Two identical circular loops P and Q each of radius r are lying in parallel planes such that they have common axis. The current through P and Q are I and 4I respectively in clockwise direction as seen from O. The net magnetic field at O is :

Solution

Physics sir · 4 steps

Same distance from both loops, so the 4I loop wins; just get each loop's field direction right first.

1. Field on the axis. For a loop of radius r, the field at a point on its axis a distance x from the centre is given below. Here O is at x = r from both loops.

   B = (μ₀ I r²)/2(r² + x²)^(3/2) = (μ₀ I r²)/2(2r²)^(3/2) = (μ₀ I)/(4√2 r)

2. Two magnitudes. Loop P gives B_P = (μ₀ I)/(4√2 r). Loop Q carries 4I, so B_Q = (4μ₀ I)/(4√2 r).
3. Directions. If you see a current going clockwise, the field it makes points away from you. Looking at P from O, the field at O points towards P. Looking at Q from O, it points towards Q. So the two fields are opposite.
4. So. Bₙₑₜ = B_Q - B_P = (3μ₀ I)/(4√2 r), along the stronger field, towards Q. That is option (b).

The idea to keep. Clockwise as you look at a loop means its axial field points away from you. Find each loop's direction first, then add or subtract the magnitudes.

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