Blundell Thermal Physics · Problem 10.2

The thermal diffusion equation

Physics — Statistical Mechanics · heat-sources-cylindrical-wire

Problem

(a) A cylindrical wire of thermal conductivity

\kappa

, radius

a

and resistivity

\rho

uniformly carries a current

I

. The temperature of its surface is fixed at

T_0

using water cooling. Show that the temperature

T(r)

inside the wire at radius

r

is given by

T(r) = T_0 + \frac{\rho I^2}{4\pi^2 a^4 \kappa}(a^2 - r^2).

(b) The wire is now placed in air at temperature

T_{\rm air}

and the wire loses heat from its surface according to Newton's law of cooling (so that the heat flux from the surface of the wire is given by

\alpha(T(a) - T_{\rm air})

, where

\alpha

is a constant). Find the temperature

T(r)