College Physics

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 15

The Electric Field - all with Video Answers

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Chapter Questions

Problem 1

(a) Construct a graph of the magnitude of the $\vec{E}$ field-versus-position for the $\vec{E}$ field created by a point-like object with charge $+Q .(b)$ Using the same set of axes, draw a graph for the field produced by an object of charge $+2 Q .(\mathrm{c})$ Using the same set of axes, draw a graph for the field produced by an object of charge $-2 Q$ .

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 2

A uranium nucleus has 92 protons. (a) Determine the magnitude of the $\vec{E}$ field at a distance of $0.58 \times 10^{-12} \mathrm{m}$ from the nucleus (about the radius of the innermost electron orbit
around the nucleus). (b) What is the magnitude of the force exerted on an electron by this $\vec{E}$ field? (c) What assumptions did you make? If the assumptions are not valid, will the magnitude in part (b) be an overestimate or underestimate of the actual value?

Dading Chen

Numerade Educator

Problem 3

The electron and the proton in a hydrogen atom are about $10^{-10} \mathrm{m}$ from each other. What quantities related to the $\vec{E}$ field can you determine using this information?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 4

Use the superposition principle to draw $\vec{E}$ field lines for the two objects whose charges are given. Consider all objects to be point-like and choose the distance you want between them:
$$
(\mathrm{a})+q \text { and }+q ;(\mathrm{b})+q \text { and }+3 q ;(\mathrm{c})+q \text { and }-q
$$

Paul A.

California State Polytechnic University, Pomona

Problem 5

Use the superposition principle to draw $\vec{E}$ field lines for the following objects whose charges are given. Consider all objects to be point-like and choose the distance you want between them: $(a)+q$ and $-3 q ;(b)+q,+q,$ and $-3 q .$

Rashmi Sinha

Numerade Educator

Problem 6

$\vec{E}$ field lines for a field created by an arrangement of charged objects are shown in Figure $P 15.6 .$ (a) Where are these objects located, and what are the signs of their electric charge? (b) What else can you determine using the information? Give two examples.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 7

Two objects with charges $+4.0 \times 10^{-9} \mathrm{C}$ and $+3.0 \times 10^{-9} \mathrm{C}$ are 50 $\mathrm{cm}$ from each other. Find a location where the $\vec{E}$ field due to these two charged objects is zero.

Harper Auman

Numerade Educator

Problem 8

$\mathrm{A}+4.0 \times 10^{-9}-\mathrm{C}$ charged object is 6.0 $\mathrm{cm}$ along a horizontal line toward the right of a $-3.0 \times 10^{-9}-$ C charged object. Determine the $\vec{E}$ field at a point 4.0 $\mathrm{cm}$ to the left of the negative charge.

Harper Auman

Numerade Educator

Problem 9

$\mathrm{A}+4.0 \times 10^{-9}-\mathrm{C}$ charged object is 4.0 $\mathrm{cm}$ along a horizontal line toward the right of a $-3.0 \times 10^{-9}-\mathrm{C}$ charged object. Determine the $\vec{E}$ field at a point 3.0 $\mathrm{cm}$ directly above the negative charge.

Harper Auman

Numerade Educator

Problem 10

A distance $d$ separates two objects, each with charge $+q .$ Determine an expression for the $\vec{E}$ field at a point that is a distance $d$ from both charges.

Harper Auman

Numerade Educator

Problem 11

A point-like charged object with a charge $+q$ is placed in an external horizontal uniform electric field $\vec{E}_{0}$ that points to the right. Determine an expression for the net $\vec{E}$ field at a distance $d$ from the charge (a) to the right of it and along a horizontal line going through the charge $+q$ and (b) to the left of it along that same line.

Daniel Alva

Numerade Educator

Problem 12

A 3.0 -g aluminum foil ball with a charge of $+4.0 \times 10^{-9} \mathrm{C}$ is suspended on a string in a uniform horizontal $\vec{E}$ field. The string makes an angle of $30^{\circ}$ with the vertical. What information about the $\vec{E}$ field can you determine for this situation?

