Chapter 31, Problem 31.49P

An L-R-C series circuit is connected to an ac source of constant voltage amplitude V and variable angular frequency ω.

1. (a) Show that the current amplitude, as a function of ω, is

$I=\frac{V}{\sqrt{R^2+{(\omega L-\text{1/}\omega C)}^2}}$

1. (b) Show that the average power dissipated in the resistor is

$P=\frac{V^{2}R\text{/2}}{R^2+{(\omega L-\text{1/}\omega C)}^2}$

1. (c) Show that I and P are both maximum when $\omega =\text{1/}\sqrt{LC}$, the resonance frequency of the circuit, (d) Graph P as a function of ω for V = 100 V, R = 200 Ω, L = 2.0 H, and C = 0.50 μF. Compare to the light purple curve in Fig. 31.19. Discuss the behavior of I and P in the limits ω = 0 and ω → ∞.