PROBLEM $3-[10$ POINTS $\backslash (]\backslash )[40$ points$]$ Non$-$magnetic metal disk under time$-$dependent external magnetic field.

A non$-$magnetic metal disk of radius $a,$ thickness $l,$ and conductivity $$ is located parallel to

the $xy$ plane, and centered at the origin. There is a slowly varying but time$-$dependent

external uniform magnetic field vec$(B)=B_{0}cos(|t)hat(z).$

$($a$)$ Find the induced current density vec$(J)$ in the disk. $[20$ points$]$

$($b$)$ A very small circular wire loop, of radius $b,$ and with resistance $R,$ is located parallel

to the disk, and centered above it at the point $(x,y,z)=(0,0,h),$ with $ha,b,l.$

Find the current induced in the wire loop due to the magnetic field of the disk. $[20$ Two thin non$-$magnetic metal plates of width $\backslash ($ w $\backslash )$ and length $\backslash ($ l $\backslash )$ are separated by a gap of thickness $\backslash ($ h$$