An electromagnet consists of 200 loops and has a length of $10.0 \mathrm{~cm}$ and a cross-sectional area of $5.00 \mathrm{~cm}^2$. Find the resonant frequency of this electromagnet when it is attached to the Earth (treat the Earth as a spherical capacitor).

In the RLC circuit in the figure, $R=60 \Omega, L=3 \mathrm{mH}$, $C=4 \mathrm{mF}$, and the source of time-varying emf has a peak voltage of $120 \mathrm{~V}$. What should the angular frequency, $\omega$, be to produce the largest current in the resistor? a) $4.2 \mathrm{rad} / \mathrm{s}$ b) $8.3 \mathrm{rad} / \mathrm{s}$ c) $204 \mathrm{rad} / \mathrm{s}$ d) $289 \mathrm{rad} / \mathrm{s}$ e) $5000 \mathrm{rad} / \mathrm{s}$ f) $20,000 \mathrm{rad} / \mathrm{s}$

a) A loop of wire $5.00 \mathrm{~cm}$ in diameter is carrying a current of $2.00 \mathrm{~A}$. What is the energy density of the magnetic field at its center? b) What current has to flow in a straight wire to produce the same energy density at a point $4.00 \mathrm{~cm}$ from the wire?

An RLC circuit has a capacitor, a resistor, and an inductor connected in parallel, as shown in the figure, and connected to a time-varying source of emf providing $V_{\text {rms }}$ at frequency $f$. Find an expression for $I_{\text {rms }}$ in terms of $V_{\text {rmss }}, f, L, C$, and $R$.

The discussion of RL. RC, and RLC circuits in this chapter has assumed a purely resistive resistor, one whose inductance and capacitance are exactly zero. While the capacitance of a resistor can generally be neglected, inductance is an intrinsic part of the resistor. Indeed, one of the most widely used resistors, the wire-wound resistor, is nothing but a solenoid made of highly resistive wire. Suppose a wire-wound resistor of unknown resistance is connected to a DC power supply. At a voltage of $V=10.0 \mathrm{~V}$ across the resistor, the current through the resistor is $1.00 \mathrm{~A}$. Next, the same resistor is connected to an $\mathrm{AC}$ power source providing $V_{\text {rms }}=10.0 \mathrm{~V}$ at a variable frequency. When the frequency is $20.0 \mathrm{kHz}$, a current, $I_{\text {rms }}=0.800 \mathrm{~A}$, is measured through the resistor. a) Calculate the resistance of the resistor. b) Calculate the inductive reactance of the resistor. c) Calculate the inductance of the resistor. d) Calculate the frequency of the AC power source at which the inductive reactance of the resistor exceeds its resistance.

In the RC high-pass filter shown in the figure, $R=10.0 \mathrm{k} \Omega$ and $C=0.0470 \mu \mathrm{F}$. What is the $3.00-\mathrm{dB}$ frequency of this circuit (where $\mathrm{dB}$ means basically the same for ? That is, at what frequency does the ratio of output voltage to input voltage satisfy $20 \log \left(V_{\text {out }} / V_{\text {in }}\right)=-3.00$ ?

A $5.00-\mu \mathrm{F}$ capacitor in series with a $4.00-\Omega$ resistor is charged with $9.00-\mathrm{V}$ battery for a long time by closing the switch (position $a$ in the figure). The capacitor is then discharged through an inductor $(L=40.0 \mathrm{mH})$ by closing the switch (position $b$ ) at $t=0$. a) Determine the maximum current through the inductor. b) What is the first time at which the current is at its maximum?

A transformer with 400 turns in its primary coil and 20 turns in its secondary coil is designed to deliver an average power of 1200 . W with a maximum voltage of $60.0 \mathrm{~V}$. What is the maximum current in the primary coil?

What is the resistance in an RLC circuit with $L=65.0 \mathrm{mH}$ and $C=1.00 \mu \mathrm{F}$ if the circuit loses $3.50 \%$ of its total energy as thermal energy in each cycle?

A $360-\mathrm{Hz}$ source of emf is connected in a circuit consisting of a capacitor, a $25-\mathrm{mH}$ inductor, and an $0.80-\Omega$ resistor. For the current and voltage to be in phase what should the value of $C$ be?

What are the maximum values of (a) current and (b) voltage when an incandescent 60 -W light bulb (at $110 \mathrm{~V}$ ) is connected to a wall plug labeled $110 \mathrm{~V}$ ?

A series RLC circuit has a source of time-varying emf providing $12.0 \mathrm{~V}$ at a frequency $f_0$, with $L=7.00 \mathrm{mH}$, $R=100 . \Omega$, and $C=0.0500 \mathrm{mF}$. a) What is the resonant frequency of this circuit? b) What is the average power dissipated in the resistor at this resonant frequency?

The AM radio band covers the frequency range from $520 \mathrm{kHz}$ to $1610 \mathrm{kHz}$. Assuming a fixed inductance in a simple LC circuit, what ratio of capacitance is necessary to cover this frequency range? That is, what is the value of $C_{\mathrm{h}} / C_1$, where $C_{\mathrm{h}}$ is the capacitance for the highest frequency and $C_1$ is the capacitance for the lowest frequency?

When you turn the dial on a radio to tune it, you are adjusting a variable capacitor in an LC circuit. Suppose you tune to an AM station broadcasting at a frequency of $1000 . \mathrm{kHz}$, and there is a $10.0$-mH inductor in the tuning circuit. When you have tuned in the station, what is the capacitance of the capacitor?