A plane loop is shaped ion the from as shown in the figure with radii `a` and `b` and is placed in a uniform time varying magnetic field `B=B_(0)sinomegat` . Find the amplitude of the current induced in the loop if its resistance per unit length is equal to `lambda` . The inductance of the loop negligible.

Explanation

If the magnetic field is increaseing, flux in both loops are increasing so direction of current is anticlockwise but this wire is continuous so single current will flow. Since emf in bigger loop, if in bigger loop current is anticlockwise, it will be clockwise in smaller loop.
Flux in bigger loop
`phi_(1)=B.pia^(2)=B_(0)pia^(2)sinomegat`
`e_(1)=(dphi_(1))/(dt)=B_(0)pia^(2)omegacosomegat`
Flux in smaller loop
`phi_(2)=B.pib^(2)=B_(0)pib^(2)sinomegat`
`e_(2)=(dphi_(2))/(dt)=B_(0)pib^(2)omegacosomegat`
Net emf `e=e_(1)-e_(2)=piB_(0)omega(a^(2)-b^(2))cosomegat`
Total resistance of wire `R=2pi(a+b)lambda`
Induced current `I=(e)/(R)=(piB_(0)omega(a^(2)-b^(b))cosomegat)/(2pilambda(a+b))`
`I=(B_(0)omega(a-b)cosomegat)/(2lambda)`
I will be maximum when `cosomegat=1`
Current amplitude `I_(0)=(B_(0)omega(a-b))/(2lambda)`