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Force and Torque on a Current Loop

A current loop in a uniform magnetic field feels no net force (the forces on opposite sides cancel) but it does feel a torque, and that torque is what turns every electric motor. The loop behaves like a little bar magnet with magnetic moment $\mathbf{m} = N I A\,\hat{\mathbf{n}}$, and the field twists it with $\boldsymbol{\tau} = \mathbf{m}\times\mathbf{B}$, of size $N I A B\sin\theta$, largest when the loop faces along the field and zero when its moment is aligned with $\mathbf{B}$. Left free, the loop swings like a pendulum and settles with $\mathbf{m}$ parallel to $\mathbf{B}$. Add a commutator that flips the current every half turn, right where the torque would otherwise reverse, and the loop spins continuously: that is a direct-current motor, and the lumpy $|\sin\theta|$ torque is the torque ripple real motors smooth out with many windings.

Figure 1. Force and torque on a current loop. Top: the loop in a uniform field (blue), the force couple F = I L x B on its axis-parallel sides (red), the current (green), and the magnetic moment m (purple). Middle: the torque versus orientation, N I A B sin(theta), zero where m is aligned with B. Bottom: the time series, the free loop settling to alignment or the motor spinning up to terminal speed. Method: symplectic integration of Im theta'' = tau - gamma theta'. Source: Griffiths, Introduction to Electrodynamics, 5th ed., Sec. 6.1.3.
free (pendulum)
field B1.1
current I1.4
load0.25

WHAT TO TRY

  • Watch the free loop swing and settle with its moment $\mathbf{m}$ (purple) aligned along $\mathbf{B}$; that is the stable zero of the torque curve.
  • Note the force couple (red) is largest when the loop faces the field (torque maximal) and does no turning when $\mathbf{m}$ points along $\mathbf{B}$ (torque zero).
  • Toggle to motor: the commutator flips the current each half turn, the current and forces reverse, and the loop spins one way continuously.
  • Raise the load and the motor settles to a lower terminal speed; raise the field or current and it spins faster, as the dashed terminal line shows.