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Chapter 30: Problem 62

A \(360-\mathrm{Hz}\) source of emf is connected in a circuit consisting of a capacitor, a \(25-\mathrm{mH}\) inductor, and an \(0.80-\Omega\) resistor. For the current and voltage to be in phase what should the value of \(C\) be?

Short Answer

Expert verified
Answer: The capacitance needed for the current and voltage to be in phase is approximately 6.19 * 10^-8 F (farads).

Step by step solution

01

Write down the known parameters.

The given parameters are: - Frequency (f): 360 Hz - Inductance (L): 25 mH = 25 x 10^-3 H - Resistance (R): 0.80 Ω We need to find the capacitance (C) for current and voltage to be in phase.
02

Convert the frequency to angular frequency.

To work with impedance, we need to convert the frequency to angular frequency (ω) using the formula: ω = 2πf Plugging the given frequency (f = 360 Hz) into the formula, we get: ω = 2π(360) ≈ 2261.95 rad/s
03

Write the condition for current and voltage to be in phase.

For the current and voltage to be in phase, the capacitive reactance (X_C) must equal the inductive reactance (X_L). Their expressions are: X_C = 1/(ωC) and X_L = ωL Setting them equal to each other: 1/(ωC) = ωL
04

Solve for capacitance C.

Now, we can solve for C in the equation: 1/(ωC) = ωL C = 1/(ω^2 * L) Substitute the values of ω and L: C = 1/[(2261.95)^2 * (25 * 10^-3)] C ≈ 6.19 * 10^-8 F The capacitance (C) required for the current and voltage to be in phase should be approximately 6.19 * 10^-8 F (farads).

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Most popular questions from this chapter

The phase constant, \(\phi\), between the voltage and the current in an AC circuit depends on the ______. a) inductive reactance b) capacitive reactance c) resistance d) all of the above

The transmission of electric power occurs at the highest possible voltage to reduce losses. By how much could the power loss be reduced by raising the voltage by a factor of \(10 ?\)

A series circuit contains a \(100.0-\Omega\) resistor, a \(0.500-\mathrm{H}\) inductor, a 0.400 - \(\mu\) F capacitor, and a time-varying source of emf providing \(40.0 \mathrm{~V}\). a) What is the resonant angular frequency of the circuit? b) What current will flow through the circuit at the resonant frequency?

Why is RMS power specified for an AC circuit, not average power?

The discussion of \(\mathrm{RL}, \mathrm{RC},\) and \(\mathrm{RLC}\) circuits in this chapter has assumed a purely resistive resistor, one whose inductance and capacitance are exactly zero. While the capacitance of a resistor can generally be neglected, inductance is an intrinsic part of the resistor. Indeed, one of the most widely used resistors, the wire-wound resistor, is nothing but a solenoid made of highly resistive wire. Suppose a wire-wound resistor of unknown resistance is connected to a DC power supply. At a voltage of \(V=10.0 \mathrm{~V}\) across the resistor, the current through the resistor is 1.00 A. Next, the same resistor is connected to an AC power source providing \(V_{\mathrm{rms}}=10.0 \mathrm{~V}\) at a variable frequency. When the frequency is \(20.0 \mathrm{kHz},\) a current, \(I_{\mathrm{rms}}=0.800 \mathrm{~A},\) is measured through the resistor. a) Calculate the resistance of the resistor. b) Calculate the inductive reactance of the resistor. c) Calculate the inductance of the resistor. d) Calculate the frequency of the AC power source at which the inductive reactance of the resistor exceeds its resistance.
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