∫SJ.dS=-dQ/dt and thus ∇.J=-∂ρ/∂t.
where the integral is taken over the closed surface S.
However I’m a little confused about the conditions of steady currents:
The book I’m using sets dQ/dt=0 and ∂ρ/∂t=0 in these cases. I don’t understand this very well, as I am possibly blinded by the fact usually I=dQ/dt and so a steady current would mean dQ/dt=constant. I believe that things are different here because we are dealing with the charge flowing out of a volume. I also can’t understand the second condition. If somebody could explain them I would be grateful.
I also have seen somewhere else:
For steady currents, ∂J/∂t=0. This seems to say J=constant, so surely I is then constant which disagrees with the above. So how does this agree with the above? Finally I have the condition ∂ρ/∂t for stationary charges, which makes sense. This then leads to ∇.J=0 as above – but is it consistent that ∂ρ/∂t is zero for constant current and stationary charges?
Everything is a bit jumbled up in my head at the moment, so if somebody could explain each of those four conditions physically that would be helpful, thanks.