College Physics 2013

A bulb in a table lamp has a current of 0.50 A through it. Determine two physical quantities related to the electrons passing through the wires leading to the bulb.

A long wire is connected to the terminals of a battery. In $8.0 \mathrm{s},$ $9.6 \times 10^{20}$ electrons pass a cross section along the wire. The electrons flow from left to right. What physical quantities can you determine using this information?

A typical flashlight battery will produce a 0.50 -A current for about 3 $\mathrm{h}$ before losing its charge. Determine the total number of electrons that have moved past a cross section of wire connecting the battery and lightbulb.

$*$ Four friends each have a battery, a bulb, and one wire. They use the materials to try to light their bulb. Two succeed and two do not. What possible circuit arrangements could they have made?

Draw a circuit that has a battery, a lightbulb, and connecting wires. Draw a schematic of a water flow system that allows a continual circulation of water and compare the function of each element of the circuit to each element of the water flow system.

Add another battery to the circuit described in Problem 5. In how many different ways can you do this? Draw analogous changes in the water flow system.

Add another lightbulb to the circuit with one battery described in Problem $5 .$ In how many different ways can you connect the second bulb? What do you need to add to the water flow system to make it similar to the circuit?

$* \mathrm{A} 9.0-\mathrm{V}$ battery is connected to a resistor so that there is a 0.50 -A current through the resistor. For how long should the battery be connected in order to do 200 $\mathrm{J}$ of work while separating charges?

A graph of the electric potential versus location in a series circuit with 1.0 $\mathrm{A}$ of current is shown in Figure $\mathbf{P} 16.9 .$ Draw a circuit in which such change could occur.

Sketch a potential-versus- location graph for the circuit shown in Figure $P 16.10$ . Start at A and move clockwise around the circuit.

BIO Electric currents in the body A person accidentally touches a 120 - V electric line with one hand while touching a ground wire with the other hand. Determine the current through the body when the hands are dry $(100,000-\Omega \text { resistance) and when wet }(5000-\Omega$ resistance). If the current exceeds about 10 $\mathrm{mA}$ , muscular contractions may prevent the person from releasing the
wires $-$ a dangerous situation. Is the person in danger with dry hands? With wet hands? Explain.

(a) An automobile light has a $1.0-$ A current when it is connected to a $12-\mathrm{V}$ battery. What can you determine about the light using this information? (b) What potential difference is needed to produce a current of 5.0 $\mathrm{mA}$ through a $2.0-\Omega$ resistor?

$*$ If a long wire is connected to the terminals of a $12-\mathrm{V}$ battery, $6.4 \times 10^{19}$ electrons pass a cross section of the wire each second. Make a list of the physical quantities whose values you can determine using this information and determine three of them.

Determine the current through a $2.5-\Omega$ flashlight filament when connected across two $1.5-\mathrm{V}$ batteries in series (a potential difference of 3.0 $\mathrm{V} )$

$*$ You have a circuit with a $50-\Omega,$ a $100-\Omega,$ and a $150-\Omega$ resistor connected in series. (a) Rank the current through them from highest to lowest. (b) Rank the potential difference across them from highest to lowest. Explain your rankings.

$*$ You have a circuit with a $50-\Omega,$ a $100-\Omega,$ and a $150-\Omega$ resistor connected in parallel to the same battery. (a) Rank the current through them from highest to lowest. (b) Rank the
potential difference across them from highest to lowest.

$*$ You connect a $50-\Omega$ resistor to a $9-V$ battery whose internal resistance is 1$\Omega .$ (a) Determine the electric power dissipated by the resistor. $(b)$ Will a lightbulb with a $10-\Omega$ resistance dissipate more or less power if connected to the same battery?

$* *$ EST Making tea You use an electric teapot to make tea. It takes about 2 $\min$ to boil 0.5 $\mathrm{L}$ of water. (a) Estimate the power of the heater. What are your assumptions? (b) Estimate
the current through the heater. State your assumptions.

$*$ If a long wire is connected to the terminals of a $12-\mathrm{V}$ battery, $6.4 \times 10^{19}$ electrons pass a cross section of the wire each second. What is the rate of work being done by the battery?

