Energy balance and Laplace’s force couple

Power of the Laplace force

The power developed by the Laplace force on a conductor moving at velocity v is given by:

P = F . v = I * L * B * v (perpendicular case)

This power is supplied by the electric generator via electrical work, and is entirely converted into mechanical power in the absence of friction: this is the direct consequence of the principle of conservation of energy.

Laplace torque on a coil

For a rectangular coil of area S, through which a current I flows, immersed in a magnetic field B, the magnetic moment is defined as m = I * S (a vector perpendicular to the coil). The Laplace torque exerted on the loop is given by:

Gamma = m ^ B, or in magnitude Gamma = m * B * sin(alpha)

where alpha is the angle between the magnetic moment and the field B.

Equilibrium position

The torque tends to align m with B (stable equilibrium when alpha = 0) or to oppose it (unstable equilibrium when alpha = pi). This principle is used in motors, galvanometers and compasses.

Angle alpha	Torque Gamma	Type of equilibrium
0	0	Stable
pi/2	maximum	-
pi	0	Unstable

A common pitfall

The maximum torque is not achieved when m is aligned with B, but when they are perpendicular (alpha = pi/2). Many students confuse the condition for equilibrium (zero torque) with the condition for maximum torque.