When a series of capacitors is not just capacitors in series

Abstract

We explore the problem of two arbitrarily charged capacitors disposed in series with a resistor. It is well known that if their initial charges were zero, then the capacitors would behave physically as a single capacitor with capacitance given by a very usual formula. We consider the case for non-zero initial charges, where not only the formula breaks down, but also the idea of 'equivalent capacitor' is meaningless. Then, what kind of thing could be considered equivalent to two arbitrarily charged capacitors in series? In the present work, we answer that question and find that this equivalent thing is a capacitor together with a constant voltage source. This weird answer opens the possibility of two interpretations that seem to be in contradiction. We then solve this paradox showing that it is just an apparent ambiguity.