$* *$ Three friends are arguing with each other. Aidan says that a battery will send the same current to any circuit you connect to it. Cathy says that the battery will produce the same potential difference at its terminals independent of the circuit connected to it. Eugenia says that the battery will always produce the same electric power independent of the circuit connected to it. Describe experiments that you can perform to convince all of them that their opinions are wrong in at least one situation.

$*$ You have a $40-\mathrm{W}$ lightbulb and a $100-\mathrm{W}$ bulb. What do these readings mean? Can we say that the 100 - W lightbulb is always brighter? Explain and give examples.

$*$ Does a $60-$ W lightbulb have more or less resistance than a 100 - W bulb? Explain.

$*$ (a) Write two loop rule equations and one junction rule equation for the circuit in Figure $P 16.23$ .
(b) Use these equations to determine the current in each branch of the circuit for the case in which
$R_{1}=0 \Omega, R_{2}=18 \Omega, R_{3}=9 \Omega$ and $\varepsilon=60 \mathrm{V}$

$*$ Repeat the previous problem for the case in which $R_{1}=50 \Omega, R_{2}=30 \Omega, R_{3}=15 \Omega,$ and $\varepsilon=120 \mathrm{V}$

$*$ The current through resistor $R_{1}$ in Figure $\mathrm{P} 16.23$ is 2.0 $\mathrm{A}$ . Determine the currents through resistors $R_{2}$ and $R_{3}$ and the emf of the battery. $R_{1}=4 \Omega, R_{2}=10 \Omega,$ and $R_{3}=40 \Omega$

$*$ Use Kirchhoff's rules to prove in general that the currents $I_{2}$ and $I_{3}$ through resistors $R_{2}$ and $R_{3}$ shown in Figure $P 16.23$ satisfy the relation $I_{2} / I_{3}=R_{3} / R_{2}$ .

$*$ (a) Write Kirchhoffs loop rule for the circuit shown in Figure $P 16.27$ for the case in which $\quad \varepsilon_{1}=20 \mathrm{V}, \varepsilon_{2}=8 \mathrm{V}$ $R_{1}=30 \Omega, R_{2}=20 \Omega, \quad$ a n $\mathrm{d}$ $R_{3}=10 \Omega .(\mathrm{b})$ Determine the current in the circuit. (c) Using this value of current, start at position A and move clockwise around the circuit, calculating the electric potential change across each element in the circuit (be sure to indicate the sign of each change).
(d) Add these potential changes around the whole circuit. Explain your result.

$* \%$ Repeat parts (a) and (b) of the previous problem for the case in which $\varepsilon_{1}=12 \mathrm{V}, \varepsilon_{2}=3 \mathrm{V}, R_{1}=R_{2}=1 \Omega,$ and $R_{3}=16 \Omega .(\mathrm{c})$ Determine the potential difference from A to B. (d) Draw a potential-versus-position graph starting at any location in the circuit and returning to the same point.

(a) Determine the value of $\varepsilon_{1}$ so that there is a clockwise current of 1.0 $\mathrm{A}$ in the circuit shown in Figure P16.27, where $\varepsilon_{2}=12 \mathrm{V}, R_{1}=2 \Omega, R_{2}=1 \Omega,$ and $R_{3}=12 \Omega .(\mathrm{b})$ Determine the potential difference from $\mathrm{B}$ to $\mathrm{A}$ . (c) Draw a graph representing the potential at different points of the circuit.

(a) Write the loop rule for two different loops in the circuit shown in Figure $P 16.30$ and the junction rule for point A. Solve the equations to find the current in each loop when $\varepsilon_{1}=3 \mathrm{V}, \varepsilon_{2}=6 \mathrm{V}, R_{1}=10 \Omega, R_{2}=20 \Omega,$ and $R_{3}=30 \Omega$ (b) Determine the potential difference from A to B. Check your answer by taking a different path
from A to $\mathrm{B}$ (c) Sketch a potential-versus-position graph for each of the three loops, assum-
ing the potential is zero at B.

