opposite sense. 

• A. $\dfrac{3\mu_0 i}{4\sqrt{2}R}$ Towards P
• B. $\dfrac{3\mu_0 i}{4\sqrt{2}R}$ Towards Q
• C. $\dfrac{3\mu_0 i}{2\sqrt{2}R}$ Towards P
• D. $\dfrac{3\mu_0 i}{2\sqrt{2}R}$ Towards Q

Hint:

When currents flow in opposite directions, subtract magnetic fields and use the stronger one to determine the direction.

Answer 1

The Correct Option is A

Step 1: Magnetic field on the axis of a circular current loop.
The magnetic field on the axis of a circular loop of radius $R$ at a distance $x$ from its center is given by:
\[ B = \dfrac{\mu_0 i R^2}{2(R^2 + x^2)^{3/2}} \]
Step 2: Distance of midpoint O from each ring.
The separation between the two rings is $2R$, therefore midpoint O lies at a distance $R$ from each ring.
Step 3: Magnetic field due to left ring (current $i$).
\[ B_1 = \dfrac{\mu_0 i R^2}{2(2R^2)^{3/2}} = \dfrac{\mu_0 i}{4\sqrt{2}R} \]
Step 4: Magnetic field due to right ring (current $4i$).
\[ B_2 = \dfrac{\mu_0 (4i) R^2}{2(2R^2)^{3/2}} = \dfrac{\mu_0 i}{\sqrt{2}R} \]
Step 5: Net magnetic field at midpoint O.
Since currents are in opposite directions, the magnetic fields oppose each other.
\[ B_{\text{net}} = B_2 - B_1 = \dfrac{3\mu_0 i}{4\sqrt{2}R} \]
Step 6: Direction of magnetic field.
The magnetic field due to the ring carrying current $4i$ is stronger, hence the net magnetic field is directed towards P.