# Is it possible to apply a torque without a moment arm?

In some statics problems, the question may say something like "a torque is applied about Point B". I've always assumed this was a simplification and the torque was created using a force and a moment arm. Recently I learned that circularly polarized light can exert a torque on a quarter-wave plate as the plate converts the beam's angular momentum to linear momentum. In this case I can't determine where the moment arm is. Is it possible that there isn't one?

• Did you mean "circularly polarized light" rather than "like"? – psmears Jun 28 '18 at 13:10
• You have specifically addedd the "classical-mechanics" tag, but your example is not quite classical. Might want to fix that inconsistency (not hyper important, but, you know... ;) ). – AnoE Jun 28 '18 at 15:49

The answer by Geoffrey provides a good explanation for mechanical torque, but the origin of photon spin is not mechanical.

Photon torque has been measured, and there is no lever arm. See here and here and here.

• Thank you very much. That helps a lot. Do you know if there is a description of this torque using the wave model of light? I'm curious to know how the electric fields interact with the electrons to cause this effect. – James Jun 28 '18 at 1:19
• The articles provide a semi-classical analysis. – Peter Diehr Jun 28 '18 at 1:21
• Although both the linked articles mention it, i feel that in any answer like this, (which is a good one), one should always cite the classic and original Beth experiment Richard A. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light", Phys. Rev., 50, #2, pp115-125 1936 – Selene Routley Sep 3 '19 at 8:26

Torque is defined as $\vec\tau = \vec r \times \vec F$, where $\vec r$ if what you call the "lever arm." This displacement vector (the lever arm) points from the axis of rotation to where the force is applied. This means that torque does not even makes sense as a concept without a chosen axis of rotation and the resulting lever arm. In fact, torque is in no sense absolute: the same force can create different torques depending on where the axis of rotation is chosen to be.

If there is a torque applied to an optical apparatus due to changing the angular momentum of some light, then how might an experimenter measure such a torque? He or she might attach said optical apparatus to a device that is free to rotate against some force gauge which has been calibrated to measure foot-pounds (or some other torque unit) within this apparatus. The lever arm for that applied torque would then be the distance from where the light hits the apparatus to the axis of rotation. Obviously, there would be some experimental error and the measurement would likely be more subtle than this, but the upshot is that if torque is being measured it only makes sense to say that it is measured with respect to some axis of rotation.

• Thanks for the answer! In the optical example, I believe the torque is applied about the center of the beam. Is that wrong? If I fix the plate so that it is free to rotate only about its center and then I shine the beam in an off-center location, will the plate begin to spin about its center? Or will it be stuck trying to spin around the center of the beam (in the off-center location)? – James Jun 28 '18 at 1:00
• Never mind. Peter answered my question. Thank you. – James Jun 28 '18 at 1:24
• I'm not so sure about this answer. Torque is defined as $\vec{r}\times\vec{F}$ in classical mechanics, but it's also equal to $\mathrm{d}\vec{L}/\mathrm{d}t$, and the latter equation could potentially apply to systems with intrinsic angular momentum where there truly is no lever arm. – David Z Jun 28 '18 at 8:32
• @James I've edited my answer, so you should be able to remove the acceptance and move it to Peter's answer if you want. – Geoffrey Jun 28 '18 at 8:42
• @DavidZ That's a good point. I glanced through the references that Peter linked, and admittedly it is almost all new to me. Although I should say that the original question mentions the torque being applied to a quarter wave plate, so presumably the torque that is produced by the light on that device would manifest classically as a force acting at some lever arm. – Geoffrey Jun 28 '18 at 9:13

A more general definition of torque is the rate of change of angular momentum, which does not explicitly involve a - time independent - lever arm. When circularly polarised light interacts with a device , the intrinsic angular momentum of the light may be transferred to the device giving rise to a torque. This was famously demonstrated in 1936 using a torsion balance.

This experiment also presents us with a paradox. Plane circularly polarised light represents the classical limit of photons with fully aligned spin - all parallel or antiparallel to the propagation direction. However, using the Poynting vector $P$ and the standard expression of total electromagnetic angular momentum $J = \vec r \times \vec P$, one finds zero angular momentum along the propagation direction. This paradox is resolved in my paper .