Question

What force is produced when a battery causes an electric current to flow trough a circuit made of copper wire

Answer 1

You'll hear that force called different things in different places. It
may be called "electromotive force", "EMF", "potential difference",
or "voltage".

It's just a matter of somehow causing the two ends of the wire
to have different electrical potential. When that happens, the
free electrons in the copper suddenly have a burning desire to
travel ... away from the end that's more negative, toward the end
that's more positive, and THAT's an "electric current".