# Why doesn't a charged particle moving with constant velocity produce electromagnetic waves?

A charged particle moving with an acceleration produces electromagnetic waves. Why doesn't a charged particle moving with a constant velocity produce electromagnetic waves? As far I understand, the electric and magnetic fields in space will still be time-dependent, if a charged particle is moving with constant velocity, so they could have given rise to electromagnetic waves, but they don't.

Also, why do accelerating charged particles produce electromagnetic waves? What is Nature's intention behind this phenomena?

• "What is Nature's Intention behind this phenomena?" This question is not answerable. – BioPhysicist Sep 15 '18 at 17:11
• Also "Why?" is a philosophical question, physics intends to answer how something happens and not why it happens. – Triatticus Sep 15 '18 at 21:33
• Would you expect a charged particle with a constant velocity of zero to produce electromagnetic waves? – PM 2Ring Sep 15 '18 at 21:58
• For the second part of this question, see related. – knzhou Sep 17 '18 at 15:08
• Perhaps you can think about how a charge moving at a constant velocity is just a stationary charge in a different inertial frame – user1379857 Sep 18 '18 at 3:41

## 5 Answers

Riemannium's answer tackles why you need acceleration to form EM waves. I will hit from a different way that I think gets at your question title as to why charges moving at a constant velocity do not produce EM waves. In the subsequent discussion all mentioned reference frames are inertial reference frames.

The easiest way to reason that charges moving at a constant velocity relative to us will not emit radiation is to observe that we can always boost to a frame moving along with the charge. Then we will just see a stationary charge with just a constant electric field.

Now, it wouldn't make sense that we don't see an EM wave in our frame, but someone moving by at some us would. If an EM wave exists in one inertial frame it must exist in all inertial frames. Therefore, it must be that a charge moving at a constant velocity (in some inertial reference frame) cannot produce an EM wave.

• If we see in a frame that a charge is moving with constant velocity then it would produce a time varying electric and magnetic field. But would they be perpendicular as in EM waves? – Antonios Sarikas Oct 25 at 20:00
• @AntoniosSarikas I suggest asking a question in a new post. The comments of an answer are not the place to ask and answer new questions, even if they are related. – BioPhysicist Oct 26 at 0:04

Okay, I'll try with a poor but "intuitive" explanation.

According to relativity theories, "it is impossible to tell if you're at rest or moving with constant velocity".

We know that a charge at rest does not emit any wave.

If you were moving at constant velocity and you saw a static charge emitting a wave, you'd think "this charge is not actually static because it emitting waves, so I'm seing static because I'm moving with the same velocity as it, so I am not at rest".

That would violate one of the most basic principles of physics: you cannot tell if the train is moving forward, or the landscape is moving backwards, provided that $\vec{v}$ is constant.

Check that the opposite would lead to the existence of "priviledged observers", or "observers who are at absolute rest". This doesn't make sense.

So we must discard the idea of charges moving at constant velocity emitting waves. It must be accelerated charges, only because there isn't any other option.

• and this begs the question, why don't charges at rest in a gravitational field emit radiation? – JEB Sep 15 '18 at 22:25
• If they are at rest, a gravitational field is irrelevant because it is being compensated. Otherwise they'd be accelerating. So think of rest charges in vaccuum. They produce an electric field given by Coulomb's law, and no magnetic field. Okay so... where's the wave in Coulomb's law? There's just a radial electrostatic field... – FGSUZ Sep 15 '18 at 22:28
• so do charges "at rest" in accelerating elevators radiate? – JEB Sep 15 '18 at 22:31
• I'm not getting your point. Rest charges do not emmit waves, where does the elevator come from? If you want charges to be at rest, do not put them in an accelerating elevator haha. If they're at rest, they don't radiate. If they're accelerating, they radiate, regardless of wheter the acceleration comes from an elevator or not. – FGSUZ Sep 15 '18 at 22:39
• Oh now I get you. I was talking about inertial frames all teh time. If you're in a gravitational field, you're no longer able to apply what I told you. On the other hand, you've got the answer in that wikipedia's article. – FGSUZ Sep 15 '18 at 22:49

Ok so if you take maxwell equations and manipulate them a little you can get \begin{aligned} \frac{\partial B}{\partial t} \quad & = -\quad \nabla\times E, \\[5pt] \frac{\partial E}{\partial t} \quad & = \frac{1}{c^2} \nabla\times B - \frac{1}{\epsilon_0}J \\[5pt] . \end{aligned} You see the left hand side guarantees you that a varying magnetic field will generate an electric field, and a varying (means it changes in time) electric field will produce a magnetic field. The definition of electric field is given by the force a test charge feels from another charge. more exactly, $$E = k q_1 \frac{\vec{x} - \vec{x_1}}{|\vec{x} - \vec{x_1}|^3}$$ where q1 is the charge that gives you the electric field in point x, and this q1 charge is situated at the point x1.

nvm, ignore everything...

