How do I calculate the temperature rise in a copper conductor?
Purely theoretically with no cooling at all:
\$ P=I^2*R(T) \$
\$ E(t)=\int{P dt}\$
\$ T=T0+dT \$
\$ dT=\frac{E(t)}{m*C} \$
\$ m=V*density \$
\$ V=l*A \$
\$ R(T)=l/A*r(T) \$  

The above can be condensed into a linear approximation:
\$ R(T)~=l/A*(r+T*\alpha) -&gt; R(dT)~=l/A*(r0+dT*\alpha) \$

combining all this:  \$dT ~= \int{I^2*l/A*(r0+dT*\alpha) dt}/(l*A*density*C) = I^2/(A^2*density*C)*\int{r0+dT*\alpha dt} \$

if \$ dT*\alpha &lt;&lt; r0 \$ then \$ dT ~= I^2*r0*dt/(A^2*density*C) \$

unless I messed up something :) and it would melt eventually

I: current, R:resistance, P: power, T: temperature, t:time, E:energy, m:mass, V:volume, l:length, A: cross section area of wire, C:heat capacity of copper

Of course some kind of heat transfer always exists: conduction, convection, radiation. A good rule of thumb is to allow 2.5A/mm^2 on a copper wire in a coil with multiple layers, 4..5 A/mm^2 for single layer (without heat insulation) and 8..9 A/mm^2 will require active cooling.