# How is the energy loss by an accelerating charge expressed in the equations of motion?

I understand how, and why, an accelerating charge emits radiation, and loses energy in the process, as well as the Larmor formula for the power, and its derivation.

However, in classical mechanics, when we want to find the motion of a particle, the process is (theoretically) simple - find all forces on the particle as a function of position and time, plug in Newton's second law and solve the differential equation. It seems to me as if simply because a particle is charged, this process just ceases to work.

When talking about a charged particle, it is simply not enough to find all forces and solve Newton's law - now we need to somehow take into account the energy lost by the particle to radiation. However no one seems to mention how to take that into account, as a term in an equation of motion. It seems like all we know is the total power radiated by the particle - there is no concrete, complete description of how that loss affects its motion.

Of course, in some simple cases we can easily guess how the particle will be affected: for example, for an electron circling in a uniform magnetic field, it is obvious that energy loss due to radiation will cause the electron to spiral inward. You can, at least theoretically, write the Lorentz force acting on the particle, due to the uniform field, plug it into Newton's second law, and also write the Larmor formula, take that loss into account and find a complete description of the particle's motion.

However still, when we write simply Newton's second law here, with the only force being the Lorentz force, we find a circular, not spiral, solution; we have to synthetically add in the Larmor formula to truly recover the particle's motion. Contrary to classical mechanics where Newton's law is a complete description of the motion.

• Why is Newton's law no longer enough, or put another way, what term am I missing that will fix the electron's motion, add an additional "force" the will cause it to spiral inward?
• Is there a general expression for that force? And how does one in general write the equation of motion for a charged particle (or a system of)?

I suspect the answer has something to do with the EM field changing not instantaneously but at the speed of light, since Newton's law of gravitation clearly yields stable, closed orbits and propagates instantaneously. Perhaps I should phrase my question this way:

• How does the fact that the EM field propagates at a finite speed affect the way we have to write the equations of motion for a charged particle (as opposed to the equations of motion for a massive particle, acted upon by an instantaneous, newtonian gravitational field)?

One last note: perhaps this has something to do with Maxwell's equations being Lorentz-invariant and not Galilean? However I find this unlikely to be the case since the electron doesn't have to move in a relativistic speed in order to experience this effect. In any case, this is also why I allowed myself to use Newton's second law and not its (special) relativistic expression - would using that fix the problem?

• This is a good question. Jul 30 '17 at 12:57
• See: radiation reaction in the index of Jackson, and Rohrlich, Classical Charged Particles. Jul 30 '17 at 15:36
• See also Landau and Lifshitz vol. 2 §65,75--77. Jul 30 '17 at 15:43

There is nothing missing, the problem is caused by leaving out the interaction of the charge with the electromagnetic field generated by itself. This so-called "self-force" is difficult to treat because the potentials associated with it are formally infinite for a point charge. Now, for a particle at rest, the self-force must vanish by symmetry, which then implies by Lorentz invariance that it vanishes for a particle moving at constant velocity. When a charge is accelerating it no longer vanishes, it will then yield the back reaction effect of the emitted electromagnetic radiation.

The problem of how to treat the self-force rigorously within the framework of classical electromagnetism was until recently unsolved, there were only heuristic approaches that were known to suffer from problems. E.g. the Abraham–Lorentz force takes into account the self-force, but this comes at the price of pre-acceleration. When we switch on an electric field at some time, the charge will start to accelerate just before the field was switched on.

It was only recently that a rigorous derivation of the self-force was given, see this article. Here one regularizes the infinities due to point charges by replacing them with bodies of finite size and then considering the full solution of the equation of motion and then take the limit where the body shrinks to zero, but also the charge and mass are scaled to zero in this limit.

• This is a good explanation, however does this mean that the lorentz expression of the force is simply not correct? Because how I was taught, a point charge simply does not produce any field at the point it is in. Or do you mean that the lorentz expression is true, but the particle only interacts with the fields that it created in another point, after it has moved to that point? But that seems very counterintuitive since the field it created moves at the speed of light, while the particle might move at a nonrelativistic speed and still feel that effect even though it "won't be able to make it". Jul 30 '17 at 14:23
• I get the impression I may be making a mess of it, but I don't get what is wrong with that mental picture I have (where the fields the particle itself creates cn't really influence it because they move much faster than the particle and thus the particle can't "get to them"). Jul 30 '17 at 14:24
• For a thorough treatment see Rohrlich, Classical Charged Particles. I cannot recommend this book enough. Jul 30 '17 at 15:37

Why is Newton's law no longer enough, or put another way, what term am I missing that will fix the electron's motion, add an additional "force" the will cause it to spiral inward? Is there a general expression for that force? And how does one in general write the equation of motion for a charged particle (or a system of)?

