A 25.0-mH inductor, a 2.00- $\mu \mathrm { F }$ capacitor, and a certain resistor are connected in series across an ac voltage source at $1.00 \mathrm { kHz }$. If the impedance of this circuit is $200 \Omega$, what is the value of the resistor?
A) $100 \Omega$
B) $552 \Omega$
C) $200 \Omega$
D) $579 \Omega$
E) $184 \Omega$

Question

Quizplus · Accepted Answer

The impedance $Z$ in a series RLC circuit is given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$, where $X_L = 2\pi fL$ is the inductive reactance, $X_C = \frac{1}{2\pi fC}$ is the capacitive reactance, and $R$ is the resistance. Given $f = 1.00 \times 10^3 \, \text{Hz}$, $L = 25.0 \times 10^{-3} \, \text{H}$, and $C = 2.00 \times 10^{-6} \, \text{F}$, we find $X_L = 2\pi(1.00 \times 10^3)(25.0 \times 10^{-3}) = 157 \, \Omega$ and $X_C = \frac{1}{2\pi(1.00 \times 10^3)(2.00 \times 10^{-6})} = 79.6 \, \Omega$. The impedance is given as $200 \, \Omega$, so solving for $R$ in the equation $200 = \sqrt{R^2 + (157 - 79.6)^2}$ yields $R \approx 184 \, \Omega$.

A 250-MH Inductor, a 2 μF\mu \mathrm { F }μF Capacitor, and a Certain Resistor Are Connected in Series Across

A 250-MH Inductor, a 2 $\mu \mathrm { F }$ Capacitor, and a Certain Resistor Are Connected in Series Across