Accelerating electrons don’t radiate!

In the 19th Century it was believed that X-rays were generated by the sudden deceleration of electrons as they slammed into “plum-pudding” atoms. This led to a search for a working model that could compute the radiation of a decelerating/accelerating electron. Many models were created but they all had serious flaws. However, in 1911 Rutherford found that atomic nuclei were in fact tiny, and it was clear that it was not impact that was causing the radiation. In fact we now know that the electrons decelerate by interacting with the massive electromagnetic fields inside atoms, and the radiation is not caused by imapct deceleration at all. However, the idea that Electromagnetic Theory requires accelerating or decelerating charged particles to radiate is a long time dying; it was necessary only until 1911. However, many of the old 19th Century derivations are being resurrected today.

The problem is generic - if charged particles radiate under acceleration under any conditions, then both Electromagnetic Theory and Quantum Mechanics have to firstly explain how that radiation comes about, and secondly, explain why it does not happen inside an atom.

So let us have a look at the basic problem.

Is it ever possible that accelerating electrons radiate?

To answer this most basic question, let us look at a simple thought experiment, where a particle is first accelerated out of our rest frame, then decelerated back into it. Make the acceleration and deceleration have equal (but of course opposite) magnitudes, and bear in mind that from the point of view of the electron there is only acceleration out of its rest frame, so when it is decelerated back into our rest frame it simply “perceives” the same acceleration, although in the reverse direction direction. This is because there are no “special viewpoints” in a relativistic universe, so every rest frame is valid, and so to see if an isolated electron radiates under acceleration we must evaluate it in its own rest frame.

If an electron radiates as it accelerates, extra work must be done in generating this radiation. The only way this can happen is if its inertia is increased. Equally, it must also radiate as it decelerates, similarly increasing its inertia; this is because the electron has no concept of deceleration - it is simply accelerating in another direction.

First consider the energies associated with the acceleration- note we use a very basic energy model, so that both electromagnetic field theory and quantum mechanics apply...

E_in = E_k + E_r

The input energy (the work we do) E_in to accelerate an electron from our rest frame to a new rest frame is equal to the final kinetic energy of the electron = E_k plus the energy radiated during acceleration = E_r. No problem so far; this much has always been part of the theory.

But now brake the electron from its new rest frame back into our own in exactly the same way. The electron in its accelerating/decelerating frame sees no difference between rest frames and radiates E_r as before. This radiation increases its inertia, so that we get...

E_k + E_r = E_out

Hence E_in = E_out (after all the acceleration and deceleration are identical from the electron’s point of view). But on the way two bursts of radiation (2.E_r ) have been radiated away. So we have recovered all the original input energy, but we have gained 2.Er extra energy that has been radiated from the system, so the Principle of Conservation of Energy has been violated.

This is a very basic result and is not tied to any particular theory - if any particle radiates anything as a result of acceleration in a relativistic universe, then the Principle of Conservation of Energy is violated. This applies to putative gravity waves as much as to electromagnetic radiation. This does not mean that (say) an electron orbiting in a cyclotron does not lose energy, but rather that the energy loss mechanism is not a result of an individual particle’s accelerative radiation.

Let us now look at a few examples of arguments used to support the idea that accelerating charges radiate:-

Radio dipole antenna

The theory: You will sometimes find the occasional statement on the Internet and elsewhere that it is the acceleration of the electrons as they rush up and down the aerial that gives rise to electromagnetic radiation. This is sometimes quoted as a proof that electrons radiate under acceleration.

What issues does this raise?

• It is not possible to create any mathematical model that will allow accelerating electrons to radiate at a single frequency as a radio aerial does; the electron cannot “know” the size of the aerial during its acceleration - the same acceleration can occur for low power at high frequency, and high power at low frequency, and the electron therefore cannot know what frequency it is expected to radiate.
• Most mathematical models that have been created to describe the theory require the electron to radiate easily at high accelerations but very little at low, radiation often being proportional to some power of acceleration. This means that it would be almost impossible to radiate on long wave, but easy at microwave frequencies.

It is in fact well known from Maxwell’s equations that aerials actually work in a very simple way, although perhaps a little difficult to follow at first; this is because the near-field (near the antenna) and far-field fields are very different. The near-field patterns are very complex.