$* *$ Determine the value of $R_{2}$ shown in Figure $\mathrm{P} 16.30,$ so that the current through $R_{3}$ equals twice that through $R_{2}$ . The values of other circuit elements are $\varepsilon_{1}=12 \mathrm{V}, \varepsilon_{2}=15 \mathrm{V}, R_{1}=15 \Omega,$ and $R_{3}=30 \Omega$

$*$ Determine (a) the equivalent resistance of resistors $R_{1}, R_{2},$ and $R_{3}$ in Figure $\mathbf{P} 16.32$ for $R_{1}=28 \Omega, R_{2}=30 \Omega$ and $R_{3}=20 \Omega$ and (b) the current through the battery if $\varepsilon=10 \mathrm{V}$

$\stackrel{*}{*}(\text { a) Determine the equivalent resistance }$ of resistors $R_{1}, R_{2},$ and $R_{3}$ in Figure $\mathrm{P} 16.32$ for $R_{1}=50 \Omega, R_{2}=30 \Omega$ and $R_{3}=15 \Omega .(\mathrm{b})$ Determine the current through $R_{1}$ if $\varepsilon=120 \mathrm{V} .(\mathrm{c})$ Use Kirchhoffs loop rule and your result from part (b) to determine the current through $R_{2}$ and through $R_{3}$ .

$*$ Determine the equivalent resistance of the resistors shown in Figure $P 16.34$ if $R_{1}=60 \Omega, R_{2}=30 \Omega, R_{3}=20 \Omega$ $R_{4}=20 \Omega, R_{5}=60 \Omega, R_{6}=20 \Omega$
and $R_{7}=10 \Omega$

$*$ Determine the equivalent resistance of the resistors in Figure $\mathrm{P} 16.34$ if $R_{1}=R_{2}=20 \Omega$ $R_{3}=10 \Omega, R_{4}=25 \Omega, R_{5}=$ $30 \Omega, R_{6}=10 \Omega,$ and $R_{7}=50 \Omega$

$*$ Determine $(\mathrm{a})$ the equivalent resistance of the resistors in the circuit in Figure $\mathrm{P} 16.36$ and (b) the current through the battery.

$*$ Write a problem for which the following mathematical statement can be a solution:
$$
1 \mathrm{A}=\frac{\varepsilon-(1 \mathrm{A})(3 \Omega)}{R}
$$

$*$ Home wiring A simplified electrical circuit for a home is shown in Figure $P 16.38$ (a) Determine the
currents through currents through the circuit breaker, the circuit breaker, the lightbulb, the microwave oven, and the toaster. (b) Determine the electric power used by each appliance.

$* *$ (a) Write Kirchhoffs rules for two loops and one junction in the circuit shown in Figure $P 16.39 .$ (b) Solve the equations for the current in each branch of the circuit when $\varepsilon_{1}=10 \mathrm{V}, \varepsilon_{2}=2 \mathrm{V}, R_{1}=50 \Omega$ $R_{2}=200 \Omega,$ and $R_{3}=20 \Omega$ (c) Determine the potential difference across resistor $R_{3}$ from point $\mathrm{B}$ to point $\mathrm{A}$

$*$ * BIO Electric eel The South American eel can generate electric current that can stun and even kill nearby fish. The eel has 140 parallel rows of electric cells $(0.15 \mathrm{V} \text { per cell })$
Each row has 5000 such cells for a total emf per row of $5000(0.15 \mathrm{V})=750 \mathrm{V}$ . Each row of cells also has about 1250$\Omega$ of internal resistance. Because each row has the same emf and the rows are connected together on each side, the eel's circuit can be represented as shown in Figure $P 16.40-a 750-\mathrm{V}$ emf source in series with $140 ~ 1250-\Omega$ parallel internal resistances all connected across an external resistance (the seawater from the front of the eel to its back) of about 800$\Omega$ Can the eel produce enough current to be dangerous to a person?
Explain your answer. Identify all assumptions that you made.

$*$ Home wiring $\mathrm{A} 120-\mathrm{V}$ electrical line in a home is connected to a $60-\mathrm{W}$ lightbulb, a 180 - W television set, a $300-\mathrm{W}$ desktop computer, a 1050 - W toaster, and a $240-\mathrm{W}$ refrigerator. How much current is flowing in the line?

