Speed of Light in Vacuum in the Case of a Lumped Electric Circuit

Abstract

Further to previous work by the author which claimed an infinite speed of light in vacuum in a problem of electromagnetism, in this paper the same result of an infinite speed of light in vacuum is reported in a lumped element electric circuit. A circuit consisting of resistors, capacitors and an independent voltage source with specific initial conditions is considered in a laboratory frame $K$ where the circuit is at rest. The relativistic transformation of the circuit is obtained when it is observed from an inertial frame $K^{\prime}$ . The circuit is analyzed by the Laplace transform technique. The fact that the electric charge on a capacitor has continuous time derivatives at the origin of the time axis leads to the conclusion that the Lorentz factor $\gamma=1/\sqrt{1-v^{2}/c^{2}}$ equals 1 for any value of $v$ , the speed of $K^{\prime}$ with respect to $K$ . This means that $c$ , the speed of light in vacuum, is infinite. A heuristic argument is exploited to claim that the circuit can be considered lumped when observed from both $K$ and $K^{\prime}$ . Also it is shown that the convection current densities due to charges on capacitor plates observed from $K^{\prime}$ have no influence on the Kirchhoff's voltage and current law equations so that the infinite speed of light in vacuum result remains applicable. The result of this paper implies that the instantaneous action-at-a-distance effect can be observed for the system under study, i.e. even though currents and voltages of the circuit in frame $K$ are functions of time, these functions can instantaneously be measured from $K^{\prime}$ , because $t=t^{\prime}$ .