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Power

From the point of view of the power supply,^21.5the circuit is a "black box" that "resists" the applied voltage with a rather weird "back ${\cal{EMF}}$" ${\cal E}_{\rm back}$ given by $R_{\rm eff}$ times the current $I$; ${\cal E}_{\rm back}$ is given by the sum of both terms in Eq. (11) or the sum of the two vectors in Fig. 21.2. The power dissipated in the circuit is the product of the real part of the applied voltage^21.6 and the real part of the resultant current^21.7

$$P(t) = \Re {\cal E} \times \Re I = \Re \left( {\cal E}_0 e^{i \omega t} \right) \Re \left( I_0 e^{i \omega t} \right) \cr$$ (21.12)

which oscillates at a frequency $2 \omega$ between zero and its maximum value

$$P_{\rm max} = {\cal E}_0^2 \Re \left( {1 \over R_{\rm eff}} \right)$$ (21.13)

so that the average power drain is^21.8

$$\langle P \rangle = {1\over2} {\cal E}_0^2 \left[ R \over R^2 + X_C^2 \right] \; .$$ (21.14)

A little more algebra will yield the practical formula

$$\langle P \rangle = {\cal E}_{\rm rms} I_{\rm rms} \cos \phi$$ (21.15)

where ${\cal E}_{\rm rms} = {\cal E}_0/\sqrt{2}$, $I_{\rm rms}$ is the root-mean-square current in the circuit,

$$\cos \phi = {R \over Z}$$ (21.16)

is the "power factor" of the $RC$ circuit and

$$Z \equiv \sqrt{R^2 + X_C^2}$$ (21.17)

is the impedance of the circuit.^21.9

This gets a lot more interesting when we add the "inertial" effects of inductance to our circuit. Stay tuned.

Footnotes

. . . supply,^21.5

Please forgive my anthropomorphization of circuit elements; these metaphors help me remember their "behaviour".

. . . voltage^21.6

The imaginary voltage component doesn't generate any power.

. . . current^21.7

Neither does the imaginary part of the current.

. . . is^21.8

I have used ${\displaystyle {1 \over x + i y} = {x - i y \over x^2 + y^2} }$ to obtain the real part of $1/R_{\rm eff}$.

. . . circuit.^21.9

Expressing the average power dissipation in this form allows one to think of an AC $RC$ circuit the same way as a DC $RC$ circuit with the power factor as a sort of "fudge factor".