Chapter 11 Electromagnetic Oscillations and Alternating Current

Learning Objectives:

In this chapter you will basically learn:

\(\bullet\) An LC oscillator, sketch graphs of the potential difference across the capacitor and the current through the inductor as functions of time, and indicate the period T on each graph.

\(\bullet\) Explain the analogy between a block–spring oscillator and an LC oscillator.

\(\bullet\) For an LC oscillator, apply the relationships between the angular frequency \(\omega\) (and the related frequency f and period T) and the values of the inductance and capacitance.

\(\bullet\) Starting with the energy of a block–spring system, explain the derivation of the differential equation for charge q in an LC oscillator and then identify the solution for q(t).

\(\bullet\) For an LC oscillator, calculate the charge q on the capacitor for any given time and identify the amplitude Q of the charge oscillations.

\(\bullet\) Starting from the equation giving the charge q(t) on the capacitor in an LC oscillator, find the current i(t) in the inductor as a function of time.

\(\bullet\) From the expressions for the charge q and the current i in an LC oscillator, find the magnetic field energy \(U_B(t)\) and the electric field energy \(U_E(t)\) and the total energy.

\(\bullet\) Draw the schematic of a damped RLC circuit and explain why the oscillations are damped.

\(\bullet\) Apply the relationship between the angular frequency of a given damped RLC oscillator and the angular frequency v of the circuit if R is removed.

\(\bullet\) Distinguish alternating current from direct current.

\(\bullet\) For an ac generator, write the emf as a function of time, identifying the emf amplitude and driving angular frequency.

\(\bullet\) For an ac generator, write the current as a function of time, identifying its amplitude and its phase constant with respect to the emf.

\(\bullet\) In a driven (series) RLC circuit, identify the conditions for resonance and the effect of resonance on the current amplitude.

\(\bullet\) For each of the three basic circuits (purely resistive load, purely capacitive load, and purely inductive load), draw the circuit and sketch graphs and phasor diagrams for voltage v(t) and current i(t).

\(\bullet\) For the three basic circuits, apply equations for voltage v(t) and current i(t).

\(\bullet\) On a phasor diagram for each of the basic circuits, identify angular speed, amplitude, projection on the vertical axis, and rotation angle.

\(\bullet\) For each basic circuit, identify the phase constant, and interpret it in terms of the relative orientations of the current phasor and voltage phasor and also in terms of leading and lagging.

\(\bullet\) For each basic circuit, apply the relationships between the voltage amplitude V and the current amplitude I.

\(\bullet\) Calculate capacitive reactance \(X_C\) and inductive reactance \(X_L\).

\(\bullet\) Apply the relationship between current amplitude I, impedance Z, and emf amplitude \(\mathcal{E}_m\).

\(\bullet\) Apply the relationships between phase constant f and voltage amplitudes \(V_L\) and \(V_C\), and also between phase constant f, resistance R, and reactances \(X_L\) and \(X_C\).

\(\bullet\) For the current, voltage, and emf in an ac circuit, apply the relationship between the rms values and the amplitudes.

\(\bullet\) Apply the relationship between average power \(P_{avg}\), rms current \(I_{rms}\), and resistance R.

\(\bullet\) Apply the relationship between the power factor \(\cos\phi\), the resistance R, and the impedance Z.

\(\bullet\) Apply the relationship between the average power \(P_{avg}\), the rms \(\mathcal{E}_{emf}\) , the rms current \(I_{rms}\), and the power factor \(\cos\phi\).

\(\bullet\) Identify what power factor is required in order to maximize the rate at which energy is supplied to a resistive load.

\(\bullet\) For power transmission lines, identify why the transmission should be at low current and high voltage.

\(\bullet\) Identify the role of transformers at the two ends of a transmission line.

\(\bullet\) Calculate the energy dissipation in a transmission line.

\(\bullet\) Apply the relationship between the voltage and number of turns on the two sides of a transformer.

\(\bullet\) Distinguish between a step-down transformer and a step-up transformer.

\(\bullet\) Apply the relationship between the current and number of turns on the two sides of a transformer.

\(\bullet\) Apply the relationship between the power into and out of an ideal transformer.

\(\bullet\) Identify the equivalent resistance as seen from the primary side of a transformer.

\(\bullet\) Explain the role of a transformer in impedance matching.

