Which forces violate Newton's third law of motion?

Question

Forces arising from magnetic fields do violate Newton's third law of motion under certain circumstances. What other forces violate the third law?

Answer 1

Anytime you hear that a force "violates Newton's Third Law," what that really means is that the action-reaction pair for that force has been misidentified. In the case of electromagnetic forces (for example,the case of two moving charges), it is incorrect to say that pairs of charges form action-reaction pairs. Rather, the action-reaction pairs are formed between each charge and the electromagnetic field itself. In electromagnetism, charges don't directly interact with other charges; rather, they interact with the electromagnetic field at their location. In other words, every action by the field on the charge $\mathbf{F}_{fq}$ has an equal and opposite reaction on the field by the charge $\mathbf{F}_{qf}$. Put formally, Newton's Third Law says that

$$\mathbf{F}_{fq}=-\mathbf{F}_{qf}$$

where

$$\mathbf{F}_{fq}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$$

and

$$\mathbf{F}_{qf}=\int \frac{d\mathbf{g}}{dt} dV=\int\frac{d}{dt}(\epsilon_0\mathbf{E}\times\mathbf{B})dV$$

where the integral is over all space. As you can see, the reason this works is because the electromagnetic field itself carries momentum density $\mathbf{g}$. Exerting a force on the field is equivalent to changing its total momentum.

Newton's Third Law is intimately related with conservation of momentum, and it is never violated as long as the interacting objects are correctly identified. If you incorrectly assume that two objects form an action-reaction pair (i.e. directly exert forces on each other), then Newton's Third Law may appear to be violated, but this is because the law has been misapplied.

Answer 2

Any force which has a finite speed of propagation "violates Newton's Third Law" in this sense, at least momentarily. Imagine two particles $A$ and $B$ some large distance apart along the $x$-axis, exerting a central force on each other. Since there is a finite speed of propagation of the force, each particle feels a force according to the position of the other particle some time ago; if we imagine that these particles have been at rest for a long time, though, then their positions "some time ago" are the same as their current positions.

Now move particle $A$ a small distance off the $x$-axis. When $A$ moves off-axis it will feel a force from particle $B$, which is still on the $x$-axis; and since the force is central, this means that $A$ will feel a force that is not aligned in the $x$-direction. However, until the "news" of particle $A$'s displacement reaches $B$, $B$ will still feel a force pointing towards the former location of $A$, and this force will point along the $x$-axis. Since the forces on $A$ and $B$ do not point in the same direction, they can't possible be equal and opposite.

This discrepancy is usually addressed by taking into account the momentum of the field that mediates the force between $A$ and $B$. When we move $A$, it creates waves in this field, and these waves carry momentum. Assuming that the laws governing this field do not have any explicit dependence on position, it can be shown that the momentum of the particle plus the momentum of the waves obeys $\vec{F}_\text{tot} = d\vec{p}/dt$. Since Newton's Third Law is equivalent to saying that momentum is conserved in closed systems, it is rescued once we take the field momentum into account.