Harper Auman

Numerade Educator

Problem 13

(a) If the string in the previous problem is cut, how long will it take the ball to fall to the floor 1.5 $\mathrm{m}$ below the level of the ball? (b) How far will the ball travel in the horizontal direction while falling? Indicate all the assumptions that you made.

Harper Auman

Numerade Educator

Problem 14

Using Earth's $\vec{E}$ field for flight Earth has an electric charge on its surface that produces a $150-\mathrm{N} / \mathrm{C} \vec{E}$ field, which points down toward the center of Earth. Estimate the electric charge that a person would need so that the electric force that the field exerts on the person will support the person in the air above Earth. Indicate all of your assumptions. Is this a reasonable idea? Explain.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 15

An electron moving with a speed $v_{0}$ enters a region where an $\vec{E}$ field points in the same direction as the electron's velocity. What will happen to the electron in this field? Answer the question qualitatively and quantitatively (using symbols).

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 16

A 1.0 -g aluminum foil ball with a charge of $1.0 \times 10^{-9} \mathrm{C}$ hangs freely from a $1.0-\mathrm{m}$ -long thread. What happens to the ball when a horizontal $5000-\mathrm{N} / \mathrm{C} \vec{E}$ field is turned on? Answer the question as fully as possible.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 17

A $0.50-\mathrm{g}$ oil droplet with charge $+5.0 \times 10^{-9} \mathrm{C}$ is in a vertical $\vec{E}$ field. In what direction should the $\vec{E}$ field point and what magnitude should it have so that the droplet moves at constant speed?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 18

An electron is ejected into a horizontal uniform $\vec{E}$ field at a parallel horizontal velocity of 1000 $\mathrm{m} / \mathrm{s} .$ Describe everything you can about the motion of the electron.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 19

Equation Jeopardy 1 The equations below describe one or more physical processes. Solve the equations for the unknowns and write a problem statement for which the equations are a satisfactory solution.
$$
\begin{array}{l}{-(0.10 \mathrm{kg})(9.8 \mathrm{N} / \mathrm{kg})+T \sin 53^{\circ}=0} \\ {+\left(-1.0 \times 10^{-6} \mathrm{C}\right) E_{x}+T \cos 53^{\circ}=0}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 20

Equation Jeopardy 2 The equations below describe one or more physical processes. Solve the equations for the unknowns and write a problem statement for which the equations are a satisfactory solution.
$$
\begin{array}{l}{\left(+1.0 \times 10^{-5} \mathrm{C}\right)\left(-4.0 \times 10^{4} \mathrm{N} / \mathrm{C}\right)=(0.20 \mathrm{kg}) a_{x}} \\ {0=(8.0 \mathrm{m} / \mathrm{s})+a_{x} t}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 21

During a lightning flash, $-15 \mathrm{C}$ of charge moves through a potential difference of $8.0 \times 10^{7} \mathrm{V} .$ Determine the change in electric potential energy of the field-charge system.

Harper Auman

Numerade Educator

Problem 22

(a) Sketch a $V$ -versus-position graph for the electric potential created by a point-like object with charge $+Q .(\mathrm{b})$ Using the same set of axes, draw a graph for an object with charge $+2 Q$ and for an object with charge $-2 Q$ .

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 23

A horizontal distance $d$ separates two objects each with charge $+q .$ Determine the value of the electric potential at a point that is located at a distance $d$ from each charge.

Harper Auman

Numerade Educator

Problem 24

Two objects with charges $-q$ and $+q$ are separated by a ditance $d .$ Determine an expression for the $V$ field at a point that is located at a distance $d$ from each charge.

Harper Auman

Numerade Educator

Problem 25

Four objects with the same charge $-q$ are placed at the corners of a square of side $d .$ Determine the values of the $\vec{E}$ field and $V$ field in the center of the square.

