OverviewThis paper proposes a photon/electron kinetic energy exchange as a "thought experiment". It shows that the exchange of kinetic energy may not be simply billiard-ball mechanics, but possibly a more subtle and interesting process. The mechanism involved is that where an electron’s magnetic dipole interacts with an external magnetic field. Magnetic interaction Consider an electron traveling across the page from left to right in Figure 1. It starts in free space from the left, enters a fixed magnetic field, then exits it again to the right. (In the normal course of events the electric charge of the electron will cause its path to curve out of the plane of the page as soon as it encounters the fixed field. For this analysis we are not interested in the interaction of the electron’s electric field and so may either ignore it or imagine a transverse electric field that exactly compensates for it.) Because the magnetic field of the electron is dipole in form (that is, has two magnetic poles, a North and a South) it is commonly referred to as a “magnetic dipole ”, or often simply just a “dipole”.
Figure 1. Electron dipole traveling through magnetic fieldThis leaves us with the interaction of the electron’s magnetic dipole and the fixed magnetic field. There are four important configurations, and for each assume that the electron enters the fixed from the left with an initial kinetic energy of Ekin:-
Magnetic configuration 1Consider first the situation where the electron enters and exits the fixed field with its magnetic dipole having North pointing up the page, in opposition to the fixed magnetic field. As it moves into the field from the left the electron’s North pole will approach the fixed North pole, and be repelled by it. Equally the electron’s South pole will approach the fixed South pole and be repelled by that too. Hence the electron as a whole will be repelled and slow down, losing kinetic energy. Where has this kinetic energy gone? The electron’s dipole, in opposition to the fixed field, is in a higher magnetic field energy configuration than the case where the dipole and fixed field are separated. So the lost kinetic energy has become magnetic potential energy. If we term the potential energy change +Epot then the residual kinetic energy is... Ekin - (+Epot) While in the fixed field the electron’s North pole is repelled equally to left and right by the fixed field all around it, and the same is true for the South pole. Hence there is no net repulsion or attraction while in the fixed field. But what happens when the electron leaves the fixed field to the right? Again the North pole of the electron repels the North pole of the fixed field, and the South repels the South, but this time the electron is trammeling away from the fixed field so the effect of the repulsion is to accelerate it - its kinetic energy increases. This increase in kinetic energy comes from the loss of the magnetic potential energy that was created when the electron first entered the fixed field. So the final kinetic energy is Ekin and the final potential energy change is zero. The net effect is that the electron slows down on approaching the fixed field, and accelerates away as it exits, recovering its original kinetic energy in full.
Magnetic configuration 2The next situation is where the electron’s North pole points down the page on entry and exit - that is, in alignment with the fixed field. As it moves into the field from the left the electron’s North pole will be attracted to the fixed field’s South pole, and its South pole to the field’s North pole. Hence the electron as a whole is attracted by the fixed field and accelerates into it, gaining kinetic energy. This kinetic energy comes from a drop in magnetic potential energy, because the configuration where the electron is in the fixed field and aligned with it is a lower total magnetic energy than that where the dipole and field are separated. With the potential energy change of -Epot, the new kinetic energy is now... Ekin - (-Epot) While in the fixed field the electron’s North pole is attracted equally to left and right by the fixed field all around it, and the same is true for the South pole. Hence there is no net repulsion or attraction while in the fixed field. On exit the electron’s North pole attracts the fixed field’s South pole and vice-versa, but the result now is to slow the electron down as it travels away from the fixed field, reducing the kinetic energy to its original level. The overall effect is that the electron accelerates on approaching the fixed field and slows down again to its original kinetic energy on exit.
