Quantum Theory of a Damped Electrical Oscillator and Noise. II. The Radiation Resistance

The results of a previous paper are extended to include damping and noise due to a radiation resistance. The average electromagnetic field energy of an oscillator of natural frequency ω, with inductance L, coupled to a radiation resistance R, as a function of time t is given by $U=[1/2{}{ω}+{{}{ω}}{exp({{}{ω}}{kT}){-}1}][1{-}{e}^{{-}{(R/L)}ᵗ}]+{U}₀{e}^{{-}{(R/L)}ᵗ}.$ Using first-order perturbation theory and quantum statistics, an expression is derived for the radiation resistance of a circuit. This agrees exactly with the classical value.For an oscillator damped by a radiation resistance the vacuum fluctuation noise, and available noise power are shown to be the same as for an oscillator damped by a lumped resistance. All of the previous results are shown to be applicable to cavity resonators.