A thin electric heater is placed between a long circular rod $($A$)$ and a coaxial tube $($B$).$ The inner and outer radii of the tube are $20$ mm and $40$ mm$,$ respectively. The heat transfer coefficient of the rod is given as kA $=0\mathrm{.}15$ W$/$mK$,$ and that of the tube is given as kB $=1\mathrm{.}5$ W$/$mK$.$ The outer surface of the tube is in contact with a fluid having a temperature of $-15$C and a convection coefficient of $50$ W$/$m$2$K$.$ It is stated that the thermal contact resistances between the cylinder surfaces and the heater can be neglected.

$($a$)$ Calculate the power required per unit length by the electric heater to maintain the outer surface temperature of the tube at $5$C$.$

$($b$)$ Determine the core temperature of the rod.