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Chapter 21: Problem 36

A series combination of a resistor and a capacitor are connected to a \(110-\mathrm{V}\) rms, \(60.0-\mathrm{Hz}\) ac source. If the capacitance is \(0.80 \mu \mathrm{F}\) and the rms current in the circuit is $28.4 \mathrm{mA},$ what is the resistance?

Short Answer

Expert verified
Answer: The resistance in the circuit is approximately 2146.72Ω.

Step by step solution

01

Determine the impedance

Since we know the RMS voltage and RMS current, we can use the equation \(V = IZ\) to find the impedance. Rearranging for \(Z\), we get \(Z = \frac{V}{I}\). Plugging in the given values, \(Z = \frac{110\,\text{V}}{28.4\times10^{-3}\,\text{A}} = 3873.24\,\Omega\).
02

Determine the reactance of the capacitor

The reactance of the capacitor (\(X_C\)) can be found by using the formula \(X_C = \frac{1}{2\pi fC}\). Plugging in the given values for frequency and capacitance, we get \(X_C = \frac{1}{2\pi(60.0\,\text{Hz})(0.80\times10^{-6}\,\text{F})} = 3316.52\,\Omega\).
03

Determine the resistance

Finally, we can use the impedance formula, \(Z = \sqrt{R^2 + X_C^2}\), to find the resistance. Rearranging for \(R\), we get \(R = \sqrt{Z^2 - X_C^2}\). Plugging in the values for \(Z\) and \(X_C\), we get \(R = \sqrt{(3873.24\,\Omega)^2 - (3316.52\,\Omega)^2} = 2146.72\,\Omega\). The resistance in the circuit is approximately \(2146.72\,\Omega\).

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Most popular questions from this chapter

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.
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Two ideal inductors \((0.10 \mathrm{H}, 0.50 \mathrm{H})\) are connected in series to an ac voltage source with amplitude \(5.0 \mathrm{V}\) and frequency \(126 \mathrm{Hz}\). (a) What are the peak voltages across each inductor? (b) What is the peak current that flows in the circuit?
A 40.0 -mH inductor, with internal resistance of \(30.0 \Omega\) is connected to an ac source $$\varepsilon(t)=(286 \mathrm{V}) \sin [(390 \mathrm{rad} / \mathrm{s}) t]$$ (a) What is the impedance of the inductor in the circuit? (b) What are the peak and rms voltages across the inductor (including the internal resistance)? (c) What is the peak current in the circuit? (d) What is the average power dissipated in the circuit? (e) Write an expression for the current through the inductor as a function of time.
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