Harper Auman

Numerade Educator

Problem 26

Spark jumps to nose An electric spark jumps from a person's finger to your nose and through your body. While passing through the air, the spark travels across a potential difference of $2.0 \times 10^{4} \mathrm{V}$ and releases $3.0 \times 10^{-7} \mathrm{J}$ of electric potential energy. What is the charge in coulombs, and how many electrons flow?

Harper Auman

Numerade Educator

Problem 27

Two $-3.0 \times 10^{-6}-\mathrm{C}$ charged point-like objects are separated by 0.20 $\mathrm{m}$ . Determine the potential (assuming zero volts at infinity) at a point (a) halfway between the objects and (b) 0.20 $\mathrm{m}$ to the side of one of the objects (and 0.40 $\mathrm{m}$ from the other) along a line joining them.

Harper Auman

Numerade Educator

Problem 28

The potential difference from the cathode (negative electrode) to the screen of an old television set is $+22,000 \mathrm{V}$ . An electron leaves the cathode with an initial speed of zero. Determine everything you can about the motion of the electron in the TV set using this information.

Harper Auman

Numerade Educator

Problem 29

Imagine a $10,000$ -kg shuttle bus that carries $a+15-$ C charged sphere on its roof. Through what potential difference must the bus travel to acquire a speed of 10 $\mathrm{m} / \mathrm{s}$ , assuming no friction force exerted on the bus?

Harper Auman

Numerade Educator

Problem 30

Electric field in body cell The electric potential difference across the membrane of a body cell is $+0.070 \mathrm{V}$ (higher on the outside than on the inside). The cell membrane is $8.0 \times 10^{-9} \mathrm{m}$ thick. Determine the magnitude and direction of the $\vec{E}$ field through the cell membrane. Describe any assumptions you made.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 31

Energy used to charge nerve cells A nerve cell is shaped like a cylinder. The membrane wall of the cylinder has $\mathrm{a}+0.07$ -V potential difference from the inside to the outside of the wall. To help maintain this potential difference, sodium ions are pumped from inside the cell to the outside. For a
typical cell, $10^{9}$ ions are pumped each second. (a) Determine the change in chemical energy each second required to produce this increase in electric potential energy. (b) If there are roughly $7 \times 10^{11}$ of these cells in the body, how much chemical energy is used in pumping sodium ions each second? (c) Estimate the fraction of a person's metabolic rate used to pump these ions.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 32

Equation Jeopardy 3 The equation below describes one or more physical processes. Solve the equation for the unknown and write a problem statement for which the equation is a satisfactory solution.
$$
0=(1 / 2)(100 \mathrm{kg})(6.0 \mathrm{m} / \mathrm{s})^{2}+\left(2.0 \times 10^{-4} \mathrm{C}\right) \Delta V
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 33

Equation Jeopardy 4 The equation below describes one or more physical situations. Solve the equation for the unknown and write a problem statement for which the equation is a satisfactory solution.
$$
\begin{array}{l}{\left(9.0 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\right)\left(+2.0 \times 10^{-5} \mathrm{C}\right) /(2000 \mathrm{m})} \\ {+\left(9.0 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\right)\left(-2.0 \times 10^{-5} \mathrm{C}\right) /(1000 \mathrm{m})=\Delta V}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 34

A metal sphere has no charge on it. A positively charged object is brought near, but does not touch the sphere. Show that this object can exert a force on the sphere even though the sphere has no net charge. How can you test your answer experimentally?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 35

A Van de Graaff generator of radius 0.10 $\mathrm{m}$ has a charge of about $-1.0 \times 10^{-6} \mathrm{C}$ on it. The Van de Graaff generator is then turned off and grounded. How many excess electrons remain on its dome?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 36

A metal ball of radius $R_{1}$ has a charge $\frac{1}{2} Q .$ Later it is connected to a metal ball of radius $R_{2} .$ What is the fraction of the charge $Q$ that remains on the first ball?

