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Physics 122, Fall 2023

Bonus Homework Problem Set

Due on Monday December 11 at 10:50 am. Send the electronic copy of your homework via BlackBoard, appending the solutions to the cover sheet provided. Follow ALL guidelines outlined in the cover sheet. Note: This HW set is optional. If you do turn it in, the set can be used to replace a low-score HW set or make up for a missed HW set.

Problem 1. A circuit consists of m inductors, n resistors and l capacitors. They are all connected in series, forming a large loop. Prove that the behavior of the circuit is the same as that of a simple RLC circuit (i.e., a circuit with 1 inductor, 1 resistor and 1 capacitor connected in series).

Problem 2. Consider an oscillating series RLC circuit. We denote the internal energy of the system as U, and the energy lost per cycle as ∆U.

a) Show that ∆U/U ≈ x R/L , where x is a coefficient (Hint: a Taylor series is involved!).

b) Calculate the coefficient x.

Problem 3. A series RLC circuit with R = 220Ω, L = 150mH, C = 24.0µF connects to a 400 Hz AC generator with emf amplitude Em = 220V. Find (a) the impedance Z; (b) the capacitor’s reactance XC ; (c) the current amplitude I;

Problem 4. Consider two electric circuit components connected in parallel, with impedance Z1 = X1 + iR1 and Z2 = X2 + iR2, respectively. Calculate the equivalent impedance Zeq, equivalent resistance Req, and equivalent reactance Xeq for the circuit. Express your answers in terms of X1, X2, R1, R2 only.

Problem 5. Consider a series RLC circuit with equivalent impedance Z. It connects to an AC generator with emf V (t) = V0 sin(ωt). The phase difference between the voltage and the current is φ. Answer the questions below and express your answers in terms of Z, t, V0, ω, φ.

a) Calculate the power P(t) delivered by the generator at time t.

b) Calculate the time average of the power h P(t)i in a complete cycle.

Problem 6. Consider a series RLC circuit connected to a 60 Hz AC generator with an emf amplitude of 30V. The maximum voltage across the resistor and capacitor is only half of the maximum voltage across the inductor.

a) Calculate the phase difference between the current and the voltage.

b) Calculate the resistance R needed to obtain a maximum current of 300 mA.

Problem 7. Calculate the resonant frequency f of the RLC circuit shown in the figure.