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: Identify symmetry and direction of magnetic fields

The midpoint O lies on the common axis of both circular current-carrying rings. Hence, the magnetic field due to each ring at O will be along the axis.

Using the right-hand thumb rule, the directions of magnetic fields due to the two rings are opposite because the currents flow in opposite directions.

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Step 2: Use proportionality of axial magnetic field

For a circular current loop, the magnetic field on its axis at a fixed distance is directly proportional to the current flowing in the loop:

B ∝ i

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Step 3: Compare magnetic fields of the two rings

Both rings have the same radius R and the same distance R from the midpoint O. Therefore, the ratio of magnetic fields depends only on their currents.

Left ring current = i
Right ring current = 4i

Thus,

B₂ : B₁ = 4 : 1

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Step 4: Write expressions using a common factor

Let the magnetic field due to current i at distance R be:

B₁ = μ₀ i / (4√2 R)

Then,

B₂ = 4 × B₁ = μ₀ i / (√2 R)

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Step 5: Calculate net magnetic field

Since the magnetic fields are in opposite directions, the net field is:

B_net = B₂ − B₁

B_net = (μ₀ i / √2 R) − (μ₀ i / 4√2 R)

B_net = 3μ₀ i / (4√2 R)

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Step 6: Direction of magnetic field

Since the ring carrying current 4i produces a stronger magnetic field, the net magnetic field at O is directed towards P.

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Final Answer:

The net magnetic field at the midpoint is
B = 3μ₀i / (4√2 R), directed towards P.