An infinitely long straight wire carrying current I is bent in a planar shape as shown in the diagram. The radius of the circular part is r. The magnetic field at the centre O of the circular loop is :

Solution

Physics sir · 4 steps

Split the wire into a full loop and one infinite straight wire, then watch which way each field points.

1. Break it up. The gap between B and D is tiny, so AB and DE together act as one infinite straight wire carrying I along +ĵ. The rest, from B up through C and back down to D, is a full circle of radius r. The straight wire touches the circle at the bottom, so O is a distance r from it.
2. Straight wire. The infinite wire gives (μ₀ I)/(2π r) at O. The current runs along +ĵ and O lies above the wire, along +k̂. The field points along ĵ × k̂ = +î, which is out of the page.
3. Circular loop. The current goes up the left side, right across the top at C, and down the right side. That is clockwise as seen from +î, the side you are looking from. So the loop gives (μ₀ I)/2r into the page, along -î.

   B⃗ = (μ₀ I)/(2π r)î - (μ₀ I)/2rî = -μ₀/2πI/r(π - 1) î

4. So. The loop's field is larger and points the other way, so the two fields partly cancel. The net field is -μ₀/2πI/r(π-1)î, which is option (b).

The idea to keep. For a bent wire, find the field of each standard piece and fix its direction with the right-hand rule before you add. Opposite directions mean you subtract magnitudes.

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