Determine (a) the equivalent resistance, (b) the current through the battery, and (c) the power supplied by the battery for the circuit shown in Figure $P 16.42 .$ (d) Use Kirchhoff's loop rule to determine the currents through the $60-\Omega, 40-\Omega,$ and $30-\Omega$ resistors.

$*$ Tree lights Nine tree lights are connected in parallel across a $120-\mathrm{V}$ potential difference. The cord to the wall socket carries a current of 0.36 $\mathrm{A}$ . (a) Determine the resistance of one
of the bulbs. (b) What would the current be if the bulbs were connected in series?

$* *$ Two lightbulbs use 30 $\mathrm{W}$ and $60 \mathrm{W},$ respectively, when connected in parallel to a $120-\mathrm{V}$ source. How much power does each bulb use when connected in series across the $120-\mathrm{V}$ source, assuming that their resistances remain the same?

$*$ Three identical resistors, when connected in series, transform electrical energy into thermal energy at a rate of 15 $\mathrm{W}$ $(5 \mathrm{W} \text { per resistor). Determine the power consumed by the }$ resistors when connected in parallel to the same potential difference.

$* *$ Impedance matching A battery has an emf of 12 $\mathrm{V}$ and an internal resistance of 3$\Omega .$ (a) Determine the power delivered to a resistor $R$ connected to the battery terminals
for values of $R$ equal to $1,2,3,4,5,$ and 6$\Omega .$ (b) Plot on a graph the calculated values of $P$ versus the different values of $R$ . Connect the points by a smooth curve. Confirm that the maximum power is delivered when $R$ has the same resistance as the internal resistance of the power source $(3 \Omega \text { in this example). }$

$* \mathrm{A} 100-\mathrm{m}$ -long copper wire of radius 0.12 $\mathrm{mm}$ is connected across a $1.5-\mathrm{V}$ battery. Make a list of the physical quantities that you can determine using this information and determine the values of three of them.

$*$ BMT subway rail resistance The BMT subway line in New York City stretches roughly 30 $\mathrm{km}$ from the Bronx to Brooklyn. The electrified rail on which it runs has a cross section of about 40 $\mathrm{cm}^{2}$ and is made of steel with a resistivity of $10 \times 10^{-8} \Omega \cdot \mathrm{m}$ . Make a list of all the physical quantities de- scribing different properties of the rail that you can determine using this information. Then calculate the values of one quantity related to the electrical properties and one quantity not related to the electrical properties.

$*$ Thermometer A platinum resistance thermometer consists of a coil of 0.10 -mm-diameter platinum wire wrapped in a coil. Determine the length of wire needed so that the coil's resistance at $20^{\circ} \mathrm{C}$ is 25$\Omega .$ The resistivity of platinum at this temperature is $1.0 \times 10^{-7} \Omega \cdot \mathrm{m} .$

As the potential difference in volts across a thin platinum wire increases, the current in amperes changes as follows: $(\Delta V, I)=(0,0),(1.0,0.112),(3.0,0.337),$ and $(6.1,0.675) .$ Plot a graph of potential difference as a function of current and indicate whether the platinum wire satisfies Ohm's law. Explain how you made your decision.

$*$ Bl0 Respiration detector A respiration detector monitors a person's breathing. One type consists of a flexible hose filled with conductive salt water (resistivity of 5.0$\Omega \cdot \mathrm{m} )$
Electrodes at the ends of the tube measure the resistance of the fluid in the tube. The tube is wrapped around a person's chest. When the person inhales and exhales, the tube stretches and contracts and its resistance changes. Determine the factor by which the resistance of the fluid changes when the hose is
stretched so that its length increases by a factor of $1.1 .$ The water volume remains constant.

$*$ A wire whose resistance is $R$ is stretched so that its length is tripled while its volume remains unchanged. Determine the resistance of the stretched wire.

Ratio reasoning Determine the ratio of the resistances of two wires that are identical except that (a) wire A is twice as long as wire $\mathrm{B},$ (b) wire A has twice the radius of wire $\mathrm{B}$ , and $(\mathrm{c})$ wire $\mathrm{A}$ is made of copper and wire $\mathrm{B}$ is made of aluminum. Be
sure to show clearly how you arrive at each answer.

