|| It seems hard to justify spending $5.00 for an LED lightbulb when an ordinary incandescent bulb costs 50¢. To see if this makes sense, compare a 60 W incandescent bulb that lasts 1000 hours to a 10 W LED bulb that has a lifetime of 15,000 hours. Both bulbs produce the same amount of visible light. If electricity costs $0.15/kWh, what is the total cost—purchase price plus energy—to get 15,000 hours of light from each type of bulb? This is called the life-cycle cost.

| A refrigerator has a 1000 W compressor, but the compressor runs only 20% of the time.

1. If electricity costs $0.15/kWh, what is the monthly (30 day) cost of running the refrigerator?

2. A more energy-efficient refrigerator with an 800 W compressor costs $150 more. If you buy the more expensive refrigerator, how many months will it take to recover your additional cost?

|| Two 75 W (120 V) lightbulbs are wired in series, then the combination is connected to a 120 V supply. How much power is dissipated by each bulb?

| BIO An electric eel develops a 450 V potential difference between its head and tail. The eel can stun a fish or other prey by using this potential difference to drive a 0.80 A current pulse for 1.0 ms. What are (a) the energy delivered by this pulse and (b) the total charge that flows?

|| You have a $2.0\text{Ω}$ resistor, a $3.0\text{Ω}$ resistor, a $6.0\text{Ω}$ resistor, and a 6.0 V battery. Draw a diagram of a circuit in which all three resistors are used and the battery delivers 9.0 W of power.

|| CALC A 2.0-m-long, 1.0-mm-diameter wire has a variable resistivity given by

where $x$ is measured from one end of the wire. What is the current if this wire is connected to the terminals of a 9.0 V battery?

||

1. To which two points in the circuit of FIGURE P28.45 should a 12 V battery be connected to dissipate the most power?

2. What is the maximum power?

   FIGURE P28.45

   

   FIGURE P28.45 Full Alternative Text

|| What is the equivalent resistance between points 1 and 2 in FIGURE P28.46?

FIGURE P28.46

|| INT A small toaster that operates at 120 V has a heating element made from a 4.4-m-long, 0.70-mm-diameter nichrome wire. The resistivity, density, and specific heat of nichrome are, respectively, $1.5\times {10}^{-6}\text{Ω}\text{m}$, $8400{\text{kg/m}}^3$, and $450\text{J}\text{/}\text{kg}\text{K}$.

1. What is the power rating of this toaster?

2. If half the heat energy is lost to the air, how long does it take the heating element to warm from $20\text{°}\text{C}$ to $450\text{°}\text{C}$, about the temperature at which it first begins to glow red?

|| A string of holiday lights can be wired in series, but all the bulbs go out if one burns out because that breaks the circuit. Most lights today are wired in series, but each bulb has a special fuse that short-circuits the bulb—making a connection around it—if it burns out, thus keeping the other lights on. Suppose a string of 50 lights is connected in this way and plugged into a 120 V outlet. By what factor does the power dissipated by each remaining bulb increase when the first bulb burns out?

|| Suppose you have resistors $2.5\text{k}\text{Ω}$, $3.5\text{k}\text{Ω}$, and $4.5\text{k}\text{Ω}$ and a 100 V power supply. What is the ratio of the total power delivered to the resistors if they are connected in parallel to the total power delivered if they are connected in series?

|| A lightbulb is in series with a $2.0\text{Ω}$ resistor. The lightbulb dissipates 10 W when this series circuit is connected to a 9.0 V battery. What is the current through the lightbulb? There are two possible answers; give both of them.

||

1. CALC Load resistor $R$ is attached to a battery of emf $\epsilon$ and internal resistance $r$. For what value of the resistance $R$, in terms of $\epsilon$ and $r$, will the power dissipated by the load resistor be a maximum?

2. What is the maximum power that the load can dissipate if the battery has $\epsilon =9.0\text{V}$ and $r=1.0\text{Ω}?$

||| For an ideal battery $(r=0\text{Ω})$, closing the switch in FIGURE P28.52 does not affect the brightness of bulb A. In practice, bulb A dims just a little when the switch closes. To see why, assume that the 1.50 V battery has an internal resistance $r=0.50\text{Ω}$ and that the resistance of a glowing bulb is $R=6.00\text{Ω}$.

1. What is the current through bulb A when the switch is open?

2. What is the current through bulb A after the switch has closed?

3. By what percentage does the current through A change when the switch is closed?

FIGURE P28.52

|| The ammeter in FIGURE P28.53 reads 3.0 A. Find $I_1,I_2$, and $\epsilon$.

FIGURE P28.53

|| What is the current in the $2\text{Ω}$ resistor in FIGURE P28.54?

FIGURE P28.54

| What are the battery current $I_{\text{bat}}$ and the potential difference $V_1-V_2$ between points 1 and 2 when the switch in FIGURE P28.55 is (a) open and (b) closed?

