<--------- charge --------->
< 15 cm
We feed mechanical energy to the device, and the device converts the energy into photons which can be observed far away.
In earlier blog posts we have asked the question where does the momentum go? The device moves at a speed considerably less than light. Its kinetic energy is converted into photons which can only carry away a fraction of the momentum lost by the device. Where is the momentum stored?
We earlier introduced the fishing float analogy for the charge. When we move the float, a complex pattern of water waves forms around the float. Waves reflect also back toward the float. The force which water exerts on the float at each moment is a very complex phenomenon. We should make a computer simulation to calculate the force.
The radio transmitter, similarly, has a "self-force" which the field of the charge exerts on the charge itself.
Now we see that some of the momentum which the device lost while accelerating forth, can be reabsorbed by the device when it moves back. The momentum was temporarily stored in the electromagnetic field of the moving charge.
In half a nanosecond, momentum and energy in the field can move at most 7.5 cm away, if they want to be reabsorbed by the device. Thus, it is the immediate vicinity of the antenna which stores the extra momentum.
The Larmor formula estimates the average energy flux from a charge doing a periodic motion. The Larmor formula has been tested empirically on radio transmitters.
If the motion of the charge is non-periodic, then the energy flux is hard to calculate and there is no reason to assume that the Larmor formula would be correct in such a case.
Quantization of the electromagnetic wave
Far away from our radio transmitter, the EM wave is very close to a plane wave. More precisely, it is a sum of two circularly polarized plane waves that rotate in opposite directions.
A plane wave is easy to quantize. We may imagine that it consists of an integer number of coherent photons. The Fourier decomposition stays almost constant. The photons are long-lived.
The waveform close to the transmitter is much more complex. It can be calculated using classical Maxwell's equations. Does the complex wave consist of photons and "virtual photons"?
We can, of course, calculate a Fourier decomposition for the complex waveform. Since the wave is interacting with the moving charge, the Fourier decomposition changes fast.
Can we meaningfully Fourier decompose the static electric field of the charge? If it is really static, the decomposition is strange. Maybe the decomposition makes sense if the charge moves?
We could interpret that the modes in the Fourier decomposition consist of photons or virtual photons which are being created and absorbed at a fast pace.
A spherical wave will always have some of the wave energy reflected back toward the center.
If a Fourier mode is a plane wave filling the entire space, then every mode is "approaching", as well as "receding" from the center.
We cannot describe the functioning of a radio transmitter just by photons moving away from the transmitter. There are always virtual photons which will be reabsorbed by the transmitter.
Where are the photons and virtual photons born in a radio transmitter?
Where are the photons or virtual photons "born" in space? The Coulomb electric field of the transmitter charge oscillates in a wide area of space. Are the quanta born at the charge? Or are they born farther away from the movements of its static electric field?
Let us consider a simple analogy. We have a drum skin which we disturb by pressing a finger agains it and moving the finger back and forth.
Where are the waves in the drum skin born? The energy comes through the finger.
Away from the finger, the drum skin obeys some kind of a partial differential equation.
As the finger moves back and forth, parts of the skin oscillate up and down. That is probably the main origin of the vertical waves in the skin.
Only part of the energy and the momentum which the finger inputs is carried away as vertical, ring-form waves. The rest is absorbed back into the finger in a backreaction.
Is there conversion from longitudinal waves to vertical waves? Suppose that the drum skin has zero friction.
All energy is at the lowest level transferred as horizontal stretching of the skin. We conjecture that we can ignore longitudinal waves.
Let us return to the radio transmitter.
1. The moving charge acts as a source in Maxwell's equations. It produces disturbance to the EM field.
2. The waves are born at the charge, but much of the energy and the momentum in the waves is reabsorbed back into the charge in a backreaction. Specifically, some of the energy in spherical waves is always reflected back.
3. Some of the waves escape as well-formed spherical waves.
4. Photons as well as virtual photons and longitudinal photons are born at the moving charge.
5. Virtual photons and longitudinal photons are quickly reabsorbed by the moving charge.
6. Virtual photons and longitudinal photons can store a considerable amount of momentum and return it back to the moving charge during the next half-cycle. The magnetic field of the moving charge resists changes in the speed of the charge. Most of the momentum is probably stored in that magnetic field and returned back to the charge when the charge changes the direction.
We defined above a virtual photon as a quantum of a wave which is quickly reabsorbed to the moving charge in the backreaction.
