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

At any instant, flux passes through the ring is given by `phi=B.A=Bacostheta=BA`
or `phi =B_0(pia^2)cosomegat`
By Faraday's law of electromagnetic induction, Magnitude of induced emf is given by `epsilon=B_0(pia^2)omegasinomegat`
This causes flow of induced current, which is given by
`I=B_0(pia^2)omegasinomegat//R^2`
Now,finding the value of current at different instants, so we have current at
`t=(pi)/(2omega)`
`I=(B_0(pia^2)omega)/(R)along hatj`
Because `sinomegat=sin (omega(pi)/(2omega))=sin,(pi)/(2)=1`
`=(pi)/(omega)I=(B(pia^2)omega)/(R)`
Here, `sinomegat=sin(omega=(pi)/(omega))=sinpi=0`
`t=(3)/(2)(pi)/(omega)`
`I=(B(pia^2)omega)/(R)along-hatj`
`sinomegat=sin(omega(3pi)/(2omega))=sin,(3pi)/2=1`