FIGURE P28.55

|| A circuit you’re building needs an ammeter that goes from 0 mA to a full-scale reading of 50 mA. Unfortunately, the only ammeter in the storeroom goes from $0\mu \text{A}$ to a full-scale reading of only $500\mu \text{A}$. Fortunately, you’ve just finished a physics class, and you realize that you can make this ammeter work by putting a resistor in parallel with it, as shown in FIGURE P28.56. You’ve measured that the resistance of the ammeter is $50.0\text{Ω}$, not the $0\text{Ω}$ of an ideal ammeter.

1. What value of $R$ must you use so that the meter will go to full scale when the current $I$ is 50 mA?

2. What is the effective resistance of your ammeter?

FIGURE P28.56

|| For the circuit shown in FIGURE P28.57, find the current through and the potential difference across each resistor. Place your results in a table for ease of reading.

FIGURE P28.57

|| For the circuit shown in FIGURE P28.58, find the current through and the potential difference across each resistor. Place your results in a table for ease of reading.

FIGURE P28.58

|| What is the current through the $20\text{Ω}$ resistor in FIGURE P28.59?

FIGURE P28.59

|| For the circuit shown in FIGURE P28.60, find the current through and the potential difference across each resistor. Place your results in a table for ease of reading.

FIGURE P28.60

|| What is the current through the $10\text{Ω}$ resistor in FIGURE P28.61? Is the current from left to right or right to left?

FIGURE P28.61

|| How much power is dissipated by the $2\text{Ω}$ resistor in FIGURE P28.62?

FIGURE P28.62

|| For what emf $\epsilon$ does the $200\text{Ω}$ resistor in FIGURE P28.63 dissipate no power? Should the emf be oriented with its positive terminal at the top or at the bottom?

FIGURE P28.63

| A 12 V car battery dies not so much because its voltage drops but because chemical reactions increase its internal resistance. A good battery connected with jumper cables can both start the engine and recharge the dead battery. Consider the automotive circuit of FIGURE P28.64.

1. How much current could the good battery alone drive through the starter motor?

2. How much current is the dead battery alone able to drive through the starter motor?

3. With the jumper cables attached, how much current passes through the starter motor?

4. With the jumper cables attached, how much current passes through the dead battery, and in which direction?

FIGURE P28.64

|| What power is dissipated by the $2\text{Ω}$ resistor in FIGURE P28.65?

FIGURE P28.65

|| How much current flows through the bottom wire in FIGURE P28.66, and in which direction?

FIGURE P28.66

| The capacitor in an RC circuit is discharged with a time constant of 10 ms. At what time after the discharge begins are (a) the charge on the capacitor reduced to half its initial value and (b) the energy stored in the capacitor reduced to half its initial value?

|| A circuit you’re using discharges a $20\mu \text{F}$ capacitor through an unknown resistor. After charging the capacitor, you close a switch at $t=0\text{s}$ and then monitor the resistor current with an ammeter. Your data are as follows:

Use an appropriate graph of the data to determine (a) the resistance and (b) the initial capacitor voltage.

|| BIO A $150\mu \text{F}$ defibrillator capacitor is charged to 1500 V. When fired through a patient’s chest, it loses 95% of its charge in 40 ms. What is the resistance of the patient’s chest?

|| A $50\mu \text{F}$ capacitor that had been charged to 30 V is discharged through a resistor. FIGURE P28.70 shows the capacitor voltage as a function of time. What is the value of the resistance?

FIGURE P28.70

|| A $0.25\mu \text{F}$ capacitor is charged to 50 V. It is then connected in series with a $25\text{Ω}$ resistor and a $100\text{Ω}$ resistor and allowed to discharge completely. How much energy is dissipated by the $25\text{Ω}$ resistor?

|| A $15\mu F$ capacitor charged to 12 V is discharged through a resistor. The energy stored in the capacitor decreases by 50% in 0.25 s. What is the value of the resistance?

|| CALC The capacitor in FIGURE P28.73 begins to charge after the switch closes at $t=0\text{s}$.

1. What is $\text{Δ}{V}_{\text{C}}$ a very long time after the switch has closed?

2. What is $Q_{\text{max}}$ in terms of $\epsilon$, $R$, and $C$?

3. In this circuit, does $I=+dQ\text{/}dt$ or $-dQ\text{/}dt?$ Explain.

4. Find an expression for the current $I$ at time t. Graph $I$ from $t=0$ to $t=5\tau$.

FIGURE P28.73

|| CALC The switch in Figure 28.38a closes at $t=0\text{s}$ and, after a very long time, the capacitor is fully charged. Find expressions for (a) the total energy supplied by the battery as the capacitor is being charged, (b) total energy dissipated by the resistor as the capacitor is being charged, and (c) the energy stored in the capacitor when it is fully charged. Your expressions will be in terms of $\epsilon$, $R$, and $C$. (d) Do your results for parts a to c show that energy is conserved? Explain.

|| The flash on a compact camera stores energy in a $120\mu \text{F}$ capacitor that is charged to 220 V. When the flash is fired, the capacitor is quickly discharged through a lightbulb with $5.0\text{Ω}$ of resistance.

