I was recently told by a physics teacher that "any system of charges in which at least some of the charges are executing some sort of accelerated motion, will radiate and lose energy". This refers to classical electrodynamics (I'm not entering into quantum mechanics), that is, this statement should either be provable or disprovable from Maxwell's equations. I tried to think of a simple system of charges that would disproof this statement, since I intuitively think that it isn't true, but I haven't come up with any. The one example I could think of was a positive charge sitting on top of a negative charge, so that the net charge is zero. No matter how they move it is obvious that there will be no radiation. But my teachers did not accept this trivial example.
So my question is whether the statement is true or not, plus a proof or a counterexample of some charges executing accelerated motion without radiating.