A hairpin like shape as shown in figure is made by bending a long current carrying wire. What is the magnitude of a magnetic field at point P which lies on the centre of the semicircle?

Solution

Physics sir · 4 steps

Break the hairpin into two semi-infinite wires and a semicircle, then check that all three fields add.

1. Straight parts. Each straight wire is semi-infinite and ends exactly at the foot of the perpendicular from P, at distance r. A semi-infinite wire gives half the field of an infinite wire, so each one gives (μ₀ I)/(4π r).
2. Semicircle. A full loop gives (μ₀ I)/2r at its centre, so a half loop gives (μ₀ I)/4r = (μ₀ I)/(4π r) × π.
3. Directions. The top wire carries current to the right with P below it, so its field at P is into the page. The bottom wire carries current to the left with P above it, so its field is also into the page. The arc runs clockwise, so its field at the centre is into the page too, and all three add.

   B = (μ₀ I)/(4π r) + (μ₀ I)/(4π r) + (μ₀ I)/4r = (μ₀ I)/(4π r)(2 + π)

4. So. The field at P is (μ₀ I)/(4π r)(2 + π), directed into the page, which is option (c).

The idea to keep. Find each piece's field on its own: a semi-infinite wire gives (μ₀ I)/(4π r), an arc of angle θ gives (μ₀ I θ)/(4π r). Then fix each direction before adding.

Answer from the NTA final answer key. Solution by PrepWiser.