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

Given, magnetic field
`vecB=B_(0) cos(omegat) hatk`
Area of coil, `A = pia^(2)` (a is the radius)
Magnetic flux linked with the coil,
`phi= BA cos 0^(@) = B_(0)cos(omegat)=pia^(2)`
Emf induced in ring is
`epsilon=-(dphi)/(dt)=B_(0)pia^(2)(d)/(dt)(cos omegat)`
`=B_(0)pia^(2)omega sin omegat`
If the resistance of the coil is R, then induced current in the coil will be
`I=(epsilon)/(R)=(B_(0)pia^(2)omega sin omegat)/(R)`
At `t=(pi)/(2omega)`
`I=(B_(0)pia^(2)omega)/(R)sin "(pi)/(2)=(B_(0)pia^(2)omega)/(R)`
Positive current means it is flowing in anticlockwise sense hence at point (a,0,0) direction of current has to be along positive Y-axis.
At `t=(pi)/(omega)`
`I=(B_(0)pia^(2)omega)/(R)sin pi=0`
At `t=(3pi)/(2omega)`
`I=(B_(0)pia^(2)omega)/(R)sin "(3pi)/(2)=-(B_(0)pia^(2)omega)/(R)`
Negative current means it is flowing clockwise. Hence at the point (a, 0, 0) direction of current has to be towards negative Y-axis.