  The subject of radiation reaction in classical electromagnetism remains
controversial over 120 years after the pioneering work of Lorentz. We give a
simple but rigorous treatment of the subject at the textbook level that
explains the apparent paradoxes that are much discussed in the literature on
the subject. We first derive the equation of motion of a charged particle from
conservation of energy and momentum, which includes the self-force term. We
then show that this theory is unstable if charged particles are pointlike: the
energy is unbounded from below, and charged particles self-accelerate
(`over-react') due to their negative `bare' mass. This theory clearly does not
describe our world, but we show that these instabilities are absent if the
particle has a finite size larger than its classical radius. For such
finite-size charged particles, the effects of radiation reaction can be
computed in a systematic expansion in the size of the particle. The leading
term in this expansion is the reduced-order Abraham-Lorentz equation of motion,
which has no stability problems. We also discuss the apparent paradox that a
particle with constant acceleration radiates, but does not suffer radiation
reaction (`under-reaction'). Along the way, we introduce the ideas of
renormalization and effective theories, which are important in many areas of
modern theoretical physics. We hope that this will be a useful addition to the
literature that will remove some of the air of mystery and paradox surrounding
the subject.
