A 120-V rms voltage at 1000 Hz is applied to a 2.00-mH inductor, a 4.00-μF capacitor and a resistor. If the rms value of the current in this circuit is 0.400 A, what is the value of the resistor?
A) 299 Ω
B) 95.1 Ω
C) 423 Ω
D) 120 Ω
E) 240 Ω

Question

Quizplus · Accepted Answer

Explanation:The impedance of the circuit is:$$Z = \sqrt{R^2 + (XL - XC)^2}$$where:* R is the resistance* XL is the inductive reactance* XC is the capacitive reactanceThe inductive reactance is:$$XL = 2πfL = 2π(1000 Hz)(2.00 mH) = 12.57 Ω$$The capacitive reactance is:$$XC = \frac{1}{2πfC} = \frac{1}{2π(1000 Hz)(4.00 μF)} = 39.79 Ω$$Substituting these values into the equation for impedance, we get:$$Z = \sqrt{R^2 + (12.57 Ω - 39.79 Ω)^2} = \sqrt{R^2 + 706.8 Ω^2}$$The rms current is:$$I = \frac{V}{Z} = \frac{120 V}{\sqrt{R^2 + 706.8 Ω^2}}$$Substituting the given value of the rms current, we get:$$0.400 A = \frac{120 V}{\sqrt{R^2 + 706.8 Ω^2}}$$Solving for R, we get:$$R = 299 Ω$$

A 120-V Rms Voltage at 1000 Hz Is Applied to a 2.00-MH