Do Either Problem # $7$ OR Problem # $8$

$7.$ A long straight cylindrical wire of cross$-$sectional radius $R$ carries a current I that is uniformly distributed over the wire's cross section.

a$)$ Derive an expression for the magnetic field inside the wire. You must clearly write down Ampere's Law, show the Amperian loop used, and calculate the total enclosed current inside the Amperian loop. $(8$ Points$)$

b$)$ Derive an expression for the magnetic field at a point outside the wire $(r>R),$ and at the surface of the wire. Once again show clearly the Amperian loop used, and the enclosed current. $(6$ Points$)$

c$)$ For a copper wire with a diameter of $2\mathrm{.}5$ mm $,$ and carrying a uniform current of $33$ Amps $,$ calculate the magnetic field at i$)$ the surface of the wire, ii$)$ outside the wire, $2\mathrm{.}5$ mm from the surface, and iii$)$ inside the wire, $0\mathrm{.}50$ mm below the surface. $(6$ Points$)$