A constant electric current flows along a uniform wire with cross-sectional radius `R` and heat conductivity coefficient `x`. A unit volume of the wire generates a thermal power `omega`. Find the temperature distribution across the wire provided the steady-state temperature at the wire surface is equal to `T_0`.

Explanation

The heat flux vector is `-k` grad `T` and its divergence equals `w`. Thus
`nabla^2 T = - (w)/(k)`
or `(1)/(r) (del)/(del r) (r (del T)/(del r)) = -(w)/(k)` in cylindrical coordinates.
or `T = b + A 1n r - (omega)/(2 x)r^2`
Since `T` is finite at `r = 0, A = 0`. Also `T = T_0` at `r = R`
so `B = T_0 + (w)/(4 K) R^2`
Thus `T = T_0 + (w)/(4 k) (R^2 - r^2)`
`r` here is the distance from the axis of wire (axial radius).