An electron moving in a magnetic field only experiences a force perpendicular to its motion - it only changes direction, not speed. If the electron doesn’t slow down, yet produces radiation, where does the energy come from?
It comes from the electric field that accelerates the electron.
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If the electron doesn’t slow down…
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It will unless it is boosted periodically, in an accelerator by electric fields at right angles to the magnetic field. The electric fields are modulated such that they don’t just keep the electrons (or whatever) moving, they accelerate them up to the desired energy before usually being smashed into something.
Why electrons don’t radiate energy, slow down and spiral into the nucleus of atoms is… because they just don’t.
I believe the slowing down is caused by residual gas, not the magnetic field. The force by the magnetic field, the Lorentz force, only acts perpendicular and doesn’t affect the speed.
The simplest answer is that energy loss can be viewed as being exerted by the so-called Abraham-Lorentz force, which can be viewed as a “recoil” force exerted back on any accelerating charged particle. This force acts in the direction of the time-derivative of the acceleration vector (i.e., the third derivative of the position vector), which for circular motion is always directly opposed to the speed of the particle. So this force does negative work on the charge, causing it to lose energy.
At least, that’s the simple answer. However, if you look at the Wikipedia article, you might notice that the derivation of the Abraham-Lorentz force actually assumes that there’s a force acting on the accelerating particle, causing the appropriate amount of energy to be lost. Actually nailing down the source of where this force comes from, even in the level of classical mechanics, is surprisingly subtle.
The best short answer that I know of (or, rather, the best answer that would be appropriate this early in a thread) is that “it interacts with its own fields”. (Bet you were always told that charges couldn’t do that, weren’t you?) The idea here is that when we deal with classical electrodynamics, we really shouldn’t be talking about “point charges”; they’re infinitely dense charge-wise, and we really shouldn’t be feeding infinities into our equations. (GIGO.) Rather, we should be talking about a small but finite blob of charge, with a definite (non-zero) volume and a certain amount of charge smeared out over that volume. If you do this in a careful way, you find that the forces on piece A of a charged accelerating body on part B of that same body don’t quite cancel out with those on part B from part A, due to the finite propagation time of “news” in electrodynamics. (No physically meaningful information about the fields can move faster than light.) This non-cancellation means that the different parts of an accelerating charge exert a net force on one another, and therefore that the whole charged object experiences a net force.
Hope this helps. Griffiths’s An introduction to electrodynamics does an admirable job in trying to explain this; it’s mostly correct in a qualitative way, though it’s by no means rigorous.
The electron, directed in a curved path by the magnetic field, emits radiation. This loss of energy would cause the electron to slow, but the energy imparted by the electric field maintains its speed.
I see, so the Lorentz force doesn’t fully describe the effect, the Abraham-Lorentz force is more accurate, and it results from the finite spatial distribution of the charge?
yoyodyne, thanks for your reply. If you had an electron in a vacuum between permanent magnets, so there is no additional energy input, wouldn’t it keep moving in a circle and emitting radiation, according to the Lorentz force? I don’t think loss of energy can be considered a reason for an object to slow, maybe conservation of momentum?
Pretty much. If you really want to use “point particles”, you have to go to the quantum-mechanical description of synchrotron radiation, which I’m not nearly as familiar with. But in that case, momentum and energy conservation are still “baked in” to the mathematics, and you’d still see the electron’s wave function losing energy and momentum to the photons it emits.
I should also say that the Abraham-Lorentz force is independent of the Lorentz force, in the sense that any charged particle that accelerates should experience an Abraham-Lorentz force; nothing says that the force accelerating said charged particle has to be electromagnetic. If I tied a charged body to a string and whirled it around above my head, or set it in orbit around the Earth, it would still go around in a circle and experience a radiation-reaction force.
The lorentz force causes the electron’s path to bend. But in response to the bending, the electron emits radiation. This causes the electron to lose energy, and slow down. As the electron slows down, it travels in smaller and smaller circles.
Loss of energy can be considered a reason for an object to slow down.
Well, technically, the string is an electromagnetic force. And I’m pretty sure the particle in orbit wouldn’t radiate, since that would be a violation of the equivalence principle.
A cleaner example might be a charged hadron (a proton, say) acted upon by the strong force.
You know, I was worried that someone was going to call me out on this, so I went back & looked over some papers I hadn’t read in a while. The bottom line is that I was mistaken in what I said above: an orbiting charge wouldn’t radiate according the Abraham-Lorentz equation. However, there is still a radiation-reaction force on a freely falling charge, caused by non-local effects in the curved spacetime. (Said effects vanish in a completely flat spacetime.)
Interested parties with a strong background in physics: see here and here.
Can it? I thought a force causing the slowing must be identified. If the Sun were to stop shining, the Earth would lose energy. Would it then slow down and spiral into the Sun?
Well, loss of kinetic energy, anyway. And when there’s a loss of kinetic energy, there will always be some force associated with that loss.
For a fundamental particle like an electron, losing energy implies slowing down; the only energy it can lose is kinetic energy, and kinetic energy is a function of speed. A composite particle (like the sun ;)) can gain or lose internal energy, because the particles inside it can move faster or slower, store energy in electromagnetic field configurations, or gain or lose mass (usually the result of themselves being composite).
Ah ok, I get your point now.