The discussion centers on the apparent violation of Newton's third law in the context of electromagnetic interactions between moving and stationary charges. It is established that while electrostatic forces are equal and opposite, the presence of magnetic forces and the momentum carried by the electromagnetic field complicates the application of Newton's third law. The conservation of momentum remains intact when accounting for the electromagnetic field's momentum, as detailed by the momentum density formula $$\frac{\vec S}{c^2}=\frac{1}{c^2}\vec E \times \vec H$$. This indicates that the electromagnetic field can carry momentum, thus preserving the fundamental laws of physics.
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How exactly is momentum transferred to fields? I know electromagnetic waves carry momentum, but in this situation I don't understand what exactly is happening. Is something sort of wave being created? If yes then how to we mathematically derive the waveIbix said:The electromagnetic field can carry momentum, and in general you need to account for that too. If you do that correctly (which is often non-trivial maths) the conservation of momentum holds.
The momentum density of the EM field is given by $$\frac{\vec S}{c^2}=\frac{1}{c^2}\vec E \times \vec H$$ The field doesn’t need to be a wave.KnightTheConqueror said:How exactly is momentum transferred to fields? I know electromagnetic waves carry momentum, but in this situation I don't understand what exactly is happening. Is something sort of wave being created? If yes then how to we mathematically derive the wave
represents the momentum density of the field, but also the energy current. The Maxwell stress tensor represents the momentum current.Dale said:The momentum density of the EM field is given by $$\frac{\vec S}{c^2}=\frac{1}{c^2}\vec E \times \vec H$$ The field doesn’t need to be a wave.
The two charges are accelerating so I expect to see EM radiation, which has momentum.KnightTheConqueror said:Suppose a charge is moving towards another charge at rest. At a given instant of time, The electrostatic force applied by either charge on the other is same, but only one is applying magnetic force on the other. Doesn't this violate Newton's third law?
That's not the fundamental issue. You could have a fixed charge and a steady current in a wire.tech99 said:The two charges are accelerating so I expect to see EM radiation, which has momentum.
Or just some stationary charges. Even if momentum density in the field is zero, the momentum current will not be. Computing the force between two charged particles by integrating the Maxwell stress tensor across a plane in between them is an exercise in my relativity lecture notes. Obviously the result comes out to the expected Coulomb law, but it is instructive for working with the Maxwell tensor and for getting a feeling for how the EM field carries momentum and energy.PeroK said:That's not the fundamental issue. You could have a fixed charge and a steady current in a wire.
Well, if you only have stationary charges then there isn’t a concern about the momentum of the charges. The forces on two stationary charges does follow N3.Orodruin said:Or just some stationary charges.
Sure there is a concern about the momentum. There are additional forces acting on the charges to keep them from moving (there must be or they would accelerate). Those forces by themselves represent momentum being added to the charges and exactly balance out the momentum added by the field interaction. No, these effects are not a Newton 3 pair, but the forces on the charges from the field and the force on the field from the charge are.Dale said:Well, if you only have stationary charges then there isn’t a concern about the momentum of the charges. The forces on two stationary charges does follow N3.
Electrostatics is a limit of electromagnetism and in electromagnetism the fields do need to be assigned momentum and energy. If you compute the corresponding stress-energy of the fields, it is non-zero. Hence, there is a computable energy and stress also in electrostatics. You may not need it for the static situation, but it is perfectly compatible with it.Dale said:Newtonian mechanics doesn’t forbid action at a distance. So because the electrostatic forces are equal and opposite there is no need to invoke field momentum or momentum flux in electrostatics. Yes, it is there in electromagnetism, but it isn’t necessary for the conservation of momentum until one of the charges is moving.
Yes. I agree completely. And the field momentum flux is only not needed for the static situation in the sense that Newtons laws don’t break without it.Orodruin said:Electrostatics is a limit of electromagnetism and in electromagnetism the fields do need to be assigned momentum and energy. If you compute the corresponding stress-energy of the fields, it is non-zero. Hence, there is a computable energy and stress also in electrostatics. You may not need it for the static situation, but it is perfectly compatible with it.
I recommend to take a look at Feynman's discussion-10.5 about conservation of momentum in case of electromagnetic field.KnightTheConqueror said:Suppose a charge is moving towards another charge at rest. At a given instant of time, The electrostatic force applied by either charge on the other is same, but only one is applying magnetic force on the other. Doesn't this violate Newton's third law?