A transformer used to step up $115 \mathrm{~V}$ to $5000 \mathrm{~V}$ has a primary coil of $100$ turns. What must be the number of turns in the secondary coil?

How much heat is required to melt a 2.5-kg block of ice at $0^{\circ} \mathrm{C}?$

Consider a parallel $R L C$ circuit with $R=$ $5.0 \Omega, L=30 \mu \mathrm{H}$, and $C=25 \mathrm{nF}$. An alternating emf with $\mathcal{E}_{\max }=15 \mathrm{~V}$ is applied. Determine the total current in the circuit when $(a)$ It operates at the resonant frequency. $(b)$ It operates at half the resonant frequency.

Consider that an emf is connected to a parallel $R L C$ circuit. In this case, the voltage $V_{\max }$ is the same across the source and each element, but the net current is the sum of the individual currents, so a phasor diagram for the currents can be drawn instead. Draw this diagram. By dividing the magnitude of each current phasor by the common value of $V_{\max }$, obtain a phasor diagram that relates impedance, reactances, and resistance. Use the geometry of the diagram to show that

$$Z=\left[\frac{1}{R^2}+\left(\frac{1}{X_L}-\frac{1}{X_C}\right)^2\right]^{-1 / 2}$$

The average power dissipated in a series $R L C$ circuit is $\bar{P}=\frac{1}{2} I_{\max }^2 R$, where $I_{\max }$ is given by Eq. ($I_{\max }=\frac{\mathcal{E}_{\max }}{Z}=\frac{\mathcal{E}_{\max }}{\sqrt{R^2+[\omega L-(1 / \omega C)]^2}}$). We can define the width of resonance in terms of the half-power points $\bar{P}_{1 / 2}$ of the resonance curve; these have half the peak value, or $\bar{P}_{1 / 2}=\frac{1}{2}\left(\frac{1}{2} \mathcal{E}_{\max }^2 / R\right)$. By setting $\bar{P}=\bar{P}_{1 / 2}$, obtain an equation for the values of $\omega$ at the half-power points. Solve the resulting quadratic equation and show that the two roots are separated by

$$\Delta \omega=\frac{R}{L}$$

Thus we can verify the equivalence of Eqs. ($\mathrm{Q}=\frac{\omega_0 L}{R}$) and ($\mathrm{Q}=\frac{\omega_0}{\Delta \omega}$), $\mathrm{Q}=\omega_0 L / R=\omega_0 / \Delta \omega$.

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 $\mathcal{E}_{\max } / \sqrt{2}$ ?

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 series $R L C$ circuit with $R=3.0 \Omega, C=20 \mathrm{pF}$, and a resonant frequency of $17 \mathrm{MHz}$ is connected to an alternating emf with $\mathcal{E}_{\max }=12 \mathrm{~V}$. $(a)$ What is the amplitude of the voltage across the inductor at resonance? $(b)$ What is the amplitude of the voltage across the resistor at resonance? $(c)$ What is the average power dissipated in this circuit? $(d)$ What is the quality factor $\mathbf{Q}$ of this circuit?

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?

If you connect a capacitor across a $115~\mathrm{V}$ outlet, does any current flow through the connecting wires? Through the space between the capacitor plates? Does the outlet deliver instaneous electric power? Average electric power?

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?

Two 40 Ω resistors are connected in parallel. In series with them are a 50 Ω resistor and a 500 Ω resistor. The total resistance of the circuit is __________.

A transformer with 1000 turns in its primary coil has to reduce the voltage from 4400 V to 220 V for home use. How many turns must there be on the secondary coil?