There are actually two independent effects. Maxwell’s equations identify two separate mechanisms; motional effects and changing-field-strength effects. A moving electric field (such as in an electric current) induces a magnetic field, and a changing electric field independently generates an associated magnetic field, simply by reason of its changing. Equally, a moving magnetic field induces an electric field, and a changing magnetic field independently generates an associated electric field, simply by reason of its changing. This is crucial to the understanding of near-field and far-field behaviour. The near-field energy is generated by moving fields, the far -field energy is electromagnetic radiation generated by changing fields. I have found the full explanation of dipole antenna operation difficult to find on the Internet so have inserted it in full here. It may need rereading a few times to get the hang of the two effects. I will use the term “induction” for the moving effect, and “associated” for the changing effect.

1. Electrons are stationary at the top end of the aerial, and a shortage of electrons at the bottom of the aerial. This creates a negative charge at the top and a positive at the bottom, so an negative up-down electric field is generated. This electric field stores energy.
2. Electrons from the top rush down the aerial. The charge on the aerial ends dissipates, so the electric field disappears. However, the moving electrons in the middle of the aerial induce a negative magnetic field that rings round the aerial. This stores the energy (that was in the electric field in step 1) in the magnetic field...
3. The electrons are stationary at the bottom of the aerial, leaving a stationary dearth of electrons at the top. This creates a positive charge at the top and a negative at the bottom, so an positive up-down electric field is generated. The energy is again stored in the electric field.
4. Electrons run up the aerial cancelling the surplus at the bottom and the dearth at the top. There is now no charge on the aerial ends, so the electric field disappears. However, the moving charges in the middle of the aerial induce a positive magnetic field that rings round the aerial. The energy is again stored in the magnetic field.

This mechanism generates energy that is stored in the induced fields and is fully recoverable from them. If the near-field was the only effect a dipole had, the impedance of a dipole would be infinite (in the absence of anything being placed in the near-field that might absorb the energy directly) since no energy is lost or radiated by this mechanism; it is simply stored alternately in the electric and magnetic fields.

Now look at what generates the electromagnetic radiation, also sometimes called ‘far-field’ radiation. Here it is the changing electric field that generates an associated magnetic field, and a changing magnetic field that generates an associated electric field. It is important to separate what we mean by the generated electric and magnetic fields in this situation - they are totally independent of the induced fields and here I will use the term “associated” to imply fields raised by the alternate field changing. These associated fields give rise to electromagnetic radiation. Bear in mind, however, that it is the same generating current flows that create both effects:-

1. Electrons are stationary at the top end of the aerial, and a shortage of electrons is stationary at the bottom of the aerial. At this point the electric field is at one of its maxima, so is momentarily not changing, and no current is flowing, so the induced magnetic field is zero. However, the induced magnetic field is changing at its fastest rate, so the associated electric field is also at its maximum.
2. Electrons from the top move down the aerial. At the point that the electric field across the aerial is zero, the rate of change of electric field is at its maximum, so the associated magnetic field peaks; the rate of change of the induced magnetic field is momentarily zero so the associated electric field is also momentarily zero.
3. The electrons are stationary at the bottom of the aerial, leaving a stationary dearth of electrons at the top. This creates a positive charge at the top and a negative at the bottom. At this point the rate of change of electric field is momentarily zero so the associated magnetic field is zero; however, although no current is flowing, the rate of change of current, and therefore the rate of change of induced magnetic field, is at its opposite (to step 1) maximum and so the associated electric field is at its opposite maximum.
4. Electrons run up the aerial cancelling the surplus at the bottom and the dearth at the top. At the point where there is no charge on the aerial ends, and the electric field disappears, the rate of change of electric field is at its opposite (to step 2) maximum, so the associated magnetic field is at its opposite  maximum. At the same time the rate of change of current, and hence of induced magnetic field, is momentarily zero, so the associated electric field is zero.

The near-field electric field strengths fall off at 1/r³ which is much faster than the 1/r² of electromagnetic radiation, so while the near-field is dominated by induction, the far-field is dominated by electromagnetic radiation (some forms of antenna can reduce the radiative fall-off to much less than 1/r²). The impedance of the aerial, which is due entirely to the far-field radiation pattern (provided nothing is placed in the near-field), is 70 ohms; without the far-field radiation the impedance would be infinite unless some object that could absorb energy was placed in the near-field to absorb energy by induction; this is because no energy would be lost (in the real world a small amount of energy would be lost through the resistance of the aerial, but this is negligible in any good design).

