The power P in W supplied to a series combination of element

The power P (in W) supplied to a series combination of elements in an ac circuit Is P = VI cos theta, where V is the effective voltage, I is the effective current, and theta is the phase angle between the current and voltage. If V = 225 mV across an RLC circuit with a resistor (R = 41.0 ohm), an inductor (L = 42.9 mH), a capacitor (C = 86.9 muF) and a frequency of 57.0 Hz determine the power supplied to these elements P = mW (Type an integer or decimal rounded to two decimal places as needed)

Solution

cos@ = R / Z

where Z = sqrt[ R^2 + (XL - Xc)^2]

XL = 2 pi f L = 2 x pi x 57 x 0.0429 = 15.35 ohm

Xc = 1 / (2 pi f C) = 1 /(2pi * 57 * 86.9 * 10^-6) = 32.13 ohm

Z = sqrt[ 41^2 + (15.35 - 32.13)^2] = 44.3 ohm

V = 225 mV = 0.225 V

and I = V/Z

P = V I cos@ = (V) ( V/Z) ( R / Z ) = V^2 R / Z^2

P = (0.225^2)(41) / (44.3)^2 = 1.06 x 10^-3 W

P = 1.06 mW ......Ans