1. Light from the flash is essentially finished after two time constants have elapsed. For how long does this flash illuminate the scene?

2. At what rate is the lightbulb dissipating energy $250\mu \text{s}$ after the flash is fired?

3. What total energy is dissipated by the lightbulb?

|| Intermittent windshield wipers use a variable resistor—one whose resistance changes as the control knob is rotated—in an RC circuit to set the delay between passes of the wipers. A typical circuit is shown in FIGURE P28.76. When the switch is closed, the capacitor (initially uncharged) begins to charge and the potential at point 2 begins to increase. A sensor (which you can assume draws no current) measures the voltage between points 1 and 2. When $\text{Δ}{V}_{12}=0\text{V,}$ a signal is sent to the wiper motor and also to another part of the circuit (not shown) that discharges the capacitor so that the cycle can start over. What value of the variable resistor causes a pass of the wipers every 12 s?

FIGURE P28.76

| The capacitors in FIGURE P28.77 are charged and the switch closes at $t=0\text{s}$. At what time has the current in the $8\text{Ω}$ resistor decayed to half the value it had immediately after the switch was closed?

FIGURE P28.77

|| BIO Large capacitors can hold a potentially dangerous charge long after a circuit has been turned off, so it is important to make sure they are discharged before you touch them. Suppose a $120\mu \text{F}$ capacitor from a camera flash unit retains a voltage of 150 V when an unwary student removes it from the camera. If the student accidentally touches the two terminals with his hands, and if the resistance of his body between his hands is $1.8\text{k}\text{Ω}$, for how long will the current across his chest exceed the danger level of 50 mA?

||| You’ve made the finals of the Science Olympics! As one of your tasks, you’re given 1.0 g of aluminum and asked to make a wire, using all the aluminum, that will dissipate 7.5 W when connected to a 1.5 V battery. What length and diameter will you choose for your wire?

||| Digital circuits require actions to take place at precise times, so they are controlled by a clock that generates a steady sequence of rectangular voltage pulses. One of the most widely used integrated circuits for creating clock pulses is called a 555 timer. FIGURE CP28.80 shows how the timer’s output pulses, oscillating between 0 V and 5 V, are controlled with two resistors and a capacitor. The circuit manufacturer tells users that $T_{\text{H}}$, the time the clock output spends in the high $(5\text{V})$ state, is $T_{\text{H}}=(R_1+R_2)C\times \text{ln}2$. Similarly, the time spent in the low $(0\text{V})$ state is $T_{\text{L}}=R_{2}C\times \text{ln}2$. You need to design a clock that will oscillate at 10 MHz and will spend 75% of each cycle in the high state. You will be using a 500 pF capacitor. What values do you need to specify for $R_1$ and $R_2?$

FIGURE CP28.80

||| The switch in FIGURE CP28.81 has been closed for a very long time.

1. What is the charge on the capacitor?

2. The switch is opened at $t=0\text{s}$. At what time has the charge on the capacitor decreased to 10% of its initial value?

FIGURE CP28.81

||| CALC The capacitor in Figure 28.38a begins to charge after the switch closes at $t=0\text{s}$. Analyze this circuit and show that $Q=Q_{\text{max}}(1-e^{-t\text{/}\tau })$, where $Q_{\text{max}}=C\epsilon$.

||| An oscillator circuit is important to many applications. A simple oscillator circuit can be built by adding a neon gas tube to an RC circuit, as shown in FIGURE CP28.83. Gas is normally a good insulator, and the resistance of the gas tube is essentially infinite when the light is off. This allows the capacitor to charge. When the capacitor voltage reaches a value $V_{\text{on}}$, the electric field inside the tube becomes strong enough to ionize the neon gas. Visually, the tube lights with an orange glow. Electrically, the ionization of the gas provides a very-low-resistance path through the tube. The capacitor very rapidly (we can think of it as instantaneously) discharges through the tube and the capacitor voltage drops. When the capacitor voltage has dropped to a value $V_{\text{off}}$, the electric field inside the tube becomes too weak to sustain the ionization and the neon light turns off. The capacitor then starts to charge again. The capacitor voltage oscillates between $V_{\text{off}}$, when it starts charging, and $V_{\text{on}}$, when the light comes on to discharge it.

1. Show that the oscillation period is

   $$T=RC\text{ln}\left({\displaystyle \frac{\epsilon -V_{\text{off}}}{\epsilon -V_{\text{on}}}}\right)$$

2. A neon gas tube has $V_{\text{on}}=80\text{V}$ and $V_{\text{off}}=20\text{V}$. What resistor value should you choose to go with a $10\mu \text{F}$ capacitor and a 90 V battery to make a 10 Hz oscillator?

FIGURE CP28.83