For the far-field radiation pattern, looking down the direction of propagation, one would see a rotating vector, with (for a vertical dipole) the electric field upwards and the magnetic field horizontal; the vector rotates from electric to magnetic, to negative electric and then negative magnetic, then back to the start. The sum of the electric and magnetic field energy densities is a constant in electromagnetic radiation, being the same at every part of the cycle. By contrast, if EM radiation was truly due to electron acceleration rather than Maxwell’s equations, the energy density would pulse at the maximum electron acceleration during the peaks of the electric field and there would be gaps in the radiated wave when the electron acceleration in the aerial fell to zero; there would be no “wave”.

Note that a single electron cannot produce Maxwellian radiation just by undergoing simple harmonic motion. You need changes in the electric and magnetic field strengths rather than alterations in motional induction.

Note:- There is in fact a certain amount of artistic licence taken in the above descriptions. The electron do not so much move along the aerial as drift along it, with each electron moving back and forth maybe only a few millimetres, depending on the wavelength and drive voltages; it does not take much of a concentration of electrons to produce large electric fields across the aerial. The overall flow of current may be likened to pushing along a row of dominoes separated by springs - the row moves together but is somewhat compacted by the forces acting on the springs.

Spiral decay

The theory: A high energy electron loses energy on entering a magnetic field that is oriented normal to its flight, and is forced to take a circular path because of induced fields. Travelling in this circular path causes a centripetal acceleration which causes the electron to radiate energy. The electron therefore follows a decaying spiral path as it loses kinetic energy and slows down. This argument is based on the statement that accelerating charged particles radiate electromagnetic energy; this process was sometimes called ‘radiation damping’ .

Figure 1: path of electron in magnetic field oriented normal to page

What issues does this raise?

• A specific problem here is that the energy loss is normal to the direction of flight of the electron - acceleration is radial, yet kinetic energy is orthogonal to it by being tangential. How does the electron, in its own private reference frame, “know” the correct vector to lose kinetic energy; assume the electron is passing right-to-left at the bottom of the above spiral - all it will see from its own reference frame is an upward acceleration, so how can it tell that it must radiate away to the left in order to slow down to the right in the reference frame of the experiment? Why should it not radiate to the right? The electron cannot decide, so whatever is causing the decay of the spiral cannot be that.
• Current will flow round a super-conducting ring indefinitely, without any decay of energy, despite the fact that the electrons experience centrifugal acceleration. This means that the centripetal acceleration is not by itself causing the spiral decay.

Electrons in atoms

The theory: The electromagnetic field model of an atom cannot be stable since a charged particle (the electron in this case) radiates when it accelerates. Unlike photon energy, this radiated energy is broadband. An electron in a circular orbit undergoes centripetal acceleration, and hence (so the theory goes) the circular orbit of an electron in an atom will decay. The basis for this theory is that a charged particle radiates electromagnetic energy in an accelerating frame, increasing the work done to accelerate it - that is, its inertia is increased. If you are interested in the details you can search out “Radiation Reaction” on the web - although don’t be surprised if you come up with something quite unrelated!

What issues does this raise?

• The biggest problem here is that it simply does not happen in real life, so if the field model requires an accelerating electron to radiate, then the model is wrong. So the real question is “Does the field model of an electron require it to radiate?”. The answer is of course “No, it doesn’t!”.

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X-Ray generation

The theory: X-rays are traditionally generated by firing electrons at high speed into a metal target. In the 19th century the radiation was believed to be generated by the sudden deceleration of electrons when they struck the metal target.

What issues does this raise?

• This idea was based on a relatively solid “plum-pudding” atomic model. In 1909 Rutherford showed that the atom was mainly space, which invalidated the whole idea, and we now know that it is the interaction of electrons with the intense fields inside an atom that gives rise to X-rays; in fact, electron microscopes look at the spectrum of the X-rays generated to determine the elements present in a sample, proving an interaction between the electron and the element’s atoms. It has nothing to do with acceleration of free electrons. Nevertheless you will still find the occasional reference on the Internet and in old text books to that old 19th-century idea.

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