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Chapter 21: Problem 37

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?

Short Answer

Expert verified
Answer: The impedance of the series RC circuit is approximately 517.44 Ohms.

Step by step solution

01

Find the reactance of the capacitor

To find the reactance of the capacitor (\(X_C\)), we can use the given values for frequency (\(f = 159 \mathrm{Hz}\)) and capacitance (\(C = 2.5 \times 10^{-6} \mathrm{F}\)) and the equation for reactance: $$ X_C = \frac{1}{2 \pi f C} $$ Plugging in the given values, we get: $$ X_C = \frac{1}{2\pi(159)(2.5\times 10^{-6})} $$ Calculate the reactance of the capacitor: $$ X_C \approx 422.15\,\Omega $$
02

Calculate the impedance

Now that we have the reactance of the capacitor, we can find the impedance of the series RC circuit. Using the equation for impedance: $$ Z = \sqrt{R^2 + X_C^2} $$ Plugging in the given resistance (\(R = 300.0\,\Omega\)) and the calculated reactance (\(X_C \approx 422.15\,\Omega\)), we get: $$ Z = \sqrt{(300.0)^2 + (422.15)^2} $$ Calculate the impedance of the circuit: $$ Z \approx 517.44\,\Omega $$ The impedance of the circuit is approximately \(517.44\,\Omega\).

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Most popular questions from this chapter

An inductor has an impedance of \(30.0 \Omega\) and a resistance of $20.0 \Omega\( at a frequency of \)50.0 \mathrm{Hz}$. What is the inductance? (Model the inductor as an ideal inductor in series with a resistor.)
A 22 -kV power line that is \(10.0 \mathrm{km}\) long supplies the electric energy to a small town at an average rate of \(6.0 \mathrm{MW} .\) (a) If a pair of aluminum cables of diameter \(9.2 \mathrm{cm}\) are used, what is the average power dissipated in the transmission line? (b) Why is aluminum used rather than a better conductor such as copper or silver?
A 6.20 -m \(H\) inductor is one of the elements in a simple RLC series circuit. When this circuit is connected to a \(1.60-\mathrm{kHz}\) sinusoidal source with an \(\mathrm{rms}\) voltage of \(960.0 \mathrm{V},\) an rms current of $2.50 \mathrm{A}\( lags behind the voltage by \)52.0^{\circ} .$ (a) What is the impedance of this circuit? (b) What is the resistance of this circuit? (c) What is the average power dissipated in this circuit?
A 0.48 - \(\mu\) F capacitor is connected in series to a $5.00-\mathrm{k} \Omega\( resistor and an ac source of voltage amplitude \)2.0 \mathrm{V}$ (a) At \(f=120 \mathrm{Hz},\) what are the voltage amplitudes across the capacitor and across the resistor? (b) Do the voltage amplitudes add to give the amplitude of the source voltage (i.e., does \(V_{\mathrm{R}}+V_{\mathrm{C}}=2.0 \mathrm{V}\) )? Explain. (c) Draw a phasor diagram to show the addition of the voltages.
In an RLC circuit, these three clements are connected in series: a resistor of \(20.0 \Omega,\) a \(35.0-\mathrm{mH}\) inductor, and a 50.0 - \(\mu\) F capacitor. The ac source of the circuit has an rms voltage of \(100.0 \mathrm{V}\) and an angular frequency of $1.0 \times 10^{3} \mathrm{rad} / \mathrm{s} .$ Find (a) the reactances of the capacitor and inductor, (b) the impedance, (c) the rms current, (d) the current amplitude, (e) the phase angle, and (f) the rims voltages across each of the circuit elements. (g) Does the current lead or lag the voltage? (h) Draw a phasor diagram.
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