AC Circuits
An inductor and a resistor are connected in series across an AC circuit. Immediately after the switch is closed, which of the following statements is true? (a) The current in the circuit is V/R. (b) The voltage across the inductor is zero. (c) The cur- rent in the circuit is zero. (d) The voltage across the resistor is V. (e) The voltage across the inductor is half its maximum value.
Answer: (c)
When a power source, AC or DC, is first connected to a RL combination, the presence of the inductor impedes the buildup of a current in the circuit. The value of the current starts at zero and increases as the back emf induced across the inductor decreases in magnitude.
What is the phase angle in a series RLC circuit at resonance?
0 degrees
The centers of two circular loops are separated by a fixed distance. For what relative orientation of the loops is their mutual inductance a maximum?
Answer: coaxial and lying in parallel planes
The mutual inductance of two loops in free space— that is, ignoring the use of cores—is a maximum if the loops are coaxial. In this way, the maximum flux of the primary loop will pass through the secondary loop, generating the largest possible emf given the changing magnetic field due to the first.
1. What is the resistance of a lightbulb that uses an average power of 75.0 W when connected to a 60.0-Hz power source having a maximum voltage of 170 V?
2. Without doing the math, what would happen to that value if power was increased to 100 W?
1. 193 Ohms
2. Resistance would decrease.
1. Does the phase angle in an RLC series circuit depend on frequency?
2. What is the phase angle for the circuit when the inductive reactance equals the capacitive reactance?
1. In an RLC series circuit, the phase angle depends on the source frequency. At very low frequency, the capacitor dominates the impedance and the phase angle is near –90°. At very high frequencies, the inductor dominates the impedance and the phase angle is near –90°.
2. When the inductive reactance equals the capacitive reactance, the frequency is the resonance frequency; the phase angle is zero.
The centers of two circular loops are separated by a fixed distance. For what relative position is their mutual inductance at a minimum?
Answer: lying in perpendicular planes, with the center of one on the axis of the other
The mutual inductance is a minimum if the magnetic field of the first coil lies in the plane of the second coil, producing no flux through the area the second coil encloses.
1. Why does a capacitor act as a short circuit at high frequencies?
2. Why does a capacitor act as an open circuit at low frequencies?
1. The capacitive reactance is proportional to the inverse of the frequency. At higher and higher frequencies, the capacitive reactance approaches zero, making a capacitor behave like a wire.
2. As the frequency goes to zero, the capacitive reactance approaches infinity—the resistance of an open circuit.
Why is the sum of the maximum voltages across each element in a series RLC circuit usually greater than the maximum applied voltage? Doesn’t that inequality violate Kirchhoff’s loop rule?
The voltages are not added in a scalar form, but in a vector form, as shown in the phasor diagrams throughout the chapter. Kirchhoff’s loop rule is true at any instant, but the voltages across different circuit elements are not simultaneously at their maximum values. Do not forget that an inductor can induce an emf in itself and that the voltage across it is 90° ahead of the current in the circuit in phase.
If the current in an inductor is doubled, by what factor is the stored energy multiplied?
Answer: 4 times larger
E=1/2LI^2
Under what conditions is the impedance of a series RLC circuit equal to the resistance in the circuit? (a) The driving frequency is lower than the resonance frequency. (b) The driving frequency is equal to the resonance frequency. (c) The driving frequency is higher than the resonance frequency. (d) always (e) never
Answer (c). At resonance the inductive reactance and capacitive reactance cancel out.
A circuit containing an AC source, a capacitor, an inductor, and a resistor has a high-Q resonance at 1000 Hz. From greatest to least, rank the following contributions to the impedance of the circuit at that frequency and at lower and higher frequencies. (a)XC at 500Hz(b)XC at 1500Hz (c)XL at 500Hz(d)XL at 1500Hz(e)R at 1000Hz
Answer: (a) > (d) > (b) > (c) > (e).
At the resonance frequency f0 = 1 000 Hz both XL and XC are equal: call their mutual value X. A high-Q value means the resonance has a small width, so XL and XC are also much larger than R at f0. Inductive reactance XL is proportional to frequency, and capacitive reactance XC is inversely proportional to frequency. In terms of X, the choices have the values: (a) f = f0 /2, so XC = 2X. (b) f = 3f0 /2, so XC = 2X/3. (c) f = f0 /2, so XL = X/2. (d) f = 3f0 /2, so XL = 3X/2. (e) R is independent of frequency, and R is less than X. Thus, we have (a) 2X > (d) 3X/2 > (b) 2X/3 > (c) X/2 > (e) less than X.
What parameters affect the inductance of a coil?
Answer: the geometry and the content of the coil.
This is similar to the parameters that determine the capacitance of a capacitor and the resistance of a resistor. With an inductor, the most important factor in the geometry is the number of turns of wire, or turns per unit length. By the “contents” we refer to the material in which the inductor establishes a magnetic field, notably the magnetic properties of the core around which the wire is wrapped.
A person pulls a vacuum cleaner at speed v across a horizontal floor, exerting on it a force of magnitude F directed upward at an angle u with the horizontal.
1. At what rate is the person doing work on the cleaner?
2. State as completely as you can the analogy between power in this situation and in an electric circuit.
1. P=Fvcosθ
2. Compare the previous equation to P = ΔVrmsIrmscosphi . One can consider the emf as the “force” that moves the charges through the circuit, and the current as the “speed” of the moving charges. The cos θ factor measures the effectiveness of the cause in producing the effect. Theta is an angle in real space for the vacuum cleaner and phi is the analogous angle of phase difference between the emf and the current in the circuit.
A certain power supply can be modeled as a source of emf in series with both a resistance of 10 V and an inductive reactance of 5 V. To obtain maximum power delivered to the load, it is found that the load should have a resistance of RL 5 10 V, an inductive reactance of zero, and a capacitive reactance of 5 V.
1. With this load, is the circuit in resonance?
2. With this load, what fraction of the average power put out by the source of emf is delivered to the load?
3. To increase the fraction of the power delivered to the load, how could the load be changed?
1. Yes. The circuit is in resonance because the inductive reactance and capcitive reactance are equal, so the total impedance Z = R.
2. Total power output by the emf Pemf = I^2Rtotal, where Rtotal = 10 Ω (source resistance) + 10 Ω (load resistance) = 20 Ω. Power delivered to the load Pload = I2RL , where RL = 10 Ω. Fraction of average power delivered to the load to average power delivered by the source of emf=.5
3. The resistance of the load could be increased to make a greater fraction of the emf’s power go to the load. Then the emf would put out a lot less power and less power would reach the load.
After the switch is closed an LC circuit, the charge on the capacitor is sometimes zero, but at such instants the current in the circuit is not zero. How is this behavior possible?
When the capacitor is fully discharged, the current in the circuit is a maximum. The inductance of the coil is making the current continue to flow. At this time the magnetic field of the coil contains all the energy that was originally stored in the charged capacitor. The current has just finished discharging the capacitor and is proceeding to charge it up again with the opposite polarity.