Joule-heated cylindrical wire
= Joule-heated cylindrical wire
{c}
{title2=$T(r)-T(a)=I^2(a^2-r^2)/(4\pi^2a^4k\sigma)$}
A long wire with uniform current <density> $I/(\pi a^2)$ generates uniform <heat> per volume $H=I^2/(\pi^2a^4\sigma)$. Radial steady conduction obeys $(rT_r)_r=-Hr/k$. Regularity at the centre removes a logarithmic solution, leaving the displayed parabolic <temperature> excess relative to the surface. Conductivity, <electrical conductivity> and current <density> are assumed constant.