Problem 53
An \(L-R-C\) series circuit has R = 500 \(\Omega\), L = 2.00 H, \(C\) = 0.500 \(\mu\)F, and \(V\) = 100 V. (a) For \(\omega\) = 800 rad/s, calculate \(V_R , V_L, V_C\), and \(\phi\). Using a single set of axes, graph \(v\), \(v_R , v_L\), and \(v_C\) as functions of time. Include two cycles of \(v\) on your graph. (b) Repeat part (a) for \(\omega\) = 1000 rad/s. (c) Repeat part (a) for \(\omega = 1250\space rad/s\).
Problem 54
A resistor, an inductor, and a capacitor are connected in parallel to an ac source with voltage amplitude \(V\) and angular frequency \(\omega\). Let the source voltage be given by \(v\) = V cos \(\omega\)t. (a) Show that each of the instantaneous voltages \(v_R\) , \(v_L\), and \(v_C\) at any instant is equal to \(v\) and that \(i\) = \(i_R\) + \(i_L\) + \(i_C\), where i is the current through the source and iR , iL, and iC are the currents through the resistor, inductor, and capacitor, respectively. (b) What are the phases of \(i_R\) , \(i_L\), and \(i_C\) with respect to v? Use current phasors to represent i, iR , iL, and iC. In a phasor diagram, show the phases of these four currents with respect to \(v\). (c) Use the phasor diagram of part (b) to show that the current amplitude I for the current i through the source is \(I =\sqrt{I_{R ^2} + (I_C - I_L)^2}\). (d) Show that the result of part (c) can be written as\( I = V/Z\), with \(1/Z = \sqrt{(1/R^2) + [\omega C - (1/\omega L)]^2}.\)
Problem 58
An \(L-R-C\) series circuit has \(R\) = 60.0 \(\Omega\), \(L\) = 0.800 H, and \(C\) = 3.00 \(\times\) 10\(^{-4}\) F. The ac source has voltage amplitude 90.0 V and angular frequency 120 rad/s. (a) What is the maximum energy stored in the inductor? (b) When the energy stored in the inductor is a maximum, how much energy is stored in the capacitor? (c) What is the maximum energy stored in the capacitor?
Problem 59
In an \(L-R-C\) series circuit, the source has a voltage amplitude of 120 V, \(R\) = 80.0 \(\Omega\), and the reactance of the capacitor is 480 \(\Omega\). The voltage amplitude across the capacitor is 360 V. (a) What is the current amplitude in the circuit? (b) What is the impedance? (c) What two values can the reactance of the inductor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain.
Problem 60
In an \(L-R-C\) series ac circuit, the source has a voltage amplitude of 240 V, \(R\) = 90.0 \(\Omega\), and the reactance of the inductor is 320 \(\Omega\). The voltage amplitude across the resistor is 135 V. (a) What is the current amplitude in the circuit? (b) What is the voltage amplitude across the inductor? (c) What two values can the reactance of the capacitor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain.
Problem 61
A resistance \(R\), capacitance \(C\), and inductance \(L\) are connected in series to a voltage source with amplitude \(V\) and variable angular frequency \(\omega\). If \(\omega\) = \(\omega$$_0\) , the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of \(R\), \(C\), \(L\), and \(V\).
Problem 62
Consider an \(L-R-C\) series circuit with a 1.80-H inductor, a 0.900-\(\mu\)F capacitor, and a 300-\(\Omega\) resistor. The source has terminal rms voltage V\(_{rms}\) = 60.0 V and variable angular frequency \(\omega\). (a) What is the resonance angular frequency \(\omega_0\) of the circuit? (b) What is the rms current through the circuit at resonance, I\(_{rms}\)-0? (c) For what two values of angular frequency, \(\omega\)1 and \(\omega\)2, is the rms current half the resonance value? (d) The quantity \(\omega\)1 - \(omega\)2 defines the resonance \(width\). Calculate I\(_{rms}\)-0 and the resonance width for R = 300 \(\Omega\), 30.0 \(\Omega\), and 3.00 \(\Omega\). Describe how your results compare to the discussion in Section 31.5. the
Problem 63
An \(L-R-C\) series circuit draws 220 W from a 120-V (rms), 50.0-Hz ac line. The power factor is 0.560, and the source voltage leads the current. (a) What is the net resistance \(R\) of the circuit? (b) Find the capacitance of the series capacitor that will result in a power factor of unity when it is added to the original circuit. (c) What power will then be drawn from the supply line?
Problem 69
What is the dc impedance of the electrode, assuming that it behaves as an ideal capacitor? (a) 0; (b) infinite; (c) \(\sqrt{2}\times10^4\Omega\); (d) \(\sqrt{2}\times10^6\Omega\).
Problem 71
The signal from the oscillating electrode is fed into an amplifier, which reports the measured voltage as an rms value, 1.5 nV. What is the potential difference between the two extremes? (a) 1.5 nV; (b) 3.0 nV; (c) 2.1 nV; (d) 4.2 nV.