Magnetic configuration 3This configuration is much more interesting. The electron’s North pole points up the page on entry and down the page on exit, and while in the magnetic field the electron radiates electromagnetic energy with no change in its kinetic energy. Identically to configuration 1, as the electron moves into the field from the left the its North pole will approach the fixed North pole, and be repelled by it. Equally the electron’s South pole will approach the fixed South pole and be repelled by that too. Hence the electron as a whole will be repelled and slow down, converting some of its kinetic energy into magnetic potential energy +Epot. Hence the kinetic energy is now... Ekin - (+Epot) ... just as with configuration 1. But here we consider something new. Immediately on entering the fixed field the electron’s North pole is repelled equally to left and right by the fixed field all around it, and the same is true for the South pole. There is no net left-to-right repulsion or attraction. However, one point not covered in the text of configuration 1 is that these repulsive forces tend to create a torque that will rotate the electron into alignment with the fixed field. If this happens the magnetic potential energy obviously changes from the +Epot of configuration 1 to -Epot of configuration 2. This lost magnetic potential energy simply goes into electromagnetic photon radiation to the tune of twice Epot, and does not affect the kinetic energy of the electron - it happens even if the electron is stationary in the fixed field. The effect is termed magnetic resonance. The exit for the electron is as in configuration 2. Kinetic energy equivalent to -Epot is lost. The sequence as the electron travels from left to right is shown below in Table 1...
Hence the net effect is first (table entries 1-2) for the electron to slow down on entry from the left losing kinetic energy to the tune of -Epot, then (table entries 2-3) to lose twice +Epot from the magnetic potential energy to photon energy while in the fixed field, and finally (table entries 3-4) to give up another -Epot kinetic energy to magnetic potential energy on exit. The reader should observe that the sum of every row in the table adds up to the same value, as required by the conservation of energy. Hence although the energy exchanges are done at different times and through the medium of an external fixed magnetic field the overall effect is to convert kinetic energy into radiated photon energy. This partially explains why free electrons do not interact with photons.
Magnetic configuration 4This is simply a reversal of configuration 3, where the electron is aligned with the fixed field on entry (North pole to South of field and vice-versa) and in opposition on exit, having been inverted while in the fixed field. The difference here is that the region of the fixed field is flooded with electromagnetic radiation of the correct frequency (the maths follows later). Just as the electron can decay from opposition to alignment, radiating twice Epot of energy, so it can be pumped up from alignment to opposition by an incoming photon of electromagnetic energy, and so we have the following sequence as the electron travels from left to right through the fixed field...
In this case the electron absorbs a photon and increases its kinetic energy as a result. Again, the change in kinetic energy occurs at a different time to the absorption of the photon by the intermediate magnetic potential energy change, but the end result is that the photon energy is converted to kinetic energy. The mathsThe exact energy involved is controlled by two parameters, namely the strength of the electron’s magnetic dipole and the strength of the fixed field. The fixed field is easy; it is defined in terms of vector field intensity B. In modern units this is in Tesla. The electron dipole field is usually given in terms of its vector magnetic moment energy, which is...
What this means is that in a fixed field of ‘B’ Tesla there is... 9.29E-24 .|B| ...joules of energy involved in rotating the dipole from perfect alignment or perfect opposition to the orthogonal orientation (this is the orientation where the electron’s dipole orientation and the fixed magnetic field orientation are at 90 degrees to each other and the torque is at a maximum- in Figure 1 the electron’s magnetic dipole would be horizontal). To move the dipole fully from alignment to opposition involves moving it from alignment to orthogonality, then from orthogonality to opposition, and so involves twice this value. This change in energy is the change in magnetic potential energy for the different orientations of the electron’s dipole in the fixed field. Hence the value of the magnetic potential energy Epot for the electron passing through the fixed field B in the previous text is just that...
The photon energy in configurations 2 and 3 is 2.Epot, and so the photon frequency f in Hertz is obtained by dividing this by Plank’s constant ‘h’...
Working this through gives the result of 48 MHz resonance at 1 meter radius, or (48/r) MHz at a radius of ‘r’. So an electron in orbit around such a central cylindrical field will absorb and radiate photon energy at the orbital frequency of (48/r) MHz. |
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