Laplace’s force and its relationship to the Lorentz force

From the microscopic to the macroscopic level

In a conductor through which a current I flows, each charge carrier experiences a magnetic Lorentz force. By summing over all the charge carriers contained within an element of length dl of the conductor, we obtain the Laplace force:

dF = I * (dl x B)

where dl is oriented in the conventional direction of the current. For a straight conductor of length L placed in a uniform field B, F = I*(L x B), with magnitude F = ILB*sin(theta), where theta is the angle between the conductor and B.

Example: the electromagnetic rail

Two parallel rails connected by a movable bar of length L, through which a current I flows, immersed in a field B perpendicular to the plane of the rails. The bar experiences a force F = ILB, which accelerates it along the rails: this is the principle behind the electromagnetic cannon (railgun) and direct-current motors.

Torque on a coil

A loop carrying a current I and having an area S, with a magnetic moment m = ISn (where n is normal to the loop), placed in a magnetic field B, experiences a torque Gamma = m x B which tends to align m with B. This is the principle behind the electric motor and the galvanometer.

Summary table

Quantity	Formula	Direction
Force on a charge	q × (v × B)	perpendicular to v and B
Laplace’s force	I ×(dl × B)	perpendicular to the wire and to B
Torque on a turn	m × B	tends to align m with B

Common pitfall

Ensure that dl is oriented in the actual direction of the current, not in an arbitrary direction; reversing it changes the sign of the force.