Action-reaction pairs with electromagnetic waves

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

"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.

Answer 5

Maxwell's electromagnetic theory is an ether theory, while the action-reaction theory represents a completely different worldview. According to the ether theory, a transmitting antenna transfers energy to the ether, which then carries the energy to a receiving antenna. Part of the energy is delivered to the receiving antenna, which thus acquires energy. The rest of the energy continues to propagate through the ether until it eventually overflows into the universe!

In contrast, the action-reaction theory — sometimes called “action at a distance” — denies the existence of a medium like the ether. Instead, every charge has invisible hands extending through space, connecting with other charges to exert forces via action and reaction. Therefore, interactions occur only between charges. For example, charge A may interact with charge B or C. But if B and C do not exist, then charge A cannot transfer energy to the ether for it to propagate. According to the action-reaction theory, if there is only one charge in the universe, it cannot radiate electromagnetic energy. If there is another charge B, then A can only interact with B. If there are $ N $ charges, each charge can interact with the other $ N - 1 $ charges, but cannot transfer energy to the ether to be carried away, even infinitely far!

Supporters of the ether theory include Maxwell and his followers. On the other hand, the action-reaction camp has a large following, including Weber, Neumann, Gauss, Kirchhoff, and Lorentz during Maxwell's era; Dirac's self-energy theory; Wheeler-Feynman's absorber theory; the action theory by Schwarzschild and others; Cramer’s transactional interpretation of quantum mechanics; and Stephenson's advanced wave theory. The author of this paper also supports the action-reaction theory.

For a long time, these two worldviews have only been debated on a philosophical level, without a definitive mathematical test. Even Wheeler and Feynman acknowledged problems in Maxwell’s theory in their absorber theory, considering electromagnetic fields described by Maxwell’s equations to be mere records of interaction, not real entities. Yet even they admitted that practical computations still had to rely on Maxwell's theory. The theory is so successful that even those who support the action-reaction perspective cannot deny its utility, nor would they show any disrespect toward Maxwell.

A compelling example lies in quantum theory's concept of wavefunction collapse. Quantum waves include electromagnetic waves satisfying Maxwell's equations, matter waves satisfying Schrödinger's equation, and electron waves satisfying Dirac's equation — all are probability waves that collapse upon measurement. While wavefunction collapse is standard in quantum mechanics, it is rarely considered in classical electromagnetism. However, proponents of quantum theory generally believe all waves, including those satisfying Maxwell's equations, should collapse. Yet they cannot convince classical electromagnetism scholars and engineers of this view.

In 1987, the author proposed the Mutual Energy Theorem, which is the Fourier transform of Welch’s reciprocity theorem (1960). While it is indeed a reciprocity theorem, the author also regards it as an energy theorem. In 1989, building on this theorem, the author proposed a version of Huygens’ principle. In 2017, based on that, the Mutual Energy Flow Theorem was developed, which the author discovered reflects action and reaction. Additionally, the so-called self-energy flow describes energy propagating through the ether. The author found that either energy is transferred through action and reaction (mutual energy flow) or through the ether (self-energy flow) — not both. If both flows transferred energy, the result would be double the actual energy.

The self-energy flows are given by:

$$\boldsymbol{S}_{11} = \int_{t=-\infty}^{\infty} dt \oint (\boldsymbol{E}_1 \times \boldsymbol{H}_1) \cdot \hat{n} \, d\Gamma, \quad$$

$$\boldsymbol{S}_{22} = \int_{t=-\infty}^{\infty} dt \oint (\boldsymbol{E}_2 \times \boldsymbol{H}_2) \cdot \hat{n} \, d\Gamma$$

The mutual energy flow is: $$ \boldsymbol{S}_{12} = \frac{1}{2} \int_{t=-\infty}^{\infty} dt \oint (\boldsymbol{E}_1 \times \boldsymbol{H}_2 + \boldsymbol{E}_2 \times \boldsymbol{H}_1) \cdot \hat{n} \, d\Gamma $$

Maxwell's theory assumes both mutual and self-energy flows transmit energy. However, textbooks usually neglect the mutual flow because it involves both delayed ($ \xi_1 = \boldsymbol{E}_1, \boldsymbol{H}_1 $) and advanced ($ \xi_2 = \boldsymbol{E}_2, \boldsymbol{H}_2 $) waves — and advanced waves are typically rejected. Without advanced waves, mutual energy flow loses its physical meaning and is considered imaginary.

But the author believes advanced waves are real physical entities. Prominent action-reaction proponents such as Dirac, Wheeler, Feynman, and Cramer all supported the existence of advanced waves. If advanced waves are real, then so is the mutual energy flow, which must be responsible for energy transmission — making the ether unnecessary.

