Re: If electromagnetic waves are massless, how can they transfer momentum?

Patrick,

I think the conceptual problem we have with light carrying momentum is rooted in the fact that we first learn momentum in classical mechanics as p=mv. In this formulation, obviously if something has no mass, it can't carry momentum, right?

Of course, this is a limited way to look at it. There really is not a problem with electromagnetic (EM) waves transferring momentum.

Anybody who has watched a surfer or stood too close to the speakers at a rap concert can tell you that waves carry momentum. Recall that the most general formulation of Newton's second law is not F=ma, but F=dp/dt, force equals the change of momentum with time. If there is a force, there is a transfer of momentum. Since electric and magnetic fields transmit forces, they transfer momentum. When two charged particles interact, momentum is transferred from one to the other via the electric and magnetic fields. We don't usually think of those fields as "carrying" the momentum from one to the other, since we treat the force as instantaneous; however, that is a perfectly valid way of looking at it. In fact, since force cannot be transferred faster than the speed of light, an exact treatment of such an interaction _has_ to take into account the time delay while the fields "carry" the force/momentum from one particle to the other. EM waves are essentially just self-sustaining, travelling fields. As such, they are able to carry momentum over long distances where we can't approximate the transfer as an instantaneous force.

When an electron wiggles in the sun, it's a good 8 minutes before that wiggle in the electric fields propagates to earth and makes an electron in the leaf of a tree wiggle, perhaps breaking a chemical bond in a chloroplast and starting a reaction that converts carbon dioxide and water into a bit of fruit and some oxygen. The basic interaction is no different than the one where an electron in the sun wiggles and causes all the other electrons nearby to start wiggling. The difference is just distance and time.

Feynman discusses the momentum of EM waves in "Feynman Lectures on Physics", Vol I, Sec. 34-9. He points out that a simple oscillating electric field cannot transfer momentum -- it just wiggles a charged particle up and down; however, when you add the magnetic field everything changes. The force from a B field is of course proportional to the velocity of the particle. In an electromagnetic wave, the electric component starts the particle moving with a velocity in the direction of the field. The magnetic force then moves the particle in the direction vxB, which is in the same direction as ExB, which happens to be the direction the wave is travelling!

Now, it's not quite as simple as Feynman makes it sound (it rarely is!). Both fields are oscillating, and the particle motion is actually 90 degrees out of phase with the electric field, so the magnetic force is not always pushing the particle forward. A full solution requires solving coupled differential equations relating the up-down motion to the forward-and back motion. I think Feynman's discussion plays pretty fast and loose with the mathematical rigor, which was a career-long habit of his but rarely prevented him from getting the right answer. When you take the time average of the motion, you get a small net force in the direction of travel of the wave -- the wonderous radiation pressure.

Sadly, most of the simplifying assumptions you might make in order to calculate this problem for an AP physics class tend to eliminate the radiation pressure effect. At least, when I try to simplify the problem I can't get it out. Maybe someone more clever can do it without doing a full solution, but as long as you relay on me for your physics, I'm afraid your students will have to be satisfied with the qualititative discussion above.

Hope it is satisfying enough to get you through the class, but not enough that they think they "have all the answers," and stop asking questions.

Have fun.

Jay H. Hartley