A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of radius R, thickness d (d<0sin wt flows through the coil. If the resistivity of the material of cylindrical shell is p(rho), find the induced current in the shell.

The megnetic field in the solenoid is given by
B=(μ0)niB=(μ0)ni
mplies B=(mu_0)ni_(0) sin omega t [:' I=(I_0)=sin omega t (given)]Themag≠ticfluxl∈kedwiththeso≤niodThemag≠ticfluxl∈kedwiththeso≤niod(phi)=vec(B).vec(A) =B A cos 90^(@)=(mu_(0)ni_(0)sin omega t)(pi a^2):. The rate of change of magnetic flux through the solenoiddϕdt=π(μ0)n(a2)i0(ω)tdϕdt=π(μ0)n(a2)i0(ω)t
The same rate of change of flux is linked with the cylindrical shell. By the priciple of electromagnetic inductio, the induced emf produced in the cylindrical shell is

e=−dϕdt=−(π)(μ0)n(a2)(i0)(ω)cos(ω)te=-dϕdt=-(π)(μ0)n(a2)(i0)(ω)cos(ω)t...(i) The resistance offered by the cylinder shell to the flow of induced current I will be
R=(ρ)lAR=(ρ)lA
here,l=2πRandA=L×dl=2πRandA=L×d
∴R=(ρ)2πRLd∴R=(ρ)2πRLd...(ii)
The induced current I will be
I=|e|R=[πμ0n(a2)I0ωcosωt]×Ldρ×2πRI=|e|R=[πμ0n(a2)I0ωcosωt]×Ldρ×2πR
⇒I=μ0na2Ld(i0)ωcosωt2ρR⇒I=μ0na2Ld(i0)ωcosωt2ρR.