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

An electric heater operating with a $115~\mathrm{V}$ AC power supply delivers $1200 \mathrm{~W}$ of heat. $(a)$ What is the rms current through this heater? $(b)$ What is the maximum instantaneous current? $(c)$ What is the resistance of this heater?

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?

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)?

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

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?