electromagnetism - What does E/M field look like when I close a circuit?

Suppose that we have a charged capacitor with two pins: $C_+$ and $C_-$.

Suppose that we have a long wire with fixed geometry, that is already connected to the pin C+.

Let the distance along the wire be called $s$.

I closed a circuit by connecting another end of the wire to the pin $C_-$ in the moment $t=0$.

What is the general way to find electrical current in wire as a function of $s$, $t$ and geometry ?

Is there a difference between if the first end of wire was connected to the $C_+$ pin, or unconnected at all?