An electrical current of $700$ A flows through a stainless steel cable having a diameter of $19mm$ and an electrical resistance of $6{10}^{-4}\frac{}{m}($i$.$e$.,$ per meter of cable length$).$ The cable is in an environment having a temperature of $30C,$ and the total coefficient associated with convection and radiation between the cable and the environment is approximately $25\frac{W}{m^2}*K.$

$($a$)$ If the cable is bare, what is its surface temperature, in $C?$

$($b$)$ If a very thin coating of electrical insulation is applied to the cable, with a contact resistance of $0\mathrm{.}02m^2*\frac{K}{W},$ what are the insulation and cable surface temperatures, both in $C?$

$($c$)$ There is some concern about the ability of the insulation to withstand elevated temperatures. What thickness, in m$,$ of this insulation $)=(0\mathrm{.}5\frac{W}{m}*K$ will yield the lowest value of the maximum insulation temperature? What is the value of the maximum temperature, in $C,$ when this thickness is used?