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Chapter 30: Problem 34

A capacitor with capacitance \(C=5.00 \cdot 10^{-6} \mathrm{~F}\) is connected to an AC power source having a peak value of \(10.0 \mathrm{~V}\) and \(f=100 . \mathrm{Hz} .\) Find the reactance of the capacitor and the maximum current in the circuit.

Short Answer

Expert verified
Answer: The reactance of the capacitor is approximately 318.31 惟, and the maximum current in the circuit is approximately 0.0314 A.

Step by step solution

01

Calculate the angular frequency

The angular frequency (蠅) of the AC power source can be calculated using the frequency (f) given by the formula: 蠅 = 2蟺f Plugging in the given value of frequency f = 100 Hz, we have: 蠅 = 2蟺(100) = 200蟺 rad/s
02

Calculate the reactance of the capacitor

The reactance (X_C) of the capacitor can be calculated using the capacitance (C) and angular frequency (蠅) with the formula: X_C = 1/(蠅C) Plugging in the given value of capacitance C = 5.00 脳 10^{-6} F and the calculated angular frequency 蠅 = 200蟺 rad/s, we have: X_C = 1/(200蟺 脳 5.00 脳 10^{-6}) = 1000/蟺 惟 鈮� 318.31 惟
03

Calculate the maximum current in the circuit

With the reactance of the capacitor (X_C) and the peak voltage (V_peak) of the AC power source, we can now calculate the maximum current (I_max) in the circuit using Ohm's law: I_max = V_peak / X_C Plugging in the given value of peak voltage V_peak = 10.0 V and the calculated reactance X_C 鈮� 318.31 惟, we have: I_max = 10.0 V / 318.31 惟 鈮� 0.0314 A Thus, the reactance of the capacitor is approximately 318.31 惟, and the maximum current in the circuit is approximately 0.0314 A.

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Most popular questions from this chapter

A 10.0 - \(\mu\) F capacitor is fully charged by a 12.0 - \(\mathrm{V}\) battery and is then disconnected from the battery and allowed to discharge through a 0.200 - \(\mathrm{H}\) inductor. Find the first three times when the charge on the capacitor is \(80.0-\mu \mathrm{C},\) taking \(t=0\) as the instant when the capacitor is connected to the inductor.

An electromagnet consists of 200 loops and has a length of \(10.0 \mathrm{~cm}\) and a cross-sectional area of \(5.00 \mathrm{~cm}^{2}\). Find the resonant frequency of this electromagnet when it is attached to the Earth (treat the Earth as a spherical capacitor)

The time-varying current in an LC circuit where \(C=10.0 \mu \mathrm{F}\) is given by \(i(t)=(1.00 \mathrm{~A}) \sin (1200 . t),\) where \(t\) is in seconds. a) At what time after \(t=0\) does the current reach its maximum value? b) What is the total energy of the circuit? c) What is the inductance, \(L\) ?

A radio tuner has a resistance of \(1.00 \mu \Omega\), a capacitance of \(25.0 \mathrm{nF}\), and an inductance of \(3.00 \mathrm{mH}\) a) Find the resonant frequency of this tuner. b) Calculate the power in the circuit if a signal at the resonant frequency produces an emf across the antenna of \(V_{\mathrm{rms}}=1.50 \mathrm{mV}\)

In Solved Problem 30.1 , the voltage supplied by the source of time-varying emf is \(33.0 \mathrm{~V}\), the voltage across the resistor is \(V_{R}=I R=13.1 \mathrm{~V}\), and the voltage across the inductor is \(V_{L}=I X_{L}=30.3 \mathrm{~V}\). Does this circuit obey Kirchhoff's rules?
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