Two similar coils of radius R and number of turns N are lying concentrically with their planes at an angle `60^@` to each other. The currents flowing in them are I and 2I respectively. The resultant magnetic induction at the centre will be

Options

A. `(mu_0NI)/(R)`
B. `(sqrt3mu_0NI)/(2R)`
C. `(sqrt5mu_0NI)/(2R)`
D. `(sqrt7mu_0NI)/(2R)`

Answer

D

Explanation

Magnetic field induction at centre O due to coil carrying current I is `B_1=(mu_0NI)/(2R)`
Magnetic field induction due to other coil at O, carrying current `2I` is

`B_2=(mu_0N2I)/(2R)=(mu_0NI)/(R)=2B_1`
Angle between `vecB_1` and `vecB_2` is `theta=60^@`.
Resultant magnetic field induction at O is
`B=sqrt(B_1^2+B_2^2+2B_1B_2cos 60^@)`
`=sqrt(B_1^2+(2B_1)^2+2B_1xx2B_1XY_2)`
`=sqrt7B_1=(sqrt7mu_1NI)/(2R)`