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Chapter 21: Problem 53

A series \(R L C\) circuit has \(R=500.0 \Omega, L=35.0 \mathrm{mH},\) and $C=87.0 \mathrm{pF} .$ What is the impedance of the circuit at resonance? Explain.

Short Answer

Expert verified
In an RLC circuit with a resistance of 500.0 Ω, an inductance of 35 mH, and a capacitance of 87 pF, the impedance at the resonant frequency is purely resistive and equal to the resistance value, which is 500.0 Ω.

Step by step solution

01

Calculate the resonant frequency

To find the resonant frequency, we use the formula: $$f_{res} = \frac{1}{2\pi\sqrt{LC}}$$ where L is the inductance and C is the capacitance.
02

Plug in given values and solve

In our case, \(L=35\times 10^{-3} H\) and \(C=87\times 10^{-12} F\). Substituting these values in the formula, we get $$f_{res} = \frac{1}{2\pi\sqrt{(35\times 10^{-3})(87\times 10^{-12})}}$$ Now, calculate the value of \(f_{res}\).
03

Determine inductive and capacitive reactance at resonance

At resonance, inductive reactance (XL) and capacitive reactance (XC) are equal. $$X_{L} = X_{C}$$ $$2\pi f_{res} L = \frac{1}{2\pi f_{res} C}$$
04

Calculate the impedance of the circuit at resonance

At resonance, the impedance is purely resistive and equal to the resistance value. So, the impedance at resonance, Z, is given by the resistance value. $$Z = R = 500.0 \Omega$$ The impedance of the RLC circuit at resonance is 500.0 \(\Omega\).

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Most popular questions from this chapter

The field coils used in an ac motor are designed to have a resistance of $0.45 \Omega\( and an impedance of \)35.0 \Omega$ What inductance is required if the frequency of the ac source is (a) \(60.0 \mathrm{Hz} ?\) and (b) $0.20 \mathrm{kHz} ?$
An \(R L C\) series circuit has a resistance of \(R=325 \Omega\) an inductance \(\quad L=0.300 \mathrm{mH}, \quad\) and \(\quad\) a capacitance $C=33.0 \mathrm{nF} .$ (a) What is the resonant frequency? (b) If the capacitor breaks down for peak voltages in excess of \(7.0 \times 10^{2} \mathrm{V},\) what is the maximum source voltage amplitude when the circuit is operated at the resonant frequency?
A coil with an internal resistance of \(120 \Omega\) and inductance of $12.0 \mathrm{H}\( is connected to a \)60.0-\mathrm{Hz}, 110-\mathrm{V}$ ms line. (a) What is the impedance of the coil? (b) Calculate the current in the coil.
A \(300.0-\Omega\) resistor and a 2.5 - \(\mu\) F capacitor are connected in series across the terminals of a sinusoidal emf with a frequency of $159 \mathrm{Hz}$. The inductance of the circuit is negligible. What is the impedance of the circuit?
The circuit shown has a source voltage of \(440 \mathrm{V}\) ms, resistance \(R=250 \Omega\), inductance \(L=0.800 \mathrm{H},\) and capacitance $C=2.22 \mu \mathrm{F} .\( (a) Find the angular frequency \)\omega_{0}$ for resonance in this circuit. (b) Draw a phasor diagram for the circuit at resonance. (c) Find these rms voltages measured between various points in the circuit: \(V_{a b}, V_{b c}, V_{c d}, V_{b d},\) and \(V_{c d},\) (d) The resistor is replaced with one of \(R=125 \Omega\). Now what is the angular frequency for resonance? (e) What is the rms current in the circuit operated at resonance with the new resistor?
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