# Where does an electromagnetic wave's energy come from?

tldr; doing work on a neutron would only increase kinetic energy, but doing work on a proton would increase kinetic energy as well as emit energy in the form of EM waves. How does that work?

I know how electromagnetic waves occur, due to the change in electric/magnetic field caused by an acceleration of a charged particle. That part is apparent, but since EM waves carry energy; where exactly does it come from according to the laws of thermodynamics?

My only guess is that it comes from the kinetic energy of the charged particle but that would mean that more work would be required to accelerate a charged particle compared to an uncharged particle due to conservation of energy, but then that wouldn't make much sense as this force pushing back on the accelerating ion would be coming from seemingly nowhere and why can't this force just be 0 or exactly the same as the force being applied so that either all the energy goes to kinetic energy or is radiated away?

• This cannot be explained in a tweet. You should study Griffiths electrodynamics and come back with any questions that you have while doing so. en.m.wikipedia.org/wiki/…. Commented May 11, 2022 at 23:23
• see answers to similar question physics.stackexchange.com/questions/707856/… Commented May 12, 2022 at 3:45
• physics.stackexchange.com/questions/707873/… I believe you will enjoy answers here, as far as I could gather, the inertia is higher for charged objects due to an effective REST mass increase, + the energy lost from abrahamlorentz force. Commented May 12, 2022 at 8:25

This is a famously messy problem. The most developed classical model is that of Abraham and Lorentz. Whether this is satisfactory depends a lot on your taste. In practice, it's not used much. For antenna theory, for example, a common approach is to treat radiation damping as a perturbation: you calculate the fields and currents as if there was no radiation, calculate the energy loss from the resulting radiation, and then insert an effective "radiation resistance" to model that loss.

• Is the Abraham/Lorentz model fully satisfactory in a classical setting? Or does it have shortcoming even there? Commented May 11, 2022 at 23:26
• @Jagerber48 Well, as math it's prone to crazy, unphysical solutions. As a physicist, you may choose to ignore those. In most experimental settings it fails to offer any important insight, but some find it useful. Commented May 12, 2022 at 16:45

One way to think about it is to realize that any work done on charges, (imagine touching it with something) means some type of electromagnetic interaction.

And when charges interact and move, the Lorentz force on each one doesn't follow the $$3^{rd}$$Newton Law, due to the magnetic part.

Without EM waves, momentum and energy would not be conserved.

since EM waves carry energy; where exactly does it come from according to the laws of thermodynamics?

From other available energy around the radiating body. Energy conservation is local, if some energy is put into "EM radiation mode", it has to be taken from other energy lying around.

If the charged body is accelerated by external electric field, then the EM wave energy comes from EM energy near and around the charge. Part of it may be from internal energy of the charged body, so this internal energy may temporarily decrease as the body radiates energy away; but if the accelerated motion is periodic and the body internal state always returns to the same state, most of radiated energy can't be from internal energy, so then it comes from the EM energy around the charged body, present due to external electric field.

If the charged body is accelerated by non-EM force, such as contact mechanical force due to another body, then EM wave energy comes from work of that another body.

• What do you mean by internal energy here? Is it internal energy of the charge? If so what is that internal energy Commented Jan 11 at 15:36
• @JeffyJames Energy in the space region occupied by the charged particle. Commented Jan 11 at 16:02

Imagine we have a mechanical grabber. And this grabber can grab something and then apply a sinusoidal force to it at frequency $$\omega$$.

$$F(t) = F_0\sin(\omega t)$$

Now imagine we grab a ball in vacuum of mass $$m$$. If we apply this force to it you can pretty easily calculate its acceleration, velocity, and kinetic energy.

But now imagine you grab the same ball but it is now sitting on top of the surface of water (you grab it above water so the grabber isn't touching the water at all). Now the grabber applies the same force. But this time, as the ball is wiggled it will start emitting water waves. These water waves will carry away some of the energy that was put in by the grabber. Though the grabber is applying the same force to the ball, the ball feels some damping due to being in the water so the net force on the ball won't be as large as it was in vacuum. This means the peak kinetic energy that the ball reaches won't be as high as in the vacuum case.

So yes, it will be more challenging to accelerate a charged particle to a certain KE than an uncharged particle due to the coupling of the charged particle to the EM field. Having a charged particle is like having a ball that is ALWAYS in water, no matter what you do.

• That is what I was thinking as well, only explanation I can think of using classical physics. But this analogy also uses conservation of momentum and transferring kinetic energy to water particles that have mass. Whereas the EM waves have no mass yet they carry momentum. I think I'm in over my head tbh, I only just completed my first year of physics bachelor and was only recently introduced to maxwell equations. Commented May 12, 2022 at 5:57
• @ALK003 Well, I think this analogy is fine and helpful if you are comfortable with the fact that EM fields carry both momentum and energy. If you're not comfortable with those facts then the question of "why do EM waves carry energy and momentum" is indeed a seperate question than the one you ask in the OP. Commented May 12, 2022 at 6:01
• you've pretty much answered my main question then. Accelerating a charged particle would require more energy and that's where the energy of em waves comes from if I got it correctly. It was just never mentioned in lectures or literally anywhere else when you try and google it which is just weird, thought I was missing something. Commented May 12, 2022 at 6:06
• @ALK003, yes, accelerating a charged particle up to a certain KE will take more energy than accelerating a neutral particle up to the same KE. If this answer is helpful or answers your question then please feel free to upvote it or select is as an answer. What you say at the end is generally why StackExchange can be very helpful, it allows you to ask a question you have in the form that it is coming into your own brain. Sometimes you have to do some work to make the question appropriate, but this typically sharpens the question and helps understanding anyways! Commented May 12, 2022 at 13:35