Harper Auman

Numerade Educator

Problem 37

(a) Draw a microscopic representation of what happens to the charge distribution in a conductor placed in an external uniform electric field. For simplicity, make the conductor of some regular shape. How can you test your explanation? (b) What happens if you cut the conductor in half (in the direction perpendicular to the direction of the $\vec{E}$ field ) while keeping it in the external field? How can you test your answer? (c) Repeat steps (a) and (b) for a dieelectric.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 38

A positively charged metal ball A is placed near metal ball B. Measurements demonstrate that the force that they exert on each other is zero. Is ball B charged? Explain.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 39

Positively charged metal sphere A is placed near metal sphere B. B has a very small positive charge. Explain why the spheres could attract each other. Draw sketches of the spheres and their charge distribution to support your answer.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 40

Two small metal spheres A and B have different electric potentials (A has a higher potential). Describe in words and mathematically what happens if you connect them with a wire. What assumptions did you make?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 41

Two metal spheres of the same radius are placed close to each other. (a) Will they exert the same magnitude force on each other when they have like charges as when they have opposite charges? Explain. (b) What assumptions did you make? (c) How will you set up an experiment to test your answer?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 42

An electric dipole such as a water molecule is in a uniform $\vec{E}$ field. (a) Will the force exerted by the field cause the dipole to have a linear acceleration along a line in the direction of $\vec{E} ? \mathrm{Ex}-$ plain. (b) Will the field exert a torque on the dipole? Explain.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 43

Electric field of a fish An African fish called the aba has a charge $q=+1.0 \times 10^{-7} \mathrm{C}$
at its head and an equal magnitude negative charge $-q$ at its tail (see Figure $P 15.43 )$ . Determine the magnitude and direction of the electric field at position A and the force exerted on a hydroxide ion (charge $-e )$ at that point. The fish and ion are in water.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 44

Body cell membrane electric field (a) Determine the average magnitude of the $\vec{E}$ field across a body cell membrane. A 0.07 -V potential difference exists from one side to the other and the membrane is $7.5 \times 10^{-9} \mathrm{m}$ thick. (b) Determine the magnitude of the electrical force on a sodium ion (charge $+e )$ in the membrane. Assume that the dielectric constant is 1.0 (it is actually somewhat larger).

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 45

Earth's electric field Earth has an electric charge of approximately $-5.7 \times 10^{5} \mathrm{C}$ distributed relatively uniformly on its surface. Determine as many quantities as possible about
the electrical properties of the space around Earth. Use any additional information that you need.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 46

Geological exploration Determine the $\vec{E}$ field at position $\mathrm{A}$ in Figure $\mathrm{P} 15.46 \mathrm{due}$ to the di- Figure $\mathrm{P} 15.46$ pole charges produced by a geologist's electrodes. The dielectric constant of the soil is 8.0 and the dipole charge $q=4.0 \times 10^{-3} \mathrm{C}$ Determine the force exerted by this field on a sodium ion (charge $+e )$ at position A.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 47

You have a parallel-plate capacitor. (a) Determine the average $\vec{E}$ field between the plates if a $120-\mathrm{V}$ potential difference exists across the plates. Their separation is 0.50 $\mathrm{cm} .(\mathrm{b})$ A spark will jump if the magnitude of the $\vec{E}$ field exceeds $3.0 \times 10^{6}$ $\mathrm{V} / \mathrm{m}$ when air separates the plates. What is the closest the plates can be placed to each other without sparking?

Harper Auman

Numerade Educator

Problem 48

A capacitor of capacitance $C$ with a vacuum between the plates is connected to a source of potential difference $\Delta V .(a)$ Write an expression for the charge on each of the plates and for the total energy stored by the capacitor. (b) You then fill the capacitor with a dielectric material of dielectric constant $\kappa$ while the capacitor remains connected to the same potential difference. What are the new charge on the plates and the new energy stored by the capacitor? (c) Where did the energy
change come from?

