$3\mathrm{.}21.$ Consider steady heat conduction in a wire of circular cross$-$section with an

electrical heat source. Suppose that the radius of the wire is $R_0,$ its electrical

conductivity is $K_{**}({}^{-1}c{m}^{-1}),$ and it is carrying an electric current density of $I$

$($A cm ${?}^{-2}).$ During the transmission of an electric current, some of the electrical

energy is converted into thermal energy. The rate of heat production per unit

volume is given by $q_t=\frac{I^2}{K_c}.$ Assume that the temperature rise in the wire is

sufficiently small that the dependence of the thermal or electric conductivity on

temperature can be neglected. The governing equations of the problem are

$-\frac{1}{r}\frac{d}{dr}(rk\frac{dT}{dr})=q_e,$ for $0r{R}_0,(kr\frac{dT}{dr}){|}_r||=0=0,T(R_0)=T_0$

Determine the distribution of temperature in the wire using $($a$)$ two linear

elements and $(b)$ one quadratic element, and compare the finite element solution

at eight equal intervals with the exact solution

$T(r)=T_0+\frac{q_c{R}_0^2}{4k}[1-(\frac{r}{R_0}{)}^2]$

Also, determine the heat flow $Q=-2R_{0}Lk(d\frac{T}{d}r){|}_{R_0}||$ at the surface using $($i$)$

the temperature field and $($ii$)$ the balance equations.