Irodov Solutions → Oscillations and waves → Electric Oscillations

 4.96. An oscillating circuit consists of a capacitor with capacitance C, a coil of inductance L with negligible resistance, and a switch. With the switch disconnected, the capacitor was charged to a voltage Vm and then at the moment t = 0 the switch was closed. Find: (a) the current I(t) in the circuit as a function of time; (b) the emf of self-inductance in the coil at the moments when the electric energy of the capacitor is equal to that of the current in the coil. Free solution >> 4.98. In an oscillating circuit shown in Fig. 4.27 the coil inductance is equal to L = 2.5 mH and the capacitor have capacitances C1 = 2.0 μF and C2 = 3.0 μF. The capacitors were charged to a voltage V = 180 V, and then the switch Sw was closed. Find: (a) the natural oscillation frequency; (b) the peak value of the current flowing through the coil. Free solution >> 4.103. A circuit with capacitance C and inductance L generates free damped oscillations with current varying with time as I = Ime-βtsin ωt. Find the voltage across the capacitor as a function of time, and in particular, at the moment t = 0. Free solution >> 4.104. An oscillating circuit consists of a capacitor with capacitance C = 4.0 μF and a coil with inductance L = 2.0 mH and active resistance R = 10 Ω. Find the ratio of the energy of the coil's magnetic field to that of the capacitor's electric field at the moment when the current has the maximum value. Free solution >> 4.107. An oscillating circuit consists of capacitance C = 10 μF, inductance L = 25 mH, and active resistance R = 1.0 Ω. How many oscillation periods does it take for the current amplitude to decrease e-fold? Free solution >> 4.108. How much (in per cent) does the free oscillation frequency ω of a circuit with quality factor Q = 5.0 differ from the natural oscillation frequency ω0 of that circuit? Free solution >> 4.109. In a circuit shown in Fig. 4.29 the battery emf is equal to ξ = 2.0 V, its internal resistance is r = 9.0 Ω, the capacitance of the capacitor is C = 10 μF, the coil inductance is L = 100 mH, and the resistance is R = 1.0 Ω. At a certain moment the switch Sw was disconnected. Find the energy of oscillations in the circuit (a) immediately after the switch was disconnected; (b) t = 0.30 s after the switch was disconnected. Free solution >> 4.110. Damped oscillations are induced in a circuit whose quality factor is Q = 50 and natural oscillation frequency is ν0 = 5.5 kHz. How soon will the energy stored in the circuit decrease η = 2.0 times? Free solution >> 4.119. A circuit consisting of a capacitor with capacitance C and a resistance R connected in series was connected at the moment t = 0 to a source of ac voltage V = Vm cos ωt. Find the current in the circuit as a function of time t. Free solution >> 4.121. A circuit consisting of a capacitor and an active resistance R = 110 Ω connected in series is fed an alternating voltage with amplitude Vm = 110 V. In this case the amplitude of steady-state current is equal to Im = 0.50 A. Find the phase difference between the current and the voltage fed. Free solution >> 4.122. Fig. 4.32 illustrates the simplest ripple filter. A voltage V = V0(1 + cos ωt) is fed to the left input. Find: (a) the output voltage V'(t); (b) the magnitude of the product RC at which the output amplitude of alternating voltage component is η = 7.0 times less than the direct voltage component, if ω = 314 s-1. Free solution >> 4.136. A solenoid with inductance L = 7 mH and active resistance R = 44 Ω is first connected to a source of direct voltage V0 and then to a source of sinusoidal voltage with effective value V = V0. At what frequency of the oscillator will the power consumed by the solenoid be η = 5.0 times less than in the former case? Free solution >> 4.138. A coil with inductance L = 0.70 H and active resistance r = 20 Ω is connected in series with an inductance-free resistance R. An alternating voltage with effective value V = 220 V and frequency ω = 314 s-1 is applied across the terminals of this circuit. At what value of the resistance R will the maximum heat power be generated in the circuit? What is it equal to? Free solution >> 4.147. A circuit consists of a capacitor with capacitance C and a coil with active resistance R and inductance L connected in parallel. Find the impedance of the circuit at frequency ω of alternating voltage. Free solution >>