$(5.)($Heat Diffusion EQ$)$ An electric cable of radius $r_1$ and thermal conductivity $k_c$ is enclosed

by an insulating sleeve whose outer surface is of radius $r_2$ and experiences convection

heat transfer and radiation exchange with the adjoining air and large surroundings,

respectively. When electric current passes through the cable, thermal energy is generated

within the cable at a volumetric rate $q^{}.($Note: There is no heat generation in the

insulating sleeve. Known surface temperatures are shown in the figure.$)$

$($cont$.-)$

Write the steady$-$state forms of the heat diffusion equation for the insulation and the

cable. Verify that these equations are satisfied by the following temperature

distributions:

Insulation: $T(r)=T_{s,2}+(T_{s,1}-T_{s,2})\frac{ln(\frac{r}{r_2})}{ln(\frac{r_1}{r_2})}$

Cable: $,T(r)=T_{s,1}+\frac{(q^{})r_1^2}{4k_c}(1-\frac{r^2}{r_1^2})$