Intuitive approach: You sit on an electron, you see Electric field spreading around you but nothing else. The thing that Newton taught us is that you can't tell the difference between standing still or moving with constant velocity. Thus if you can't tell you're moving at all, from Maxwell eq, you can't have B, which is the magnetic field. If you can't have B, you can't have varying B, you can't have varying E, thus you cant have EM field, you will only detect that electrostatic field if you sit on an electron (that moves with constant velocity). fml I'm the worse explainer ever.

Classically: you need an acceleration of a charge to produce electromagnetic radiation. The electromagnetic tensor couples to charge, so if you look at Newton second law: $$\dot{P^\mu}=QF^{\mu\nu}\dot{X_{\nu}}$$ So, in order to have an electromagnetic wave, a solution of $$\square^2 A^\mu=\mu_0J^\mu$$ with $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ in form of, e.g., a plane wave, you need a non-null field strength $F_{\mu\nu}$ and thus a varying momentum, and thus an acceleration. Quantumly (much more complicated answer, but I will simplify it without quantum electrodynamics or electroweak theory): in order to have an interacting field and to have "waves" by excitations of the vacuum of the theory, you need something that vibrates...And perturbations are impossible without some time variation of the generalized momentum. For a spinor field, you also requires a proper "wave field". They fill up the complete continuum space-time.

A good loophole to my above argument should be about what causes gravitational waves, since momentum is conserved...Then, what causes gravitational waves? Perturbation of the vacuum of the theory of gravity (the metric field itself)! The source of gravity are the connection fields and the local spacetime variations, since you have energy-momentum tensor conservation, you need something else...Indeed, there are another momenta...Dipoles are sources of electromagnetic fields, you need two charges to oscillate or a single charge oscillating rapidly to produce a electromagnetic wave, you need asymmetric masses moving to get gravitational waves...And, at least, a non-null quadrupole momentum variation in order to get gravitational waves...

• The reasoning in your first paragraph doesn't make sense. Are you claiming "because electromagnetic forces cause acceleration, only acceleration can create an electromagnetic field?" That's just a non-sequitur. Even a completely stationary charge makes an electric field. A nonaccelerating moving charge makes a magnetic field. The equation really doesn't say the (very very) strong statement you think it's making. – knzhou Sep 15 '18 at 22:25
• Well, you have not understood...If you have an electromagnetic WAVE, read it please, you need a non-stationary electric (generally electromagnetic in boosted frames) field. You can NOT get any electromagnetic wave from a stationary electric field if you carefully look at the equations of motion, no way to get the wave function...Yes, it is a nonsequitur, taught by some books: electromagnetic fields and e.m. waves need to be non oscillating and that causes oscillations in the field. In vacuum, and with certain gauge selection, you get the wave equation. Any book on electromagnetism teaches it. – riemannium Sep 16 '18 at 16:23
• Non-sequitur? Of course...Unless you suppose electromagnetism is gravity in the fifth dimension (Kaluza-Klein theory) there is no explanation to the self-sustain existence of electromagnetic waves in vacuum. You need field theory and quantum theory in particular to understand electromagnetism, a section of the electroweak theory at higher energies. – riemannium Sep 16 '18 at 16:25
• Well, yes, an electromagnetic wave does involve a changing electromagnetic field. That simply does not prove, by itself, that acceleration is needed to make them. For instance, I can make water waves by moving my finger at a constant velocity. I think you're doing extremely fuzzy reasoning from words alone, without taking notice of what the underlying math actually says. – knzhou Sep 16 '18 at 16:27
• Similarly, there's no problem for electromagnetic waves existing in vacuum, and Kaluza-Klein doesn't change the situation. If you think otherwise, please back it up, preferably with actual equations. – knzhou Sep 16 '18 at 16:28

The energy radiating from an accelerating charged particle has nothing to do with space or any so called fields.It has to do with the particles inertia. It's resistance to change leads to a local build up of energy that is carried away in photons. To answer your question, a particle in space moving with constant velocity and no resistance would have no build up of energy to radiate. Interestingly acceleration isn’t always about velocity. Acceleration due to gravity can affect particles on a planets surface even though they have no velocity.