A magnetic field in a certain region is given by `B = B_(0) cos (omega t) hat k` and a coil of radius a with resistance R is placed in the x-y plane with its centre at the origin in the magnetic field, Fig. Find the magnitude and the direction of the current at (a,0,0) `at t = pi//2 omega, t = pi//omega` and `t = 3 pi//2 omega`.

Explanation

Here, `vec B = B_(0) cos (omega t) hat k`
i.e., magnetic field is alond positive z-axis. Fig
Magnetic flux linked with the ring
`phi = BA cos 0^(@) = B(0) cos omega t (pi a^(2))`
emf induced in the ring, `e = - (d phi)/(dt) = - (d)/(dt) [B_(0) pi a^(2) cos omega t] = B_(0) pi a^(2) omega sin omega t`
current induced, `I = (e)/(R ) = (B_(0) pi a^(2) omega sin omega t)/( R )`
At `t = (pi)/(2 omega)`, `I = (B_(0) pi a^(2) omega)/(R ) sin (pi)/(2) = (B pi a^(2) omega)/(R )` , along `hat j`
At `t = (pi)/(omega)`, `I = (B_(0) pi a^(2) omega)/(R ) sin pi =` zero
At `t = (3 pi)/(2 omega)`, `I = (B_(0) pi a^(2) omega)/(R ) sin ((3 pi)/(R )) = - (B_(0) pi a^(2) omega)/(R )`, along `- hat j`
Note that the direction of current at any instant is as per Lenz's law.