Definition and expression of Laplace’s force

Physical context

A conductor through which an electric current flows and which is immersed in a magnetic field experiences a force: this is the Laplace force. It results from the action of the magnetic field on moving charges (the Lorentz force), summed over all the moving charge carriers in the conductor.

Vector expression

For a wire-like circuit element of length dl, carrying a current I and immersed in a magnetic field B, the elementary Laplace force is given by:

dF = I * dl ^ B

where dl is a vector directed in the direction of the current, and ^ denotes the cross product.

For a straight conductor of length L placed in a uniform magnetic field B, we integrate:

F = I * L ^ B

Magnitude of the force

If θ is the angle between the conductor and the field B:

F = I * L * B * sin(θ)

Case	Angle θ	Consequence
Conductor parallel to B	theta = 0	F = 0
Conductor perpendicular to B	theta = pi/2	F = I * L * B (maximum)

Common pitfall

Note: Laplace’s force only exists if there is both a current AND a field that is not collinear with the conductor. A current alone, or a field alone, is not sufficient to produce a force.