Centers of two similar coils P and Q having same number of turns are located at the coordinates (0.4,0) and (0,0.3) such that the plane of coils are perpendicular to X-and Y-axis respectively. The areas of cross section of coils P and Q are in the ration `4:3` Coil P has 16 A current in clockwise direction and coil Q has `9sqrt3A` current in anticlockwise direction as seen from the origin. A small compass needle is placed at the origin. Find the deflection in the needle, assuming the earth's magnetic field negligible and the radii of the coils very small compared to their distances from the origin.

Explanation

We know that the magnetic field on the axis of a current
carrying coil of radius r and number of turns N at a distance x
from the centre of coil is given by

`B=(mu_0Nir^2)/(2(r^2+x^2)^(3//2))=(mu_0Ni(pir^2))/(2pi(r^2+x^2)^(3//2))=(mu_0NiA)/(2pi(r^2+x^2)^(3//2))`
where A =Area of the coil.
When `rlt ltx`, then `B=(mu_0NiA)/(2pix^3)`
Now magnetic field at O due to coil P is
`B_P=(mu_0Ni_PA_P)/(2pix^3) (along OX)`
Similarly, `B_Q=(mu_0Ni_QA_Q)/(2piy^3) (along YO)`
Suppose the compass needle makes an angle `theta` with x-axis, then
`tan theta = (B_Q)/(B_P) =((i_Q)/(i_P)) ((A_Q)/(A_P)) ((x^3)/(y^3)) =((9sqrt3)/16) (3/4) (0.4/0.3)^3 =sqrt3`
`implies theta=tan^-1sqrt3=60^@`