$$ F = qE + qv \times B$$
For the Lorentz force relevant to a current carrying wire, that is caused by the motion of a wire w.r.t to an exterior magnetic field $B$, the second term($qv \times B$) on the RHS is focused, while the first term($qE$) is negated.
Only when an electric force caused by an induced $E$ from a time-varying magnetic field is the first term in the RHS is considered.
But what about the force that caused current to flow from the potential difference voltage source? The voltage source can be a battery or capacitor, is $qE$ relevant to it?
Is the electric force within the Lorentz force equation related to the power supply in anyway?
The force on an electric charge depends not only on where it is, but also on how fast it is moving. Every point in space is characterized by two vector quantities which determine the force on any charge. First, there is the electric force, which gives a force component independent of the motion of the charge. We describe it by the electric field, E.