A series $R C$ circuit is often used as a high-pass filter. Consider a resistor $R$ and capacitor $C$ connected in series to an oscillating source of emf with amplitude $\mathcal{E}_{\max }$. In terms of $\mathcal{E}_{\max }$, what is the amplitude of the voltage across the resistor for a low frequency, when $\omega R C=1 / 10$ ? For a high frequency, when $\omega R C=10$ ? At what frequency is the amplitude of the voltage across the resistor equal to $\mathcal{E}_{\max } / \sqrt{2}$ ?

A motor may be modeled as a series $R L$ circuit. A particular motor operates from an ordinary $60~\mathrm{Hz}$ wall outlet with $\mathcal{E}_{\mathrm{rms}}=115 \mathrm{~V}$. The motor draws a current of $I_{\mathrm{rms}}=2.5 \mathrm{~A}$. If the effective resistance is $R=16 \Omega$, what is the inductive reactance of the circuit? What is the value of the inductance?

A series $R C$ circuit is connected to an alternating $60~\mathrm{Hz}$ emf with $\mathcal{E}_{\max }=163 \mathrm{~V}$. If $R=50 \Omega$, what value of $C$ will result in a current amplitude $I_{\max }=0.50 \mathrm{~A}$ ? What is the average rate of power dissipation in this circuit?

An underdamped series $R L C$ circuit with $R=2.2 \times 10^{-3} \Omega$ is to have an oscillation frequency of $3.5 \times 10^6 \mathrm{~Hz}$ and a quality factor $\mathcal{Q}$ of $750$ . What must be the values of the capacitance $C$ and the inductance $L$ ?

Consider an underdamped series $R L C$ circuitwith $R=0.75 \Omega, L=25 \mu \mathrm{H}$, and $C=100 \mathrm{nF}$. What is the value of the quality factor $\mathcal{Q}$ for this circuit? If the capacitor has an initial charge $Q_{\max }$, how many oscillations will occur before the charge amplitude is reduced to $(1 / e) \times Q_{\max } \approx 0.37 Q_{\max }$ ?

A radio receiver contains an $L C$ circuit whose natural frequency of oscillation can be adjusted, or tuned, to match the frequency of incoming radio waves. The adjustment is made by means of a variable capacitor. Suppose that the inductance of the circuit is $15 \mu \mathrm{H}$. Over what range of capacitances must the capacitor be adjustable if the frequencies of oscillation of the circuit are to span the range from $530 \mathrm{kHz}$ to $1600 \mathrm{kHz}$ ?

You want to construct an $L C$ circuit of natural frequency $8.0 \times 10^3 \mathrm{~Hz}$. You have available a capacitor of $0.20 \mu \mathrm{F}$. What inductor do you need?

Some electric motors operate only on DC, others only on AC. What is the difference between these motors?

The GG-$1$ electric locomotive develops $4600 \mathrm{hp}$; it runs on an rms AC voltage of $1100 \mathrm{~V}$. $(a)$ What rms current does this locomotive draw? $(b)$ Why is it advantageous to supply the electric power for locomotives at high voltage (and fairly low current)?

What is the natural frequency for an $L C$ circuit consisting of a $2.2 \times 10^{-6} \mathrm{~F}$ capacitor and an $8.0 \times 10^{-2} \mathrm{H}$ inductor?

An inductor is built in the shape of a solenoid of radius $0.20$ $\mathrm{cm}$, length $4.0 \mathrm{~cm}$, with 1000 turns. This inductor is connected to a generator supplying an rms voltage of $1.2 \times 10^{-4}$ $\mathrm{V} \mathrm{AC}$ at an angular frequency of $9.0 \times 10^3$ radians $/ \mathrm{s}$. $(a)$ What is the inductance? Assume that the solenoid can be treated as very long. $(b)$ What is the rms current flowing through the inductor? $(c)$ What is the rms energy stored in the inductor?

An inductor of $4.5 \times 10^{-2} \mathrm{H}$ and a capacitor of $0.25 \mu \mathrm{F}$ are connected in parallel to a source of alternating emf of frequency $\omega$. For what value of $\omega$ will the rms currents in the inductor and the capacitor be of equal magnitudes?

A $2.0~\mu \mathrm{F}$ capacitor and a $3.0~\mathrm{mH}$ inductor are connected in parallel with an AC power supply. At what frequency will the inductive reactance be $3$ times the capacitive reactance?

An inductor with $L=4.0 \times 10^{-2} \mathrm{H}$ is connected to a source of alternating emf. This source provides an $\operatorname{emf} \mathcal{E}=\mathcal{E}_{\max } \times$ $\cos \omega t$, with $\mathcal{E}_{\max }=0.20 \mathrm{~V}$ and $\omega=6.0 \times 10^3$ radians $/ \mathrm{s}$. $(a)$ What is the reactance of the inductor? $(b)$ What is the maximum current in the circuit? $(c)$ What is the current at time $t=0$ ? At time $t=\pi / 4 \omega$ ?