EM Wave - Where Does the Energy Go?

[Master J]

For an EM wave in vacuum, we know the energy density is given by1/2 e E^2 for the electric field, with a similar expression for the magnetic (e is permittivity of vacuum). E^2 implies that the energy oscillates as a cosine squares function if we represent the E field as E_0.cos [ kx - wt].

But since cosine squared goes between 0 and 1, where does the energy go when it is at 0?? It can't be stored in the magnetic field since they are in phase in vacuum, so where does it "go"? I hope you can understand my question!Secondly, in Poyntings theorem for the conservation of energy with EM waves, the kinetic energy density is given by E.J. Does this current density refer to a current that the E field has generated AND/OR one that was already there?Cheers guys!

 

[nonequilibrium]

Hello, I'm only following an introductory course on electromagnetism myself (Griffiths-level), so I'm no authority, but I'll do my best to try and help!

Regarding your first question:
Not a very physically insightful answer, but I believe the standard answer on your question is "The amplitude is not zero everywhere at the same time", i.e. when you're at a certain x and you wait for the t so that cos(kx-wt) = 0, this just means the energy has "moved" to another x' where cos(kx' - wt) is NOT zero.

For your second question:
I'm not sure how you'd draw the distinction between a current generated by your electric field or not. Anyhow, the J in the formula is simply any current (density) that's there (in that point), no matter how it got there.
Too maybe help demistify the "E.J" expression:
we know the electromagnetic power on a particle is [tex]\vec F_{\textrm{electromagnetic}} \cdot \vec v = q(\vec E + \vec v \times \vec B) \cdot \vec v = q \vec E \cdot \vec v[/tex]
and this gives us for the power density [tex]\rho \vec E \cdot \vec v[/tex], and [tex]\rho \vec v[/tex] is exactly what we call [tex]\vec j[/tex].
So the E.J is just an indication of the (derivative of the) work delivered by the electromagnetic force on some charge, no matter how that charge got there

 

[Bill_K]

It doesn't go anywhere. The energy in a sinusoidal electromagnetic wave just happens to be zero some places and nonzero others. All of it travels forward with the speed of light.

You must be thinking that the first cycle of the wave somehow has the job of passing its energy back to the next cycle to maintain it, but that's not the case. Each cycle is independent of the others. You could have just one cycle! The amplitude is an arbitrary function, just as the amplitude of a sound wave is an arbitrary function or in any other solution of the wave equation. All that Maxwell's Equations tell you is that E and B are transverse, that B is perpendicular to E with amplitude related to dE/dt, and that the whole thing moves forward with velocity c.

 

[sophiecentaur]

The EM wave seems to be different from other waves. The energy seems to arrive in bursts, at the peaks of an EM wave because E and H fields are In Phase in the far field but, for sound etc, the Potential and Kinetic Energies are in quadrature and the energy in the individual oscillating parts is at a constant level.
Funny that, in and around a transmitting antenna, the KE and PE are in quadrature and there is a 'steady' flow of energy outwards- it just gets 'lumpy' as you get further away. I believe the Sums but I'd love a pictorial (yes - arm waving) explanation, too.

 

[sardar]

I can provide an explanation for the phenomenon you have described. In the case of an EM wave in vacuum, the energy does not actually "go" anywhere when the electric field is at 0. This is because the energy is not physically present in the electric field itself, but rather it is stored in the surrounding space. This is known as the energy density of the vacuum, and it is a fundamental concept in quantum field theory.

When the electric field is at 0, the energy is still present in the form of an oscillating magnetic field. As you mentioned, the electric and magnetic fields are in phase in vacuum, meaning that they reach their maximum and minimum values at the same time. So when the electric field is at 0, the magnetic field is also at 0, but this does not mean that the energy disappears. It is simply transferred between the electric and magnetic fields.

As for your second question, the current density in Poynting's theorem refers to the total current, both generated by the electric field and already present in the medium. This total current is what determines the rate of change of energy density in a given volume. So both the generated current and the pre-existing current contribute to the conservation of energy in an EM wave.

I hope this explanation helps to clarify your questions. It is important to remember that the behavior of EM waves in vacuum can be quite different from their behavior in a medium, where energy can be stored and dissipated in different ways. As scientists, we continue to study and understand these phenomena in order to further our knowledge of the universe.