Hence, the author proposes the following "No Radiation Leakage Axiom": $$ \boldsymbol{S}_{11} = \int_{t=-\infty}^{\infty} dt \oint (\boldsymbol{E}_1 \times \boldsymbol{H}_1) \cdot \hat{n} \, d\Gamma = 0, \quad$$

$$\boldsymbol{S}_{22} = \int_{t=-\infty}^{\infty} dt \oint (\boldsymbol{E}_2 \times \boldsymbol{H}_2) \cdot \hat{n} \, d\Gamma = 0 $$

In the frequency domain: $$ \boldsymbol{S}_{11} = \oint (\boldsymbol{E}_1 \times \boldsymbol{H}_1^*) \cdot \hat{n} \, d\Gamma = 0 \tag{1}, \quad$$

$$\boldsymbol{S}_{22} = \oint (\boldsymbol{E}_2 \times \boldsymbol{H}_2^*) \cdot \hat{n} \, d\Gamma = 0 \tag{2} $$

However, for any transmitting antenna, Maxwell's theory gives: $$ \boldsymbol{S}_{11} = \oint (\boldsymbol{E}_1 \times \boldsymbol{H}_1^*) \cdot \hat{n} \, d\Gamma \neq 0, \quad \boldsymbol{S}_{22} = \oint (\boldsymbol{E}_2 \times \boldsymbol{H}_2^*) \cdot \hat{n} \, d\Gamma \neq 0 $$

The author believes Maxwell's theory is incorrect here and introduces the axiom that all electromagnetic waves carry only reactive power. That is, the electric and magnetic fields maintain a $90^\circ$ phase difference: $$ \boldsymbol{S}_{11} = \mathrm{Re} \left( \oint_{\Gamma} (\boldsymbol{E}_1 \times \boldsymbol{H}_1^*) \cdot \hat{n} \, d\Gamma \right) = 0 \tag{1}, \quad$$

$$\boldsymbol{S}_{22} = \mathrm{Re} \left( \oint_{\Gamma} (\boldsymbol{E}_2 \times \boldsymbol{H}_2^*) \cdot \hat{n} \, d\Gamma \right) = 0 \tag{2} $$

Where $ \mathrm{Re} $ denotes the real part.

The author later found that for electromagnetic waves with a time factor $ \exp(j\omega t) $, the magnetic field from Maxwell’s equations must be phase-shifted by $90^\circ$: $$ \boldsymbol{H}^{(r)} = -j \boldsymbol{H}^{(r)}_{\text{Maxwell}}, \quad \boldsymbol{H}^{(a)} = j \boldsymbol{H}^{(a)}_{\text{Maxwell}} $$

Here, $ \text{Maxwell} $ denotes fields computed from Maxwell’s equations; superscript $ (r) $ means retarded, $ (a) $ means advanced. Thus, only the magnetic field needs phase correction; the electric field is valid as is.

The conclusion is that action and reaction is the correct theory of electromagnetism. The ether theory (as proposed by Maxwell) is incorrect but produces nearly correct results. The necessary correction is a $ \pm 90^\circ $ phase shift to the magnetic field, depending on whether it's a retarded or advanced wave. The mutual energy flow expression becomes:

$$ - \int_{V_1} \boldsymbol{E}_2 \cdot \boldsymbol{J}_1 \, dV = \frac{1}{2} \oint_{\Gamma} (\boldsymbol{E}_1 \times \boldsymbol{H}_2 + \boldsymbol{E}_2^* \times \boldsymbol{H}_1) \cdot \hat{n} \, d\Gamma = \int_{V_2} \boldsymbol{E}_1 \cdot \boldsymbol{J}_2^* \, dV $$

In the language of action and reaction: $$ - \langle 2, 1 \rangle = \frac{1}{2} \oint_{\Gamma} (\boldsymbol{E}_1 \times \boldsymbol{H}_2^* + \boldsymbol{E}_2^* \times \boldsymbol{H}_1) \cdot \hat{n} \, d\Gamma = \langle 1, 2 \rangle $$

This mutual energy flow theorem also applies to the Schrödinger equation. The same magnetic field phase correction must be made, and once that is done, the mutual energy theorem becomes a localized law of energy conservation. This theorem reveals the energy flow of a particle, and we can regard this flow as the particle itself.

Hence, a particle is mutual energy flow.

With the mutual energy theory, waves need not be probabilistic. Since they carry only reactive power, they already resemble probability waves. Reactive waves propagate but do not lose energy to infinity — their outward energy transfer is canceled on average by inward energy flow.

The discussion in this paper is at the energy level, but similar reasoning can be extended to forces, momentum, momentum flow, and stress-energy flow — though that requires more complex theory, which readers may consult in the book The Electromagnetic Wave Theory of Photons: https://www.amazon.com/s?k=photons+electromagnetic+wave&i=stripbooks-intl-ship&crid=N6OBC65K4GQ3&sprefix=photons+electromagnetic+wave%2Cstripbooks-intl-ship%2C77&ref=nb_sb_noss