11.1 Electromagnetic Oscillations and Alternating Current:

11.2 LC Oscillations, Qualitatively, The Electrical-Mechanical Analogy:

11.3 LC Oscillations, Quantitatively:

11.4 Damped Oscillations in an RLC Circuit:

11.5 Alternating Current:

11.6 Forced Oscillations:

11.7 Three Simple Circuits:

11.8 The Series RLC Circuit:

11.9 Power in Alternating-Current Circuits:

11.10 Transformers:

Solved Problems Electromagnetic Oscillations

  1. [8] A single loop consists of inductors \((L_1, L_2, . . .)\), capacitors \((C_1,C_2, . . .)\), and resistors \((R_1, R_2, . . .)\) connected in series as shown, for example, in Fig. 31-27a. Show that regardless of the sequence of these circuit elements in the loop, the behavior of this circuit is identical to that of the simple LC circuit shown in Fig. 31-27b. (Hint: Consider the loop rule and see Problem 47 in Chapter 30.)

  2. [20] In an oscillating LC circuit in which C = 4.00 \(\mu F\), the maximum potential difference across the capacitor during the oscillations is 1.50 V and the maximum current through the inductor is 50.0 mA.What are (a) the inductance L and (b) the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value?

  3. [22] A series circuit containing inductance \(L_1\) and capacitance \(C_1\) oscillates at angular frequency \(\omega\). A second series circuit, containing inductance \(L_2\) and capacitance \(C_2\), oscillates at the same angular frequency. In terms of \(\omega\), what is the angular frequency of oscillation of a series circuit containing all four of these elements? Neglect resistance. (Hint: Use the formulas for equivalent capacitance and equivalent inductance)

  4. [25] What resistance R should be connected in series with an inductance L = 220 mH and capacitance C = 12.0 \(\mu F\) for the maximum charge on the capacitor to decay to 99.0% of its initial value in 50.0 cycles? (Assume \(\omega'\approx \omega\).)

  5. [27] In an oscillating series RLC circuit, show that \(\Delta U/U\), the fraction of the energy lost per cycle of oscillation, is given to a close approximation by \(2\pi R/\omega L\).The quantity \(\omega L/R\) is often called the Q of the circuit (for quality). A high-Q circuit has low resistance and a low fractional energy loss \((2\pi/Q)\) per cycle.

  6. [31] (a) At what frequency would a 6.0 mH inductor and a 10 \(\mu F\) capacitor have the same reactance? (b) What would the reactance be? (c) Show that this frequency would be the natural frequency of an oscillating circuit with the same L and C.

  7. [33] An ac generator has emf \(\mathcal{E} = \mathcal{E}_m \sin (\omega_dt-\pi/4)\), where \(\mathcal{E}_m\) = 30.0 V and \(\omega_d\) = 350 rad/s. The current produced in a connected circuit is \(i(t) = I \sin(\omega_dt-3\pi/4)\), where I = 620 mA. At what time after t = 0 does (a) the generator emf first reach a maximum and (b) the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer. (d) What is the value of the capacitance, inductance, or resistance, as the case may be?

  8. [49] In Fig. 31-33, a generator with an adjustable frequency of oscillation is connected to resistance R = 100 \(\Omega\), inductances \(L_1\) = 1.70 mH and \(L_2\) = 2.30 mH, and capacitances \(C_1\) = 4.00 \(\mu F\), \(C_2\) = 2.50 \(\mu F\), and \(C_3\) = 3.50 \(\mu F\). (a) What is the resonant frequency of the circuit? (Hint: See Problem 47 in Chapter 10.) What happens to the resonant frequency if (b) R is increased, (c) \(L_1\) is increased, and (d) \(C_3\) is removed from the circuit?

  9. [53] An air conditioner connected to a 120 V rms ac line is equivalent to a 12.0 resistance and a 1.30 inductive reactance in series. Calculate (a) the impedance of the air conditioner and

  1. the average rate at which energy is supplied to the appliance.

  1. [55] What direct current will produce the same amount of thermal energy, in a particular resistor, as an alternating current that has a maximum value of 2.60 A?

  2. [61] Figure 31-36 shows an ac generator connected to a “black box” through a pair of terminals. The box contains an RLC circuit, possibly even a multiloop circuit, whose elements and connections we do not know. Measurements outside the box reveal that

\[\mathcal{E}(t)=(75.0~V)\sin\omega_dt\] and \[i(t) = (1.20~A)\sin(\omega_dt+42.0^\circ\].

  1. What is the power factor? (b) Does the current lead or lag the emf? (c) Is the circuit in the box largely inductive or largely capacitive? (d) Is the circuit in the box in resonance? (e) Must there be a capacitor in the box? (f) An inductor? (g) A resistor?
  2. At what average rate is energy delivered to the box by the generator? (i) Why don’t you need to know vd to answer all these questions?
  1. [64] Figure 31-37 shows an “autotransformer.” It consists of a single coil (with an iron core). Three taps \(T_i\) are provided. Between taps \(T_1\) and \(T_2\) there are 200 turns, and between taps \(T_2\) and \(T_3\) there are 800 turns.Any two taps can be chosen as the primary terminals, and any two taps can be chosen as the secondary terminals. For choices producing a step-up transformer, what are the (a) smallest, (b) second smallest, and (c) largest values of the ratio \(V_s/V_p\)? For a step-down transformer, what are the (d) smallest, (e) second smallest, and (f) largest values of \(V_s/V_p\)?

