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Chapter 21: Problem 16

A \(0.250-\mu \mathrm{F}\) capacitor is connected to a \(220-\mathrm{V}\) rms ac source at \(50.0 \mathrm{Hz}\) (a) Find the reactance of the capacitor. (b) What is the rms current through the capacitor?

Short Answer

Expert verified
Answer: The reactance of the capacitor is approximately 1276.45 Ω, and the rms current through it is approximately 0.172 A.

Step by step solution

01

Calculate the reactance of the capacitor

To find the reactance, we use the formula \(X_C = \frac{1}{2\pi fC}\), where \(f\) is the frequency of 50 Hz and \(C\) is the given capacitance of 0.250 \(\mu F\). Plug the values into the formula: \(X_C = \frac{1}{2\pi (50)(0.250×10^{-6})}\) Now, calculate the reactance: \(X_C = \frac{1}{2\pi (50)(0.250×10^{-6})} ≈ 1276.45 \Omega\)
02

Calculate the RMS current through the capacitor

We can now use Ohm's law for capacitive AC circuits to find the RMS current. The formula is \(I = \frac{V}{X_C}\), where V is the RMS voltage (220 V) and \(X_C\) is the reactance we calculated in step 1 (1276.45 \(\Omega\)). Plug the values into the formula: \(I = \frac{220}{1276.45}\) Now, calculate the RMS current: \(I = \frac{220}{1276.45} ≈ 0.172 A\) The RMS current through the capacitor is approximately 0.172 A.

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Most popular questions from this chapter

An RLC series circuit has \(L=0.300 \mathrm{H}\) and \(C=6.00 \mu \mathrm{F}\) The source has a peak voltage of \(440 \mathrm{V}\). (a) What is the angular resonant frequency? (b) When the source is set at the resonant frequency, the peak current in the circuit is \(0.560 \mathrm{A}\). What is the resistance in the circuit? (c) What are the peak voltages across the resistor, the inductor, and the capacitor at the resonant frequency?
A 4.00 -mH inductor is connected to an ac voltage source of \(151.0 \mathrm{V}\) rms. If the rms current in the circuit is \(0.820 \mathrm{A},\) what is the frequency of the source?
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.
A capacitor is rated at \(0.025 \mu \mathrm{F}\). How much rms current flows when the capacitor is connected to a \(110-\mathrm{V}\) rms, \(60.0-\mathrm{Hz}\) line?
A variable inductor with negligible resistance is connected to an ac voltage source. How does the current in the inductor change if the inductance is increased by a factor of 3.0 and the driving frequency is increased by a factor of \(2.0 ?\)
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