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Chapter 21: Problem 55

An RLC series circuit is driven by a sinusoidal emf at the circuit's resonant frequency. (a) What is the phase difference between the voltages across the capacitor and inductor? [Hint: since they are in series, the same current \(i(t) \text { flows through them. }]\) (b) At resonance, the rms current in the circuit is \(120 \mathrm{mA}\). The resistance in the circuit is \(20 \Omega .\) What is the rms value of the applied emf? (c) If the frequency of the emf is changed without changing its rms value, what happens to the rms current? (W) tutorial: resonance)

Short Answer

Expert verified
Answer: When the frequency of the emf is changed without changing its rms value, the rms current in the circuit will decrease.

Step by step solution

01

a) Phase Difference Between Capacitor and Inductor Voltages

At the resonant frequency, the impedance of the inductor (\(j\omega L\)) and the impedance of the capacitor (\(-\frac{j}{\omega C}\)) cancel each other out. Since the imaginary parts of these impedances are equal and opposite, the voltages across the capacitor and inductor are 180° out of phase. Therefore, the phase difference between the voltages across the capacitor and inductor is 180° or π radians.
02

b) RMS Value of the Applied EMF

At the resonant frequency, the total impedance of the circuit is purely resistive, and its magnitude is equal to the resistance in the circuit. The rms value of the applied emf can be found using Ohm's Law: \(E_{rms} = I_{rms} \times R\) where \(E_{rms}\) is the rms value of the applied emf, \(I_{rms}\) is the rms current, and \(R\) is the resistance. Given the rms current (\(I_{rms}\)) is 120 mA or 0.12 A, and the resistance (\(R\)) is 20 Ohms, we can calculate the rms value of the applied emf: \(E_{rms} = 0.12 \text{ A} \times 20 \Omega = 2.4 \text{ V}\)
03

c) Change in RMS Current with EMF Frequency Change

If the frequency of the emf is changed without changing its rms value, the impedance of the inductor and capacitor will no longer perfectly cancel each other out. This means that the total impedance of the circuit will not be purely resistive, and its magnitude will be greater than the resistance in the circuit. As the impedance increases, the rms current in the circuit will decrease, according to Ohm's Law: \(I_{rms} = \frac{E_{rms}}{Z}\) where \(Z\) is the total impedance of the circuit. So, if the frequency of the emf is changed without changing its rms value, the rms current in the circuit will decrease.

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Most popular questions from this chapter

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).
A European outlet supplies \(220 \mathrm{V}\) (rms) at \(50 \mathrm{Hz}\). How many times per second is the magnitude of the voltage equal to $220 \mathrm{V} ?$
The FM radio band is broadcast between \(88 \mathrm{MHz}\) and $108 \mathrm{MHz}$. What range of capacitors must be used to tune in these signals if an inductor of \(3.00 \mu \mathrm{H}\) is used?
A hair dryer has a power rating of \(1200 \mathrm{W}\) at \(120 \mathrm{V}\) rms. Assume the hair dryer circuit contains only resistance. (a) What is the resistance of the heating element? (b) What is the rms current drawn by the hair dryer? (c) What is the maximum instantaneous power that the resistance must withstand?
A 6.20 -m \(H\) inductor is one of the elements in a simple RLC series circuit. When this circuit is connected to a \(1.60-\mathrm{kHz}\) sinusoidal source with an \(\mathrm{rms}\) voltage of \(960.0 \mathrm{V},\) an rms current of $2.50 \mathrm{A}\( lags behind the voltage by \)52.0^{\circ} .$ (a) What is the impedance of this circuit? (b) What is the resistance of this circuit? (c) What is the average power dissipated in this circuit?
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