Why does Newton's Third Law work for fields?

Question

Newton's 3rd law goes like this: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

I find the law intuitive in some cases, for instance, when a moving particle collides with a stationary particle. Since momentum is conserved in the collision (assuming its elastic), some momentum of the moving particle is transferred to the stationary particle. As a result, the momentum of the moving particle decreases upon collision, an effect of the "equal and opposite force" exerted by the stationary particle.

But when it comes to gravitational (and electric) fields, Newton's Third Law seems to hold because of the equation $F=G\frac{m_1m_2}{r^2}$. Also the previous example wouldn't make sense because the total momentum/kinetic energy of objects in a field is always changing (not conserved). Edit: The momentum is conserved, because the system here includes the earth which also experiences a force of equal magnitude. the system should be a closed system for momentum to be conserved.

The justifications for Newton's Third Law seem to vary from case to case. Would there be any way to relate the two above cases and other cases?

Answer 1

Newton's third law doesn't work for fields. There isn't even any way to state it for fields. (It's stated for instantaneous action at a distance between material particles, not for systems that include fields carrying their own momentum.) Consider the situation shown below, with two positive charges moving with the velocities indicated by the arrows.

The electric forces are not equal and opposite, because the electric field of a moving charge is not spherically symmetric. The magnetic forces are not equal and opposite, because the magnetic force acting on 1 is zero, while the magnetic force acting on 2 is nonzero and upward. Momentum is still conserved, because momentum is being taken away from the system through electromagnetic radiation.

Re the question of why Newton's third law is true, this depends on what principles you consider to be more fundamental than Newton's third law. You can't prove theorems unless you pick axioms to start with. If you ask most physicists today, they would probably say that conservation of momentum follows from translation symmetry via Noether's theorem, and Newton's third law is the only obvious way of making up a physical law that enforces conservation of momentum, in the special case of a system of point particles interacting through instantaneous action at a distance.

Within the more restrictive scenario of Newtonian systems of particles, another good justification for Newton's third law is that you need it for conservation of momentum, and without conservation of momentum you don't get conservation of energy. This is because requiring conservation of energy to hold in all frames of reference, not just in one frame, is equivalent to conservation of momentum.

Answer 2

Newton's third law is always and everywhere true in classical physics as long as it is understood the right way, and that means locally, not action-at-a-distance. The law we are talking about states that forces appear in balanced pairs, the pair we sometimes call "action and reaction". Thus if an electromagnetic force on $A$ causes $A$ to accelerate, for example, then you should expect to find another physical thing, $B$, with an electromagnetic force on it, equal and opposite to the force on $A$. The question now arises, can this $B$ be distant from $A$? Relativity answers an unequivocal "no". The $B$ has to be right next to $A$, because momentum is conserved locally not just globally. So what is this $B$ in the case of an electron being accelerated in an electrostatic field, for example? Answer it is the field itself. When a particle is accelerated by a field, the field itself gains the equal and opposite momentum, and furthermore that momentum appears in each physical location in the field exactly where the particle picked up some momentum. It might seem odd at first to think of a "force acting on a field" but if you prefer you can simply say that the field is acquiring momentum.

When two particles attract one another with equal and opposite forces, the momentum appears locally in the field, such that when it is integrated over all locations there is no net momentum of field (in the chosen frame of reference) and therefore the particles must also be acquiring equal and opposite amounts of momentum. When two interacting particles have forces that are not balanced, as in the example with charged particles whose velocity vectors are at right angles, then the field acquires a net momentum. To repeat, Newton's third law applies perfectly well at each location as long as you apply it directly to the two interacting physical things at each location, i.e. particle and field.

Electromagnetism is the best example to pick for this, because there we can work out an exact relativistic treatment without needing the tools of general relativity. The technical detail is that the 4-divergence of the stress tensor of the field is equal to minus the force per unit volume on the local charged matter: this expresses conservation of energy and momentum together. It is indeed a beautiful aspect of the theory that it captures the notion of momentum conservation so elegantly. It is worth keeping hold of the terminology "law of physics" here (i.e. naming the observation with a name such as "Newton's third law") because it draws attention to a significant and notable fact, namely that force, i.e. that which causes increase of momentum, always appears in balanced pairs. I decided to write quite fully about this in the penultimate chapter of Relativity Made Relativity Easy should you wish to get on top of this at the level of undergraduate but not graduate physics.

Answer 3

...total momentum/kinetic energy of objects in a field is always changing (not conserved).

Momentum is a vector, so when two masses accelerate toward each other, the total momentum is still preserved - for the same reason it is preserved during mechanical collisions: forces of equal magnitude are acting on each body over the same time interval.

Kinetic energy does not have to be preserved. In this example, potential energy of the two bodies is converted to their kinetic energies and the total energy is preserved.