The two-capacitor problem with radiation

We discuss the two-capacitor problem found in many introductory physics texts in which there appears to be missing energy in an ideal, zero-resistance circuit, following the sudden charging of one capacitor from another. The paradox of this missing energy is traditionally ascribed to finite-resistance wires, the initial assumption of an ideal circuit and the rapid nature of the charging notwithstanding. By treating radiative effects in the simplest approximation, we show that the paradox is really nothing more than an inappropriately applied lumped-parameter model. In particular, we show that in the zero-resistance circuit, radiation fully accounts for all of the energy lost. To explore radiative effects in more realistic circuits, we also discuss numerical examples that include a small resistance and inductance. [DOI: 10.1119/1.1435344]