Problems Electromagnetic Oscillations

Section 11-1 LC Oscillations

  1. [1] An oscillating LC circuit consists of a 75.0 mH inductor and a 3.60 \(\mu F\) capacitor. If the maximum charge on the capacitor is 2.90 \(\mu C\), what are (a) the total energy in the circuit and (b) the maximum current?

  2. [17] In Fig. 31-28, R = 14.0 \(\Omega\), C = 6.20 \(\mu F\), and L = 54.0 mH, and the ideal battery has emf \(\mathcal{E}\) = 34.0 V. The switch is kept at a for a long time and then thrown to position b. What are the (a) frequency and (b) current amplitude of the resulting oscillations?

  3. [21] In an oscillating LC circuit with C = 64.0 \(\mu F\), the current is given by i = (1.60) sin(2500t + 0.680), where t is in seconds, i in amperes, and the phase constant in radians. (a) How soon after t = 0 will the current reach its maximum value? What are (b) the inductance L and (c) the total energy?

Section 11-2 Damped Oscillations in an RLC Circuit

  1. [24] A single-loop circuit consists of a 7.20 \(\Omega\) resistor, a 12.0 H inductor, and a 3.20 \(\mu F\) capacitor. Initially the capacitor has a charge of 6.20 \(\mu C\) and the current is zero. Calculate the charge on the capacitor N complete cycles later for (a) N = 5, (b) N = 10, and (c) N = 100.

  2. [26] In an oscillating series RLC circuit, find the time required for the maximum energy present in the capacitor during an oscillation to fall to half its initial value. Assume q = Q at t = 0.

Section 11-3 Forced Oscillations of Three Simple Circuits

  1. [30] A 50.0 \(\Omega\) resistor is connected as in Fig. 31-8 to an ac generator with \(\mathcal{E}_m\)=30.0 V. What is the amplitude of the resulting alternating current if the frequency of the emf is (a) 1.00 kHz and (b) 8.00 kHz?

  2. [32] An ac generator has emf \(\mathcal{E} = \mathcal{E}_m \sin \omega_dt\), with \(\mathcal{E}_m\) = 25.0 V and \(\omega_d\) = 377 rad/s. It is connected to a 12.7 H inductor. (a) What is the maximum value of the current? (b) When the current is a maximum, what is the emf of the generator? (c) When the emf of the generator is ’12.5 V and increasing in magnitude, what is the current?

Section 11-4 The Series RLC Circuit

  1. [48] Figure 31-32 shows a driven RLC circuit that contains two identical capacitors and two switches. The emf amplitude is set at 12.0 V, and the driving frequency is set at 60.0 Hz. With both switches open, the current leads the emf by \(30.9^\circ\).With switch \(S_1\) closed and switch \(S_2\) still open, the emf leads the current by \(15^\circ\). With both switches closed, the current amplitude is 447 mA. What are (a) R, (b) C, and (c) L?

Section 11-5 Power in Alternating-Current Circuits

  1. [52] An ac voltmeter with large impedance is connected in turn across the inductor, the capacitor, and the resistor in a series circuit having an alternating emf of 100 V (rms); the meter gives the same reading in volts in each case.What is this reading?

  2. [54] What is the maximum value of an ac voltage whose rms value is 100 V?

  3. [58] For Fig. 31-35, show that the average rate at which energy is dissipated in resistance R is a maximum when R is equal to the internal resistance r of the ac generator. (In the text discussion we tacitly assumed that r = 0.)

Section 11-6 Transformers

  1. [63] A transformer has 500 primary turns and 10 secondary turns. (a) If \(V_p\) is 120 V (rms), what is \(V_s\) with an open circuit? If the secondary now has a resistive load of 15 \(\Omega\), what is the current in the (b) primary and (c) secondary?

  2. [65] An ac generator provides emf to a resistive load in a remote factory over a two-cable transmission line. At the factory a stepdown transformer reduces the voltage from its (rms) transmission value \(V_t\) to a much lower value that is safe and convenient for use in the factory. The transmission line resistance is 0.30 \(\Omega\)/cable, and the power of the generator is 250 kW. If \(V_t\) = 80 kV, what are (a) the voltage decrease \(\Delta V\) along the transmission line and (b) the rate \(P_d\) at which energy is dissipated in the line as thermal energy? If \(V_t\) = 8.0 kV, what are (c) \(\Delta V\) and (d) \(P_d\)? If \(V_t\) = 0.80 kV, what are (e) \(\Delta V\) and (f) \(P_d\)?