For a plane wave, we have a unique decomposition into photons: we just specify the wave and the number of photons.
Is there a similar decomposition into virtual photons?
In Feynman diagrams, virtual photons can carry any momentum and energy. They have a "continuous spectrum". That suggests that there is no unique decomposition.
However, the success of Feynman diagrams suggests that virtual photons are, indeed, emitted and absorbed as discrete quanta. If we take that literally, then the decomposition of an EM field is those virtual and real photons which have been emitted but not yet absorbed. We probably cannot measure what exact quanta are present, though.
Away from the finger, the drum skin obeys some kind of a partial differential equation.
As the finger moves back and forth, parts of the skin oscillate up and down. That is probably the main origin of the vertical waves in the skin.
Only part of the energy and the momentum which the finger inputs is carried away as vertical, ring-form waves. The rest is absorbed back into the finger in a backreaction.
Is there conversion from longitudinal waves to vertical waves? Suppose that the drum skin has zero friction.
All energy is at the lowest level transferred as horizontal stretching of the skin. We conjecture that we can ignore longitudinal waves.
Let us return to the radio transmitter.
1. The moving charge acts as a source in Maxwell's equations. It produces disturbance to the EM field.
2. The waves are born at the charge, but much of the energy and the momentum in the waves is reabsorbed back into the charge in a backreaction. Specifically, some of the energy in spherical waves is always reflected back.
3. Some of the waves escape as well-formed spherical waves.
4. Photons as well as virtual photons and longitudinal photons are born at the moving charge.
5. Virtual photons and longitudinal photons are quickly reabsorbed by the moving charge.
6. Virtual photons and longitudinal photons can store a considerable amount of momentum and return it back to the moving charge during the next half-cycle. The magnetic field of the moving charge resists changes in the speed of the charge. Most of the momentum is probably stored in that magnetic field and returned back to the charge when the charge changes the direction.
Is there a unique decomposition of an EM field into virtual and real photons?
We defined above a virtual photon as a quantum of a wave which is quickly reabsorbed to the moving charge in the backreaction.
For a plane wave, we have a unique decomposition into photons: we just specify the wave and the number of photons.
Is there a similar decomposition into virtual photons?
In Feynman diagrams, virtual photons can carry any momentum and energy. They have a "continuous spectrum". That suggests that there is no unique decomposition.
However, the success of Feynman diagrams suggests that virtual photons are, indeed, emitted and absorbed as discrete quanta. If we take that literally, then the decomposition of an EM field is those virtual and real photons which have been emitted but not yet absorbed. We probably cannot measure what exact quanta are present, though.
How can a moving charge know that the virtual photons that it emitted will be reabsorbed soon?
Let us consider an electromagnetic coil. It is well known that a current will create a magnetic field which, in turn, will keep the current running even if the voltage is switched off. There is considerable momentum stored in the field.
How does the coil know that it can store momentum into the magnetic field, and that the momentum will be reabsorbed? If we remove the coil, the momentum cannot linger in thin air.
The solution probably is that we cannot remove the charges which are present in the magnetic field, without making the charges to reabsorb the momentum in the magnetic field. The magnetic field of a moving charge resists changes in the motion of the charge. The momentum stored in the field will always be returned back to the charge.
In an earlier blog post we had the model where the field of a charge is an elastic object attached to the charge. Then it is clear that any momentum stored in the movements of the elastic object will always be returned back to the charge.
In an earlier blog post we had the model where the field of a charge is an elastic object attached to the charge. Then it is clear that any momentum stored in the movements of the elastic object will always be returned back to the charge.
Thus, the short-lived electromagnetic waves close to a moving charge are such that they inevitably will get reabsorbed if we try to remove the charge.
Summary
The nature of virtual photons was revealed, and we explained the functioning of a radio transmitter with the new concepts.
Virtual photons correspond to classical waves which the moving charge will reabsorb soon after they were emitted. The reabsorption is the backreaction of the charge on its own field. It is also called the "self-force" of the charge on itself.
The charge can only emit short-lived waves which certainly will get reabsorbed. Virtual photons cannot live for long.
The momentum stored in short-lived waves can be considerable. The stored momentum explains how a radio transmitter can convert kinetic energy of its charges to real photons, even though the real photons cannot carry away all the momentum.
Some of the waves which the moving charge emits will survive to a large distance. Those waves consist of real photons which the charge emitted. Far away, spherical waves look like plane waves, and there is negligible reflection back toward the source of the waves.
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