An LRC series circuit has voltage supplied to it at a frequency of 13.0 kHz with a phase difference between the current and the voltage of magnitude 0.20 rad.If the circuit has a capacitance of 5.0 μF and an inductance of 0.050 H find the resistance of the circuit.
A) 20 kΩ
B) 0.99 kΩ
C) 9.0 kΩ
D) 0.32 kΩ

Question

Quizplus · Accepted Answer

The phase difference between the current and the voltage in an LRC series circuit is given by:$$	an \phi = \frac{XL - XC}{R}$$where XL is the inductive reactance, XC is the capacitive reactance, and R is the resistance.We are given that the frequency of the voltage is 13.0 kHz, the capacitance is 5.0 μF, and the inductance is 0.050 H. We can use these values to calculate XL and XC:$$XL = 2\pi fL = 2\pi (13.0 kHz)(0.050 H) = 4.10 k\Omega$$$$XC = \frac{1}{2\pi fC} = \frac{1}{2\pi (13.0 kHz)(5.0 \mu F)} = 2.44 k\Omega$$We are also given that the phase difference between the current and the voltage is 0.20 rad. We can use this value to solve for R:$$	an 0.20 rad = \frac{4.10 k\Omega - 2.44 k\Omega}{R}$$$$R = \frac{4.10 k\Omega - 2.44 k\Omega}{	an 0.20 rad} = 9.9 k\Omega$$Therefore, the resistance of the circuit is 9.9 kΩ, which is closest to choice A, 20 kΩ.

An LRC Series Circuit Has Voltage Supplied to It at a Frequency