On an infinite flat metal surface, a uniform surface current $K$

$($current per unit width$)$ is flowing in the $+x$ direction. Above the

surface, a conductor ring with radius $a$ is held with its normal

direction in the $z-$direction. The total resistance of the ring is $R.$

a$)$ Assume that the permeability of the metal equals the

permeability of free space $({}_0).$ Find the direction and

magnitude of the magnetic field above the metal surface. $($Assume that hat$(x)$hat$(y)=$hat$(z).)$

b$)$ If the surface current is gradually shut off as $K(t)=K_0(1-\frac{t}{}),$ where $0t,$ find the

direction and magnitude of the induced electric field in the ring.

c$)$ Find the total heat that is dissipated on the ring in the entire shut$-$off process. $($An any moment,

electric power $P=i^{2}R.$ Here the changing current i follows the relation emf$-L\frac{di}{dt}=iR,$ where

$L$ is the known constant inductance of the ring.$)$

d$)$ If the surface current is shut off at a faster rate, would the total heat dissipated on the ring in the

process increase or decrease?