Energy transfer and conservation cases for pendulum motion and EM wave

[anuttarasammyak]

Let me ask a very primitive question.

To and fro motion of pendulum under gravity tells us
potential energy + kinetic energy = const.
At the top points potential energy: max kinetic energy :0
At the bottom point potential energy: 0 kinetic energy :max

EM wave is usually illustrated as https://en.wikipedia.org/wiki/Electromagnetic_radiation#/media/File:Onde_electromagnetique.svg .
As for electric field energy $E^2$ and magnetic field energy $B^2$
At the antinodes $E^2$: max $B^2$:max
At the nodes $E^2$: 0 $B^2$: 0

How do I reconcile the difference of energy transfer / conservation of the two ?

 

[Cthugha]

I am not sure I get your point. The emission of light maps pretty well to the pendulum motion. Consider a two-level system such as an atom as the emitter and the light field into which it emits. Assuming there is just one excitation present, you will find that the probability amplitude to find the excitation in the atom or in the light field oscillates back and forth initially.
So you get the following equivalents:
pendulum at the top point, maximal potential energy, no kinetic energy corresponds to: atom in the excited state, no light in the light field
pendulum at the bottom point, no potential energy, maximal kinetic energy corresponds to: atom in the ground state, one photon in the light field.

Just to make sure we are not talking about different things as people often get this wrong: The energy in an electromagnetic wave is not periodically exchanged between the electric and the magnetic field and the local energy at each point in space along the path of the light beam is not the same. The total energy consisting of the energy stored in the light field at some point and the energy of the emitter at the time that this part of the light field was emitted, is conserved. The energy flux of the light field is also conserved i. So all energy entering a region will leave it again.

 

[vanhees71]

The energy density (!) of the electromagnetic field is (in Heaviside-Lorentz units)
$$u_{\text{em}}=\frac{1}{2} (\vec{E}^2+\vec{B}^2).$$
What's conserved is the total energy. This you cannot discuss on the example of a plane-wave solution of Maxwell's equation, becaus this is not a solution that can be strictly realized in nature, because it has an infinite energy. You rather have to consider a wave packet with finite energy. Then you'd see that the total energy of the free electromagnetic field is conserved (Poynting's theorem).

 

[anuttarasammyak]

So all energy entering a region will leave it again.

With your instruction now I distinguish the two cases :
Pendulum motion is a case of energy exchange between the two STOCKS (K.E and P.E.)
EM wave is a case of energy FLOW as you said above. I made a hand drawing to show my understanding.

Thanks.

 

[Dale]

Summary:: Energy transfer and conservation cases for pendulum motion and EM wave, how they are same and different

How do I reconcile the difference of energy transfer / conservation of the two ?

I don’t understand. Why should there be any relationship between a pendulum and light? Can you formulate your question about without light without the pendulum?

 

[anuttarasammyak]

I retell my confusion in simpler case.

Energy of a harmonic oscillator
$$E_{ho}=\frac{1}{2}mv^2+\frac{1}{2}kx^2$$
and

The energy density (!) of the electromagnetic field is (in Heaviside-Lorentz units)
$$u_{em}=\frac{1}{2}(E^2+B^2).$$
What's conserved is the total energy.

have similar mathematical form of $A^2+B^2$. However
$E_{ho}$ = time const. but $u_{em}$ of EM wave varies 0 < $u_{em}$ < max. depending on time.

I wrongly thought these two phenomena should behave similarly. Sorry for the confusion.

 

[Dale]

I retell my confusion in simpler case.

Energy of a harmonic oscillator
$$E_{ho}=\frac{1}{2}mv^2+\frac{1}{2}kx^2$$
and

have similar mathematical form of $A^2+B^2$. However
$E_{ho}$ = time const. but $u_{em}$ of EM wave varies 0 < $u_{em}$ < max. depending on time.

I wrongly thought these two phenomena should behave similarly. Sorry for the confusion.

Thanks for explaining. That makes more sense now.

Note that they also have the same form as the Pythagorean theorem and I am sure many other similarly disconnected formulas. Merely having the same form does not always provide a clean analogy.

In this case one big difference is that $E_{ho}$ is the total energy, which is conserved, but $u_{em}$ is the energy density which is not conserved by itself.

 

[kuruman]

n this case one big difference is that $E_{ho}$ is the total energy, which is conserved, but $u_{em}$ is the energy density which is not conserved by itself.

To elaborate on that, I add that this stems from the difference in the sinusoidals that enter the two expressions. In the case of the pendulum, $\theta(t)$ and $\dot \theta(t)$ are multiplied by sinusoidals with a phase difference of $\pi/2$; adding their squares gives a constant at any time time $t$ which is the mechanical energy that stays in the pendulum and changes form from potential to kinetic back and forth. In the case of the EM wave, the sinusoidals describing $E(t)$ and $B(t)$ are in phase; adding their squares gives another function of time which is time-averaged over a cycle to get the energy that is transferred by the wave across a surface without changing its form.

 

[anuttarasammyak]

My confusion of harmonic oscillator and EM wave told in post #6 might be back when EM wave is quantized to a set of harmonic oscillators. I am still puzzled and do not have clear distinguished images of these two kind of harmonic oscillators.
$$\frac{1}{2}E^2+\frac{1}{2}B^2$$ varies with time and coordinate, right ?
$$\hbar \omega$$ is conserved constant energy ? They are not equal, right ?
I should appreciate your teachings.

 

[vanhees71]

Thanks for explaining. That makes more sense now.

Note that they also have the same form as the Pythagorean theorem and I am sure many other similarly disconnected formulas. Merely having the same form does not always provide a clean analogy.

In this case one big difference is that $E_{ho}$ is the total energy, which is conserved, but $u_{em}$ is the energy density which is not conserved by itself.

I think the intuition about the free em. fields and the harmonic oscillator is not so wrong. This becomes more obvious in the canonical formalism for fields, where you indeed can write the Hamiltonian of the free fields in a form which is precisely analogous to the harmonic oscillator. For a very nice discussion of this feature, which is very important for the canonical quantization of the electromagnetic field (using the complete gauge-fixing of the potentials for the free fields, the socalled radiation gauge, where the scalar potential $\Phi=0$ and $\vec{A}$ is purely transverse, i.e., fulfilling the Coulomb-gauge condition $\vec{\nabla} \cdot \vec{A}=0$), see Landau&Lifschitz vol. 4.

 

[anuttarasammyak]

Re:post #9 I attached my preliminary understanding of harmonic oscillation interpretation of EM field. Any questions, corrections and teachings are appreciated.

Where the harmonic oscillators are ? In k-space. In each k cell lies an oscillator.
What vibrates ? Fourier transform of vector potential $A_k$. Re$A_k$and Im$A_k$ interchange with angular velocity $\omega=ck$. P.E. & K.E. , total energy correspond to $(Re\ A_k)^2$ & $(Im\ A_k)^2$, $|A_k|^2$.