Abstract We consider the forces acting on electrons in magnetic field including the constraints and a condition arising from quantum mechanics. The force is calculated as the electron mass, me, multiplied
by the total time-derivative of the velocity field evaluated using the quantum mechanical many-electron wave function. The velocity field includes a term of the Berry connection from the many-body wave function; thereby, quantum mechanical effects are included.
It is shown that additional important forces besides the Lorentz force exist; they include the gradient of
the electron velocity field kinetic energy, the gradient of the chemical potential, and the `force' for producing topologically protected loop currents.
These additional forces are shown to be important in superconductivity,
electric current in metallic wires, and charging of capacitors.