Find the magnetic field at the point P in figure. The curved portion is a semicircle connected to two long straight wires.

Solution

Physics sir · 4 steps

Split the wire into three pieces, and notice that P sits right on the line of the vertical wire.

1. Top straight wire. It is semi-infinite, and P lies a distance r from it, right opposite its end. A semi-infinite wire gives half the field of an infinite one: (μ₀ i)/(4π r), directed out of the page.
2. Semicircle. P is the centre of the arc, so it gets half of a full loop's field (μ₀ i)/2r, which is (μ₀ i)/4r. The current runs anticlockwise around P, so this field also points out of the page.
3. Bottom straight wire. P lies on the line of this wire, extended upward. Every element dl⃗ is parallel to the line joining it to P, so dl⃗ × r = 0 and this wire adds nothing.

   B_P = (μ₀ i)/4r + (μ₀ i)/(4π r) = (μ₀ i)/2r(1/2 + 1/2π)

4. So. Both contributions point out of the page and add directly, which gives option (a).

The idea to keep. For bent wires, find the field of each piece separately. A semicircle gives half a loop's field, a semi-infinite wire gives half an infinite wire's field, and a wire whose line passes through the point gives zero.

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