An ac voltage source is connected in series with a 2.0-μF capacitor and a 750-Ω resistor. Using a digital ac voltmeter, the voltage source is measured to be 4.0 V rms, and the voltages across the resistor and across the capacitor are found to be 3.0 V rms and 2.7 V rms, respectively. Determine the frequency of the ac voltage source. Why is the voltage measured across the voltage source not equal to the sum of the voltages measured across the resistor and across the capacitor?

For the circuit of FIGURE EX32.32, Find V_R and V_L at resonance.

Show that the fraction of electromagnetic energy lost (to thermal energy) per cycle in a lightly damped (R² ≪ 4L/C) LRC circuit is approximately $\frac{\Delta U}{U}=\frac{2\pi R}{L\omega }=\frac{2\pi }{Q}$. The quantity Q can be defined as Q = Lω/R, and is called the Q-value, or quality factor, of the circuit and is a measure of the damping present. A high Q-value means smaller damping and less energy input required to maintain oscillations.

Suppose a series LRC circuit has two resistors, R₁ and R₂, two capacitors, C₁ and C₂, and two inductors, L₁ and L₂ all in series. Calculate the total impedance of the circuit.

An inductance coil draws 2.2 A dc when connected to a 45-V battery. When connected to a 60.0-Hz 120-V (rms) source, the current drawn is 3.8 A (rms). Determine the inductance and resistance of the coil.

A circuit contains two elements, but it is not known if they are L, R, or C. The current in this circuit when connected to a 120-V rms 60.0-Hz source is 5.6 A and lags the voltage by 55°. What are the two elements and what are their values?

For an underdamped LRC circuit, determine a formula for the energy U = UE + UB stored in the electric and magnetic fields as a function of time. Give answer in terms of the initial charge Q_o on the capacitor.

For an underdamped LRC circuit, determine a formula for the energy U = U_E + U_B stored in the electric and magnetic fields as a function of time. Give answer in terms of the initial charge Q_o on the capacitor. Show how dU/dt is related to the rate energy is transformed in the resistor, I²R.

An inductor L in series with a resistor R, driven by a sinusoidal voltage source, responds as described by the following differential equation: $V_0\sin \omega t=L\frac{dI}{dt}+RI.$ Show that a current of the form I = I₀ sin (ωt - Φ) flows through the circuit by direct substitution into the differential equation. Determine the amplitude of the current (I₀) and the phase difference Φ between the current and the voltage source.