1. Power dissipation and reactance

To be specific, consider a particle of charge q subject to the electric field

E(t) = \frac{1}{2} (E e^{−iωt} + E^{∗} e^{iωt})

and assume that the particle is also subject to a harmonic force with damping, so that it obeys

m \frac{d^{2}x}{dt^{2}} = −mω_{0}^{2}x(t) − mγ \frac{dx}{dt} + qE(t)

The solution of this equation, apart from transients, is

x(t) = \frac{1}{2} (x e^{−iωt} + x^{∗} e^{iωt})

with

m (ω_{0}^{2} − ω^{2} − iγω) x = qE

The particle’s velocity is

v(t) = \frac{1}{2} (v e^{−iωt} + v^{∗} e^{iωt})

with

m (ω_{0}^{2} − ω^{2} − iγω) v = −iωqE

and the instantaneous power supplied by the field is

\frac{dW}{dt} = qE(t)v(t) = \frac{q}{4} (Ev e^{−2iωt} + Ev^{∗} + E^{∗}v + E^{∗}v^{∗}e^{2iωt})

Following standard notation we define the mechanical impedance

Z = m (γ − iω + iω_{0}^{2}/ω)

so that

v = \frac{qE}{Z} = \frac{m (γ + iω − iω_{0}^{2}/ω)}{|Z|^{2}} qE

and we rewrite dW/dt in terms of Z. We separate the time-averaged power

⟨\frac{dW}{dt} ⟩ = \frac{q}{4} (Ev^{∗} + E^{∗}v) = \frac{1}{2}q^{2} |E|^{2} \frac{mγ}{|Z|^{2}} = \frac{1}{2} mγ |v|^{2}  (1.1)

from the fluctuating part

\frac{dW}{dt} − ⟨\frac{dW}{dt} ⟩ = \frac{q}{4} (Ev e^{−2iωt} + E^{∗}v^{∗}e^{2iωt})

which has amplitude

\frac{1}{2} q |Ev| = \frac{1}{2 |Z|} q^{2} |E|^{2} = \frac{1}{2} |Z| |v|^{2}

The general meaning of Z can be deduced from the complex power

\frac{1}{2} qEv^{∗} = \frac{1}{2Z^{∗}} q^{2} |E|^{2} = \frac{1}{2} Z |v|^{2}

The real part of this is the average power (1.1). The imaginary part is

\frac{1}{2}m (−iω + iω_{0}^{2}/ω) |v|^{2} = \frac{1}{2}(−iω)m |v|^{2} − \frac{1}{2}(−iω)mω_{0}^{2} |\frac{v}{ω} |^{2}

Since the average kinetic energy is

\frac{1}{2}m ⟨v^{2}(t)⟩ = \frac{1}{4}m |v|^{2}

and the average potential energy is

\frac{1}{2}mω_{0}^{2} ⟨x^{2}(t)⟩ = \frac{1}{4}mω_{0}^{2} |x|^{2} = \frac{1}{4}mω_{0}^{2} |\frac{v}{ω} |^{2}

we see that the imaginary part of \frac{1}{2} qEv^{∗} is equal to

−2iω [\frac{1}{2}m ⟨v^{2}⟩ − \frac{1}{2}mω_{0}^{2} ⟨x^{2}⟩]

In summary, the formula

\frac{1}{2} qEv^{∗} = \frac{1}{2} Z |v|^{2} = \frac{1}{2} mγ |v|^{2} − 2iω [\frac{1}{2}m ⟨v^{2}⟩ − \frac{1}{2}mω_{0}^{2} ⟨x^{2}⟩]

tells us that the real part of Z (the resistance R) comes from the damping, while the imaginary part of Z (the negative of the reactance X) comes from the difference between the stored kinetic energy and the stored potential energy. In formulas, Z = R − iX, with X = ωL − 1/Cω for a circuit. Compare Jackson, page 244.