# Action-reaction pairs with electromagnetic waves

Suppose that a tower is releasing radio waves. These waves are received by an antenna. The radio waves apply force to the electrons in the antenna. My question is that by newton's third law, every action must have an equal and opposite reaction. But in the above case of the antenna, I am able to identify only the action forces. Where are the reaction forces? Do the electrons also apply force to the receiver?

It is hard to discuss the interaction of the electromagnetic wave with an antenna in terms of newton laws, since the latter were designed (and applicable) for strictly mechanical systems. Let me however consider possible ways to consider various options.

Quantum point of view: photon momentum
Momentum conservation is the most obvious representation of Newtonian mechanics when talking about the electromagnetic radiation, since we identify the momentum associated with a photon $$\mathbf{p}=\hbar\mathbf{k}$$, and use this relation extensively in discussing quantum processes, e.g., the Compton scattering. What may pass as annoticed here is that radio waves emitted by a tower are not plane waves, and have mode structure spread over hundres of kilometers. In other words, the momentum of photons absorbed by the antenna is poorly defined and mostly negligeable from the point of view of accelerating electrons. The components of this momentum are necessarily perpendicular to the antenna, which absorbs the waves with electric field parallel to it (the antenna). This causes some deflection, that could be compensated by the elastic forces in the antenna, if it were significant. (The place where such photon absorption is actively exploited is laser cooling of atoms - but we speak tehre of bigger EM momentum and much smaller objects.)

Classical point of view: induced current
Quantum point of view is impractical, since we are talking about the electric field inducing a macroscopic current in the antenna. Electrons are made to oscillate with the frequency of the electric field, dissipating its energy via the antenna resistance (i.e., the Joule's heat) and passing part of their energy to the receiver circuit. In this sense the deleted answer by the OP author was probably the closest to truth: electrons emit the radiation giving the moment back to the electromagnetic field. This however would account only for the part of the force acting on the antenna. The missing element is that the motion of electrons is in fact nothing but screening of the electric field, i.e., eliminating it (which is what we implied when talking about the absorption of photons in the quantum picture). In other words, the electrons act back on the electromagnetic radiation by cancelling it.

More formal way to restate this: when discussing the interaction of an electric field with electrons in terms of Newton's laws, one usually treats the EM radiation as an external force. The consistent description is provided by the Maxwell equations, which depend on the dynamics of the currents and the charges, together with the material equations, i.e., the equations describing response of the currents and charges to the electromagnetic fields (which are essentially the Newton's equations). Only together these form a complete system of equations correctly describing the action-reaction.

Simplified discussion To clarify the concepts, we can consider a simpler situation of a metallic 2D sheet/mirror (instead of the antenna) with a plane electromagnetic wave incident on it perpendicularly.

Photon description In terms of photons, a photon reflected by the mirror us scattered elastically, i.e., changes its momentum from $$\hbar k$$ to $$-\hbar k$$, transferring momentum $$2\hbar k$$ to the mirror. An absorbed photon is a case of inelastic scattering, transferring momentum $$\hbar k$$. This is no different from the situation with gas molecules creating pressure on the walls of a container.

Classical description In classical terms the pressure on the mirror can be expressed in terms of the Poynting vector (see radiation pressure). If the mirror is perfectly conducting, the incident wave is complitely reflected. The reflected wave has the same Poynting vector, but directed in the opposite direction. The net energy flux is therefore zero. If the mirror is not perfectly reflecting (due to resistance and losses to the downstream electruc circuit), the energy loss can be described as the Joule heat due to the induced current, as covered by the Poynting theorem.

• is the deleted answer somewhat correct ? Feb 23, 2021 at 16:37
• @AStudent4ever Yes, as I say in my answer, reemitted radiation is a part of the reaction force. Feb 23, 2021 at 16:55

The photons absorbed by the antenna carry momentum, The reaction is removal of momentum from the wave. Radiation pressure is what separates the tail from a comet, and could in theory be used to drive a space ship with a sail.

• What about the force that makes the electrons move perpendicularly to the direction that the wave propagates? Feb 18, 2021 at 19:54

The thing that pushes the receiver antenna electrons has disappeared after doing the pushing work. That is why it may be difficult to locate the reaction force.

But the electrons pushed back on the EM-wave that pushed the electrons, and by pushing back caused the EM-wave to vanish.

Hmm, I forgot two things: That the EM-wave is not very massive, and that the receiving antenna contains positive charges too.

So when the EM-wave is pushing the electrons, it is also pushing the protons. So when the electrons are pushing back on the EM-wave that is pushing the electrons, the electrons are actually using the light EM-wave to push the heavy protons.

• But the E field is transversal. You need to explain a forward force. Feb 23, 2021 at 14:23
• @my2cts Well forward force was explained already, and it seemed to me that the question was more likely about the transverse force. And it still seems like that to me. Feb 23, 2021 at 19:02
• I apparently missed the explanation of the forward force. Where is it? Feb 23, 2021 at 19:03
• @my2cts The answer by R.W. Bird, that mentions radiation pressure. Feb 23, 2021 at 19:08

"Forces" are things which act on massive particles. But there is a deeper formulation than $$\vec{F}_{12} = - \vec{F}_{21}$$ which is more widely applicable. It's called the conservation of momentum, of which $$\vec{F}_{12} = - \vec{F}_{21}$$ is just one special case. When charges wiggle, and radiate off light, some of the momentum of the particle is sapped and goes into the electromagnetic field, creating a light wave moving off in some direction.

The equal and opposite force stuff can't really be made meaningful in electromagnetism, because "forces" aren't instantaneous. The influence of a particle moves through the electromagnetic field at the speed of light.