Oh Larmor! Energy in Electromagnetic Waves
This week I started the bit of my Advanced Electromagnetism module that deals with electromagnetic radiation, including deriving the famous Larmor Formula. It reminded me of this little physics riddle, which I thought I’d share again here.
As you all know, electromagnetic radiation consists of oscillating electric and magnetic fields rather like this:
(Graphic stolen from here.) The polarization state of the wave is defined by the direction of the Electric field, in this case vertically upwards.
Now the energy carried by an electromagnetic wave of a given wavelength is proportional to the square of its amplitude, denoted in the Figure by A, so the energy is of the form kA2 in this case with k constant. Two separate electromagnetic waves with the same amplitude and wavelength would thus carry an energy = 2kA2.
But now consider what happens if you superpose two waves in phase, each having the same wavelength, polarization and amplitude to generate a single wave with amplitude 2A. The energy carried now is k(2A)2 = 4kA2, which is twice the value obtained for two separate waves.
Where does the extra energy come from?
Answers through the Comments Box please!
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April 17, 2021 at 7:06 am
I’m thinking that this is due to the assumption that the magnetic field carries none of the energy when it actually carries 1/2 of the energy of the wave.
April 17, 2021 at 9:08 am
Clearly it must have previously been dark energy. 😉
April 17, 2021 at 2:16 pm
Isn’t it because the power flux is proportional to the (time-dependentt) r.,m.s. value so the A^2 formulation is not right??
April 17, 2021 at 4:53 pm
There’s destructive interference in other directions, which balances to the total energy.
April 19, 2021 at 5:12 am
Actually, that only works (as a useful approximation) if the sources are far apart compared to the wavelength. If the sources are close, the energy required to drive the sources depends on the phase difference.
April 17, 2021 at 8:45 pm
I find the opposite case to be an even more thought-provoking way to express the same problem: suppose that you superpose two waves that are identical except for a 180-degree phase difference. Each wave carries energy, but the superposition has none. Where did the energy go?
It’s fundamentally the same question, but in this case nothing seems to depend on the specific fact that the energy scales as amplitude squared, so fiddling with that fact isn’t going to save you.
April 18, 2021 at 1:56 pm
This excellent comment gets me thinking about the energy flux when two broad beams cross at a narrow angle and are 180 degrees out of phase at their crossing…
April 19, 2021 at 5:00 pm
I agree that this would be a salutary case to think through.
April 17, 2021 at 8:49 pm
I’ll add that it’s instructive to think about specific ways you might bring those waves together — a suggestion that I think anon would approve of. For instance, use an idealized beam-splitter that transmits half of an incoming wave and reflects the other half. Send in two identical waves at right angles to each other and at 45-degree angles to the beam-splitter, and see what happens.
April 18, 2021 at 9:29 pm
With the 180 phase difference idea are you talking about the E field or the H field? Is it even possible for both of them to be out of phase at the same time?
April 18, 2021 at 10:36 pm
The wording suggests that a state with “two identical waves” means having double the amplitude. But this is misleading. Sure, a state with double the amplitude has 4 times the energy. But usually we mean by “two identical waves” something like “two photons in the same single-particle state”. And the amplitude of a multi-photon state goes like the square root of the occupation number of photons. That is: Double the photons in the same state means sqrt(2) times the amplitude and twice the energy.
I.e. If we set up an experiment that produces two identical photons, we don’t end up with twice the amplitude, but with sqrt(2) the amplitude. The trick is in how the question is set up to make us into think we get double the amplitude in such an experiment.
April 18, 2021 at 10:40 pm
The answer lies in classical electromagnetism. It has nothing at all to do with quantum theory.
April 19, 2021 at 10:39 am
Let me try a classical argument:
Consider a spring. To displace the spring from its equilibrium position by x you need energy k x^2 for some k. Now consider a second separate spring. To displace both of these springs by x you need energy 2 k x^2.
But: To displace a single spring by 2x, you need energy 4 k x^2. This is twice the energy compared to the case of two separate springs. In both cases the “total displacement” is 2x. Where does the extra energy come from?
This question about springs is analogous to the EM wave question in the original post. I think this shows that nothing strange is going on. Here’s a concrete example:
Consider a mass m in the Earth’s gravitational field. If we attach this mass to a spring, it exerts a force F = g m on the spring. This gives a displacement proportional to x ~ g m/k. The spring now carries energy k x^2. If we attach two such masses to two springs, each mass will exert a force F on the spring it is attached to. The total energy in the springs is now 2 k x^2.
But: If we attach both masses to the same spring, this spring will feel a force 2F and will be displaced by 2x. The energy stored in this spring will be 4 k x^2 — twice the energy of the case where we attach the masses to separate springs. Where does the extra energy come from?
Here the answer is: From the gravitational energy of the two masses m. If we attach both masses to separate springs, the energy taken from the gravitational energy is:
gm x + gm x = 2 g m x
If we attach both masses to the same spring this becomes:
gm (2x) + gm (2x) = 4 g m x
which is indeed twice the energy of the case where both are attached to separate springs.
In the EM wave case the energy must come from whatever produces those waves. Producing two separate EM waves with amplitude A takes less energy than producing a single EM wave with amplitude 2A. Just as displacing two springs by x takes less energy than displacing a single spring by 2x. There’s nothing mysterious about it.
I guess this is one reason why “amplitude” is a less useful quantity than energy.
April 19, 2021 at 11:15 am
Phillip, I guess you’re right, this basically boils down to what you already said. I hope you don’t mind!