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Chapter 30: Problem 9

What is the impedance of a series RLC circuit when the frequency of time- varying emf is set to the resonant frequency of the circuit?

Short Answer

Expert verified
Answer: At the resonant frequency of an RLC circuit, the impedance is equivalent to the resistance (R) of the resistor in the circuit.

Step by step solution

01

Calculate the resonant frequency

In a series RLC circuit, the resonant frequency (f_r) is given by the formula: f_r = \frac{1}{2\pi\sqrt{LC}} Where L is the inductance (measured in Henries) and C is the capacitance (measured in Farads).
02

Calculate the inductive reactance at resonant frequency

Inductive reactance (X_L) is the opposition offered by the inductor coil at a specific frequency (f) and is given by the formula: X_L = 2\pi f L At the resonant frequency, replace f with f_r: X_L = 2\pi f_r L
03

Calculate the capacitive reactance at resonant frequency

Capacitive reactance (X_C) is the opposition offered by the capacitor at a specific frequency (f) and is given by the formula: X_C = \frac{1}{2\pi f C} At the resonant frequency, replace f with f_r: X_C = \frac{1}{2\pi f_r C}
04

Find the net reactance at resonant frequency

The net reactance (X_net) of the RLC circuit is the difference between inductive reactance and capacitive reactance: X_net = X_L - X_C At the resonant frequency, we know that the inductive and capacitive reactance are equal: X_L = X_C Therefore, X_net = 0
05

Calculate the impedance at resonant frequency

In a series RLC circuit, the impedance (Z) is given by the formula: Z = \sqrt{R^2 + X_{net}^2} At the resonant frequency, the net reactance is zero (X_net = 0), so the impedance is solely determined by the resistor: Z = R In conclusion, at the resonant frequency of an RLC circuit, the impedance is equivalent to the resistance (R) of the resistor in the circuit.

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Most popular questions from this chapter

An AC power source with \(V_{\mathrm{m}}=220 \mathrm{~V}\) and \(f=60.0 \mathrm{~Hz}\) is connected in a series RLC circuit. The resistance, \(R\), inductance, \(L\), and capacitance, \(C\), of this circuit are, respectively, \(50.0 \Omega, 0.200 \mathrm{H},\) and \(0.040 \mathrm{mF}\). Find each of the following quantities: a) the inductive reactance b) the capacitive reactance c) the impedance of the circuit d) the maximum current through the circuit e) the maximum potential difference across each circuit element

When you turn the dial on a radio to tune it, you are adjusting a variable capacitor in an LC circuit. Suppose you tune to an AM station broadcasting at a frequency of \(1000 . \mathrm{kHz},\) and there is a \(10.0-\mathrm{mH}\) inductor in the tuning circuit. When you have tuned in the station, what is the capacitance of the capacitor?

A transformer has 800 turns in the primary coil and 40 turns in the secondary coil. a) What happens if an AC voltage of \(100 . \mathrm{V}\) is across the primary coil? b) If the initial \(A C\) current is \(5.00 \mathrm{~A}\), what is the output current? c) What happens if a DC current at \(100 .\) V flows into the primary coil? d) If the initial DC current is \(5.00 \mathrm{~A}\), what is the output current?

A series RLC circuit is in resonance when driven by a sinusoidal voltage at its resonant frequency, \(\omega_{0}=(L C)^{-1 / 2}\) But if the same circuit is driven by a square-wave voltage (which is alternately on and off for equal time intervals), it will exhibit resonance at its resonant frequency and at \(\frac{1}{3}, \frac{1}{5}\), \(\frac{1}{7}, \ldots,\) of this frequency. Explain why.

Laboratory experiments with series RLC circuits require some care, as these circuits can produce large voltages at resonance. Suppose you have a 1.00 - \(\mathrm{H}\) inductor (not difficult to obtain) and a variety of resistors and capacitors. Design a series RLC circuit that will resonate at a frequency (not an angular frequency) of \(60.0 \mathrm{~Hz}\) and will produce at resonance a magnification of the voltage across the capacitor or the inductor by a factor of 20.0 times the input voltage or the voltage across the resistor.
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