Is Newton's third law getting violated here?

[KnightTheConqueror]

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?

 

[Ibix]

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.

 

[Dale]

So, if you limit Newton’s 3 rd law to only refer to forces on matter, yes, this is a violation. If you generalize N3 to include momentum transferred to fields then it is not a violation.

 

[KnightTheConqueror]

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.

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

 

[Dale]

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

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.

 

[Mister T]

"The point is that the third law does not always hold, and this is why modern physics has given primacy to conservation of momentum in the hierarchy of physical law."

-- Arnold B. Arons, Teaching Introductory Physics, 1997, John Wiley & Sons

 

[Orodruin]

Coincidentally, this is the topic of the lecture I will give in my special relativity class today.

Ultimately, just considering local conservation of energy and momentum means you really cannot have action at a distance compatible with special relativity as it would violate causality. Any force produced between two charged particles must be mediated by the electromagnetic field carrying momentum and energy. In particular, the Poynting vector

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.

represents the momentum density of the field, but also the energy current. The Maxwell stress tensor represents the momentum current.

Integrating the currents over a closed surface will give the total energy/momentum flowing out of (or into depending on normal direction) the enclosed volume. All forces become local and causality is fine again. We also obtain the relativistic analogy of Newton’s third law, $\partial_\mu T^{\mu\nu}_{\rm tot} = 0$.

 

[tech99]

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?

The two charges are accelerating so I expect to see EM radiation, which has momentum.

 

[PeroK]

The two charges are accelerating so I expect to see EM radiation, which has momentum.

That's not the fundamental issue. You could have a fixed charge and a steady current in a wire.

 

[Orodruin]

That's not the fundamental issue. You could have a fixed charge and a steady current in a wire.

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.

 

[Dale]

Or just some stationary charges.

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]

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.

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]

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.

 

[Orodruin]

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.

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]

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.

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.