$* *$ Electronics detective You need to determine the mass and length of the wire inside a particular electronic device. You cannot take it out. Devise a method to do this by using a battery, ammeter, voltmeter, and micrometer (a device that measures small distances.)

A battery produces a 2.0 -A current when connected to an unknown resistor of resistance $R .$ When a $10-\Omega$ resistor is connected in series with $R$ , the current drops to 1.2 $\mathrm{A}$ . (a) Determine the emf of the battery and the resistance $R$ . What assumptions did you make? Do your assumptions make each of the values you determined greater than or less than their actual values? Explain.

Bl0 Resistance of human nerve cell Some human nerve cells have a long, thin cylindrical cable (the axon) from their inputs to their outputs. Consider an axon of radius $5 \times 10^{-6} \mathrm{m}$ and length 0.6 $\mathrm{m} .$ The resistivity of the fluid inside the axon is 0.5$\Omega \cdot \mathrm{m} .$ Determine the resistance of the fluid in this axon.

" A fuse for one line in your home's electrical system melts if the electric current through it is greater than 30 A. Will the fuse melt if the following appliances are all connected in parallel to that line: a $13-\Omega$ toaster, an $18-\Omega$ dishwasher, a $24-\Omega$ refrigerator, and a $15-\Omega$ heater? The potential difference across the devices is 120 $\mathrm{V}$ .

$* B 10$ Circulatory system (a) Use Kirchhoffs rules to determine the current in each branch of the circuit shown in Figure $P 16.58$ for the case in which $\varepsilon=90 \mathrm{V}, R_{1}=10 \Omega$
$R_{2}=30 \Omega,$ and $R_{3}=60 \Omega$ (b) Repeat the current calculations if $R_{1}$ increases to $30 \Omega,$ while $R_{2}$ and $R_{3}$ remain unchanged. (c) Discuss briefly how this electrical circuit is analogous to the circulatory system of a person and the effect of a decrease in the size of a main artery on the flow of blood to other parts of the body.

$*$ Bl0 Circulatory system Suppose the electric circuit shown in Figure $\mathrm{P} 16.58$ models the circulatory system with $10,000$ arterioles in parallel, each of resistance 100$\Omega .(\mathrm{a})$ Determine the net resistance of the arterioles. (b) If the resistance of each arteriole increased by 50$\%$ because of narrowing, what now would be the net resistance of the $10,000$ arterioles? (c) By how much would the potential difference across these arteriole resistors change if the flow rate (current) is to remain the same?

$*$ BIO Arteriosclerosis Recall that the resistance of a vessel to fluid flow is inversely proportional to the radius of the vessel to the fourth power. (a) By what factor would a resistor representing the aorta shown in Figure P16.58 change if the aorta radius of a person was 0.20 times what it had been years earlier? (b) If the pressure drop across the aorta had originally been 5 $\mathrm{mm} \mathrm{Hg}$ with normal blood flow, what would the pressure drop now be in order to have the same flow rate? Note that the blood pressure will have to increase to maintain the same flow rate.

$* *$ People current Suppose that all the people on the Earth moved at the same speed around a circular track. Approximately how many times per second would each person pass the starting line if that "people current" equaled the number of electrons passing a cross section in a wire when there is a
1.0 -A current through it?

$* *$ EST Bird on a power line A bird stands on a $20,000-\mathrm{V}$ bare copper power line. A $10-\mathrm{A}$ current passes through the line. Estimate the current through the bird. Be sure to discuss
all of your assumptions and indicate clearly the method you use in making the estimate.

$* *$ A 5.0 -A current caused by moving electrons flows through a wire. (a) Determine the number of electrons that flow past a cross section each second. (b) The same number of water molecules moves along a stream. Determine the volume of water that moves each second under a bridge that passes over the stream.

