# Scenario where Newton's 3rd Law breaks down

In Kleppner and Kolenkow's Introduction to Classical Mechanics, it reads,

One system that is particularly troublesome for our present formulation of Newtonian mechanics is the electromagnetic field. Paradoxes can arise when such a field is present. For instance, two charged bodies which interact electrically actually interact via the electric fields they cre- ate. The interaction is not instantaneously transmitted from one particle to the other but propagates at the speed of light. During the propagation time there is an apparent breakdown of Newton’s third law; the forces on the particles are not equal and opposite.

In Coulomb's Law, the two forces are exactly equal and opposite - what forces is this apparent paradox referring to?

Here's the apparent paradox: Imagine two charges, $A$ and $B$ at rest. No problem with Newton's Third Law; both feel the same force magnitude. Now move $A$ radially away from $B$. The force that $A$ feels is smaller because it has moved to a region of space where the electric field due to $B$ is smaller. But $B$ doesn't yet know that $A$ has moved because of propagation delay. The field at $B$ is the same as it was before $A$ moved, so it feels an unchanged force.