Formula for Temperature Gradient due to Joule Heating I was wondering how to calculate how much the temperature will increase in a cylindrical wire due to current passing through it, I did some googling didn't find anything so I came up with the following and I am posting it here for verification or correction.
Model assumptions:
1- Symmterical cylinder on the z-axis,
2- Uniform current density in $+\hat{k}$ direction,
3- Ignoring thermal radiation.
Key Equations: $$\vec{\dot{q}}=-k\nabla T\quad (1)\\P=I^2R\quad  (2)$$
Steps:
From dimensional analysis of $\vec{\dot{q}}$ we find it is W/m², i.e power over area, which can be encapsulated in the following formula: $P=\oint\vec{\dot{q}}\cdot d\vec{S}$, unpacking that expression for our model exploiting the symmetry and using eq.(2):
$$
I^2R=\dot{q}\oint dS=\dot{q}(2\pi z r_c+\pi r_c^2)=\dot{q}\pi r_c(2z+r_c)\\
\dot{q}=\frac{I^2R}{\pi r_c(2z+r_c)}
$$
With heat flux being in the direction of the current density, substituting in eq.(1) we get:
$$
\frac{I^2R}{\pi r_c(2z+r_c)}=-k\frac{\partial T}{\partial z}\\
T=T_0-\frac{I^2R}{\pi r_c k}\int_0^L{\frac{1}{2z+r_c}dz}\\
T=T_0+\frac{I^2R}{\pi r_c k}\ln{\frac{r_c}{2L+r_c}}\quad (3)
$$
Where eq.(3) should give the -steady-state- temperature for cylindrical conductor. Right?
Glossary: [$\vec{\dot{q}}$ (heat flux density), $k$ (material's thermal conductivity, $T$ (temperature), $P$ (power), $I$ (current), $R$ (wire's resistance), $\vec{S}$ (wire's area vector), $r_c$ (wire's cross-sectional radius), $L$ (wire's length)]
 A: This Wikipedia article gives some useful equations for analyzing heat propagation. Note that it gives also a solution for heat distribution of a cylindrical wire. I think what might be complicating here is choosing the correct boundary conditions: is the wire isolated from the external world? Is it losing the heat at the surface? How much of the heat is being lost?
Another relevant point is using local equation for the Joule's heat density, rather than an equation for the total energy generated in the wire:
$$
P = IV = I^2R \longrightarrow q = \mathbf{j}\mathbf{E} = \sigma \mathbf{E}^2,
$$
where $\sigma$ is the conductivity, $\mathbf{j}$ is the current density and $\mathbf{E}$ is the local electric field.
Thus, you need to know not only the distribution of the currentd ensity in the cross-section, but how the electric field changes along the wire.
At first this may seem as complicating your life, but I hope it will lead you to think through your approximations and come up with a simple practical solution.