The horizontal $4-\Omega$ resistors in the two circuits in Figure $P 16.78$ represent the resistance of a small horizontal length of fluid inside an axon. The $15-\Omega$ resistors represent the resistance across the axon membrane for a tiny length of axon. The $10-\Omega$ resistor to the right side of each "ladder" is the effective resistance in front of the axon sections under consideration. Determine the resistance between $\mathrm{B}$ and $\mathrm{G}$ of the small length of axon shown in Figure $\mathrm{P} 16.78 \mathrm{a}$ and between $\mathrm{A}$ and $\mathrm{G}$ of the slightly longer length axon shown in Figure $\mathrm{P} 16.78 \mathrm{b}$ .
$$
\begin{array}{ll}{\text { (a) } 4.2 \Omega, 8.4 \Omega} & {\text { (b) } 10 \Omega, 10 \Omega}&{text { (c) }29 \Omega, 48 \Omega }\\ {\text { (d) } 10 \Omega, 12 \Omega} & {\text { (e) } 4.2 \Omega, 12 \Omega}\end{array} text { (c) 29 } \Omega, 48 \Omega
$$

Suppose nerve impulses travel at 100 $\mathrm{m} / \mathrm{s}$ in the axons of nerve cells from your fingers to your brain and then back again to your fingers in order to stimulate muscles that lift your fingers off the hot burner of a stove. Which answer below is closest to the time interval needed for the nerve signal transmission along the axons?
$$\begin{array}{llll}{\text { (a) } 0.01 \mathrm{s}} & {\text { (b) } 0.001 \mathrm{s}} & {\text { (c) } 0.2 \mathrm{s}}\\{\text { (d) }0.02 \mathrm{s}} &{\text { (e) } 0.002 \mathrm{s}}\end{array}$
$$

The electrical resistance across dry skin is about $100,000 \Omega .$ Suppose a person with dry skin puts one hand on a 120 -V power cord from a home wall socket while the other hand is touching a metal object at 0 $\mathrm{V}(\text { at what is called ground). }$ Which condition described below is most likely to occur?
(a) No sensation (b) Threshold of pain
(c) Cannot let go (d) Ventricular fibrillation
(e) Severe burns and shock

The electrical resistance across wet skin is about 1000$\Omega .$ Sup- pose a person with wet skin puts one hand on a $120-\mathrm{V}$ power cord from a home wall socket while the other hand is touching a metal object at 0 $\mathrm{V}(\text { at what is called ground). Which }$ condition described below is most likely to occur?
(a) No sensation (b) Threshold of pain
(c) Cannot let go (d) Ventricular fibrillation
(e) Severe burns and shock

Suppose the electrical resistance across your wet skin is about 1000$\Omega$ . Which answer below is closest to the least potential difference from one hand that will cause slight pain?
(a) 0.1 V (b) 1 V (c) 10 V
(d) 100 V (e) 1000 V

Suppose the electrical resistance across your wet skin is about 1000$\Omega .$ Which answer below is closest to the least potential difference from one hand to the other that will cause ventricular fibrillation?
(a) 0.1 V (b) 1 V (c) 10 V
(d) 100 V (e) 1000 V

When muscular contraction caused by electrical stimulation prevents a person from releasing contact with the potential difference sources, lower current can be extremely dangerous. For example, a 100 -mA current for 3 s causes about the same effect as a 900 -mA current for 0.03 s. In which of these situations is the least electric charge transferred through the body?
(a) 100 $\mathrm{mA}$ for 3 s $\quad$ (b) 900 $\mathrm{mA}$ for 0.03 $\mathrm{s}$
(c) Too little information to decide

Why is it dangerous to place a hair dryer, radio, or other electric appliance that is plugged into a wall socket near a bathtub?
(a) The water provides a conductive path for current, which heats the metal cover on the appliance and can cause burns.
(b) If the appliance is accidentally knocked into the tub while a person is bathing, large currents could pass through the person’s low-resistance body because of the 120-V potential difference that powers the appliance.
(c) There is no potential for danger, because electric appliances are grounded.
(d) a and b

Occasionally, the electric circuit that produces a coordinated pumping of blood from the four chambers of the heart becomes disturbed. Ventricular fibrillation can occur-random muscle contractions that produce little or no blood pumping. To stop the fibrillation, two defibrillator pads are placed on the chest and a large current (about 14 amps) is sent through the heart, restarting its normal rhythmic pattern. The current lasts 10 $\mathrm{ms}$ and transfers 140 $\mathrm{J}$ of electric energy to the body. Which answer below is closest to the potential difference between the defibrillator pads?
(a) 100 V (b) 400 V (c) 1000 V
(d) 5000 V (e) 10,000 V