Newton's law is not enough, because in electromagnetic theory, bodies made of several particles may experience non-zero net force due to their own particles. In Newtonian theory, internal forces always cancel each other, but in electromagnetic theory, they may not. This does not violate conservation of momentum though, since it is possible to introduce EM momentum which allows introduction of another, more general, law of conservation of momentum.

The simplest model where this effect can be seen is a pair of point charges of same sign which are held in close mutual distance by some other non-charged body.

Imagine such a pair is in uniform external electric field. Assuming that each point particle experiences EM force due to the external field and due to the other particle, and assuming the EM fields due to particles are given by the standard retarded solutions of the Maxwell equations for point charge (propagating away from the particle), it turns out that sum of forces on the whole pair does not always equal the net external force, except in special situation where the pair is at rest. The missing difference is given by the sum of internal EM forces (force of 1 on 2 + force of 2 on 1) and in contrast to Newtonian mechanics, this may not be zero if the pair moves.

The pair will be moving with acceleration due to the external field, but this acceleration is not simply (net external force)/(sum of masses), not even if moving slowly. Due to mentioned internal forces, there will be another force at play and its value and direction will depend on the state of motion of both particles.

It turns out (from detailed calculations) that for comoving particles of same sign, the net effect of their retarded interaction is this:

• apparent mass of the system increases; this increase is the greater the closer the particles are; the sense of this increase is that system will have lower acceleration than what would be expected based on Newtonian theory;

• the equation of motion for the pair as a whole contains not only electric force due to external electric field, but also another force, resisting the motion. This effect is usually called "radiation damping" or "self-force". This resisting force is such that net energy conservation is preserved, that is, the work done by the external field goes partly to

1) increase of energy of the material particles $\gamma_1m_1c^2 + \gamma_2m_2c^2$ ; 2) increase of EM energy in the space surrounding the pair; part of this goes away from the pair, part stays localized near the pair.

I suspect the answer has something to do with the EM field changing not instantaneously but at the speed of light, since Newton's law of gravitation clearly yields stable, closed orbits and propagates instantaneously. Perhaps I should phrase my question this way:

You're right, the damping force due to mutual internal forces is present only because the forces are due to retarded fields, which follow from relativistic EM theory. If Newtonian or Coulomb fields are used instead, there is no such damping effect.

How does the fact that the EM field propagates at a finite speed affect the way we have to write the equations of motion for a charged particle (as opposed to the equations of motion for a massive particle, acted upon by an instantaneous, newtonian gravitational field)?

The forces can no longer be expressed as functions of particle positions at one common time. For retarded fields, they sometimes can be written as functions of present and past positions, velocities and accelerations.

• Good, but can you do it with only one particle? Apr 26 '18 at 14:34
• If the particle is composed of smaller particles, then in principle yes, but in practice this is difficult, so I explained the simplest case where the particle is composed of two point particles. If the particle is not composed (is a point particle) then there is no consistent way to introduce EM mass/energy contributions; the most simple and consistent way to think of point particle is that it does not act on itself. Apr 26 '18 at 20:52

I have been able to see a paper analyzing the problem of the radiation of an accelerated charge, but assuming that at the base of the phenomenon there is no Newtonian force. The author makes an analysis of the Pointing Theorem in function of all the existing fields: external, induction and radiation, for the case of an accelerated charge in an external field, and shows the existence of a problem with physical causality; since the conservation of energy at a given instant would depend on acceleration values of the particle at future instants. In attempting to solve this problem of causality at the context of the Pointing theorem one has to accept a coupling between the field of radiation and the own field of the particle that leads to a dynamic equation distinct from Newtonian :

$$\frac {d}{dt} \left(a - \frac 1m F_\text{ext}\right) + b\times \left(\frac ac\right)^2 v = 0$$

$a$=acceleration, $F_\text{ext}$=Lorentz force, $v$=velocity, $c$=light's velocity, $b$=constant

The paper analyzes several movements with this law: constant electric field, constant magnetic field, Coulombian field and harmonic oscillator.

The paper is "Radiation of an accelerated charge." by E.C del Río accessible on-line in "International Journal of Electromagnetics (IJEL)" (2016); a peer-reviewed Indian journal.