I am refering to Ampere's force law, and to the beams accelerated after the cathode, so the deflection is not due to their respective cathode. In other words, do two electrons accelerating parallel to each other converge because of magnetic attraction?
Does it apply to wires only or to a beam of charge carriers too?
If coulomb interaction is stronger and make them diverge, is there at least a small attraction limiting it?
Electromagnetism – Is There Magnetic Attraction Between Two Parallel Electron Beams?
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Electromagnetism – How Does Special Relativity Explain the Attraction Between Two Parallel Currents?
As far as your comment goes, you mean there is an absolute symmetry between the 2 wires. Maybe, but one thing I must tell you that when you are considering the electrons in WIRE 1, the relativistic effects will be as follows:
- The electrons in their reference frame will consider the protons IN WIRE 2 to be in motion.
- Then due to relativistic length contraction, the electrons will observe that WIRE 2 has a higher positive charge density. So the electrons will face more Coulombic attraction from the WIRE 2 than repulsion from the electrons in that wire.
- Most importantly, where I think you are going wrong, the electrons in WIRE 1 will NEVER see the protons in WIRE 1 to be MORE as you say. That is, electrons in WIRE 1 will see that WIRE 2 has a higher positive charge density than WIRE 1 always due to special relativity and nothing else. On the other hand, in a similar fashion, electrons in WIRE 2 will see that WIRE 1 has a higher positive charge density than WIRE 2 always due to special relativity and nothing else.
Hope your doubt has been resolved.
The electron will accelerate, which, roughly speaking, will create an increasing magnetic field and therefore an electromagnetic wave.
As the electron is moving, its initial potential energy is being converted to the kinetic energy and some of it will be lost to the radiation.
Once the electron hits a positively charged plate (because that's where it would obviously end up), its kinetic energy will be transformed, roughly speaking, to heat.
Presumably, the electron was initially placed between the plates by some external force, so we can say that it got its initial potential energy (eventually lost to the radiation and heat), as it was moved to this position.
The change of the electrostatic energy of the system before the electron was moved to its initial position and after it has landed on the positive plate, depends on where this electron came from.
For instance, if the electron was taken from the positive plate, the potential energy was added to the system and then lost, so the final electrostatic energy of the system would be the same as it was before the whole experiment had started.
If the electron was taken from the negative plate, some energy was lost as the electron was moved into its initial position between the plates and some more energy was lost as the electron completed its trip to the positive plate, moving on its own. In this case the final electrostatic energy of the system would be lower than it was before the experiment. We can say that it was slightly discharged.
If the electron was moved in from the outside, the final electrostatic energy of the system would also be lowered because, at the end of the experiment, the charge on the positive plate will be reduced. The exact sequence of energy changes, in this case, depends on the initial position of the electron.
For illustration purposes, let's look at three basic subcases: A) initial position is in the middle of the gap B) initial position is quarter gap above the neutral line C) initial position is quarter gap below the neutral line
Let's assume that the voltage between the plates is 1V. So, it takes 1eV to move an electron from the positive plate to the negative or 1eV is released (to radiation and heat), when an electron moves from the negative plate to the positive.
The diagrams below show that the introduction of the electron into the system, could leave its energy unchanged (case A), increase it (case B) or decrease it (case C).
In case A, the electron is moved normal to the electric field lines and therefore no work is performed by the system or on the system and, therefore, the energy of the system does not change.
In case B, the electrons is moved into the field, i.e., it has to be pushed in by an external force, which performs the work, and therefore the energy of the system (which now includes the new electron) increases. Based on the position of the electron between the plates, quarter gap above the neutral line, the energy of the system has increased by 0.25eV.
In case C, the electron is moved against the field, i.e., it is pulled by the field, i.e., the system performs some work and, therefore the system loses some energy. Based on the position of the electron between the plates, quarter gap below the neutral line, the energy of the system has decreased by 0.25eV.
As illustrated in the diagrams, the energy loss of the system after the electron has been released depends on the initial position of the electron, but in the end, the total loss of energy by the system, in comparison with its energy before the electron has been introduced, is the same 0.5eV in all three cases and could be calculated based on the fact that the system ended up with one unit of charge less on the positive plate.
NOTE: In the case when the electron was taken from the negative plate and ended up on the positive plate, the charge on both plates was reduced by one unit or, assuming that the voltage in that case was 1V, the energy loss would be 1eV.
Best Answer
In the lab frame there is a magnetic attraction, but it will never overpower the Coulomb repulsion between the two beams.
This is easiest to see in a frame of reference which moves with the electrons themselves: there, the electrons are stationary, and the only force between them is the repulsive Coulomb force. That said, if the electrons are moving fast enough (and, since the problem is scale-free, any velocity is "fast enough"), special relativity will require some minor tweaks to how that repulsion is observed from the lab frame, because of effects coming from length contraction and time dilation.
In the lab frame, those relativistic corrections to the Coulomb repulsion can be interpreted as an additional force which is proportional to the velocities and to the charge of the electrons. This is what we know as the magnetic interaction between the two beams.
If you want to see this line of understanding in all its glory, I recommend the relativity-and-magnetism chapter ('The fields of moving charges') in Ed Purcell's Electricity and Magnetism.