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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

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) $0.20 \mathrm{kHz} ?$
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.
For a particular \(R L C\) series circuit, the capacitive reactance is $12.0 \Omega,\( the inductive reactance is \)23.0 \Omega,$ and the maximum voltage across the \(25.0-\Omega\) resistor is \(8.00 \mathrm{V}\) (a) What is the impedance of the circuit? (b) What is the maximum voltage across this circuit?
A variable inductor can be placed in series with a lightbulb to act as a dimmer. (a) What inductance would reduce the current through a \(100-\) W lightbulb to \(75 \%\) of its maximum value? Assume a \(120-\mathrm{V}\) rms, \(60-\mathrm{Hz}\) source. (b) Could a variable resistor be used in place of the variable inductor to reduce the current? Why is the inductor a much better choice for a dimmer?
Make a figure analogous to Fig. 21.4 for an ideal inductor in an ac circuit. Start by assuming that the voltage across an ideal inductor is \(v_{\mathrm{L}}(t)=V_{\mathrm{L}}\) sin \(\omega t .\) Make a graph showing one cycle of \(v_{\mathrm{L}}(t)\) and \(i(t)\) on the same axes. Then, at each of the times \(t=0, \frac{1}{8} T, \frac{2}{8} T, \ldots, T,\) indicate the direction of the current (or that it is zero), whether the current is increasing, decreasing, or (instantaneously) not changing, and the direction of the induced emf in the inductor (or that it is zero).
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