Two identical circular coils, P and Q each of radius R, carrying currents 1 A and `sqrt(3)` A respectively, are placed concentrically are perpendicualr to each other lying in the XY and YZ planes. Find the magnitude and direction of the net magnetic field at the centre of the coils.

Explanation

As per question, `R_P = R_Q = R, I_P = I and I_Q = sqrt(3)I`
Magnitude of magnetic field at common centre O due to current flowing in coil P is :
`B_P = (mu_0 I_P)/(2 R_P) = (mu_0 I)/(2R)`
Direction of `B_P` is perpendicular to the plane of coil P as shown. Similarly magnitude of magnetic field due to current flowing in coil Q is :
`B_Q = (mu_0 I_Q)/(2 R_Q) = (mu_0 (sqrt(3)I))/(2R)`
and its direction is perpendicular to the plane of coil Q (or along the plane of coil P).
`:.` Magnitude of resultant field
`B = sqrt(B_(P)^(2) + B_(Q)^2) = sqrt(((mu_0I)/(2R))^(2) + ((mu_0sqrt(3)I)/(2R))^(2)) = (mu_0 I)/(R )`
The resultant magnetic field subtends an angle `beta` from direction of `B_Q`, where
`tan beta = (B_P)/(B_Q) = 1/(sqrt(3)) implies beta = tan^(-1) ((1)/(sqrt(-3))) = 30^@`