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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

A vacuum cleaner motor can be viewed as an inductor with an inductance of \(100 . \mathrm{mH} .\) For a \(60.0-\mathrm{Hz} \mathrm{AC}\) voltage, \(V_{\mathrm{rms}}=115 \mathrm{~V}\), what capacitance must be in series with the motor to maximize the power output of the vacuum cleaner?

What is the impedance of a series RLC circuit when the frequency of time- varying emf is set to the resonant frequency of the circuit?

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