Harper Auman

Numerade Educator

Problem 49

A capacitor of capacitance $C$ with a vacuum between the plates is connected to a source of potential difference $\Delta V$ . (a) Write an expression for the charge on each of the plates and for the total energy stored by the capacitor. (b) The capacitor is then disconnected from the potential difference source and you fill it with a dielectric that has a dielectric constant $\kappa .$ What are the new charge on the plates and the new energy? (c) Where did the energy change come from or where did the energy go?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 50

How does the capacitance of a parallel plate capacitor change if you double the magnitude of the charge on its plates? If you triple the potential difference across the plates? What assumptions did you make?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 51

Axon capacitance The long thin cylindrical axon of a nerve carries nerve impulses. The axon can be as long as 1 $\mathrm{m} .$ (a) Estimate the capacitance of a $1.0-\mathrm{m}$ -long axon of radius $4.0 \times 10^{-6} \mathrm{m}$ with a membrane thickness of $8.0 \times 10^{-9}$ $\mathrm{m} .$ The dielectric constant of the membrane material is about 6.0. (b) Determine the magnitude of the charge on the inside (negative) and outside (positive) of the membrane wall if there is a 0.070 -V potential difference across the wall. (c) Determine the energy stored in this axon capacitor when charged.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 52

Sphere capacitance A metal sphere of radius $R$ has an electric charge $+q$ on it. Determine an expression for the electric potential $V$ on the sphere's surface. Use the definition of capacitance to show that the capacitance of this isolated sphere is $R / k,$ where $k$ is the constant used in Coulomb's law.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 53

You have a capacitor of capacitance $C$ with plate separation $d$ and filled with a dielectric with a dielectric constant $\kappa .$ Design three different problems that you can solve using this information. Then solve the problems.

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Problem 54

Capacitance of red blood cell Assume that a red blood cell is spherical with a radius of $4 \times 10^{-6} \mathrm{m}$ and with wall thickness of $9 \times 10^{-8} \mathrm{m} .$ The dielectric constant of the membrane is about 5 . Assuming the cell is a parallel plate capacitor, estimate the capacitance of the cell and determine the positive charge on the outside and the equal-magnitude negative charge inside when the potential difference across the membrane is 0.080 $\mathrm{V}$ .

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 55

Defibrillator During ventricular fibrillation the heart muscles contract randomly, preventing the coordinated pumping of blood. A defibrillator can often restore normal blood pumping by discharging the charge on a capacitor through the heart. Paddles are held against the patient's chest, and a $6 \times 10^{-6}-\mathrm{F}$ charged capacitor is discharged in several milliseconds. If the capacitor energy is 250 $\mathrm{J}$ , what potential difference was used to charge the capacitor?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 56

The dielectric strength of air is $3 \times 10^{6} \mathrm{V} / \mathrm{m} .$ As you walk across a synthetic rug, your body accumulates electric charge, causing a potential difference of 6000 $\mathrm{V}$ between your body and a doorknob. What can you can determine using this information?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 57

Charged cloud causes electric field on Earth The electric charge of clouds is a complex subject. Consider the simplified model shown in Figure $P 15.57 .$ A. A positive charge is near the top of the cloud and a negative charge is near the bottom. Determine the direction of the $\vec{E}$ field on Earth at point $\mathrm{P}$ below the cloud and explain it so that a classmate can understand why there is positive charge on the ground directly below the cloud.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 58

Heart's dipole charge The heart has a dipole charge distribution with a charge of $+1.0 \times 10^{-7} \mathrm{C}$ that is 6.0 $\mathrm{cm}$ above a charge of $-1.0 \times 10^{-7} \mathrm{C} .$ Determine the $\vec{E}$ field (magnitude and direction) caused by the heart's dipole at a distance of 8.0 $\mathrm{cm}$ directly above the heart's positive charge. All charges are located in body tissue of dielectric constant $7.0 .$ What is the force exerted on a sodium ion (charge $+1.6 \times 10^{-19} \mathrm{C} )$ at that point?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 59

A speck of dust of mass 0.10 \mug floats between two oppositely charged horizontal metal plates separated by 0.10 m. The potential difference between the plates is 200 $\mathrm{V}$ . The speck is not falling at first, but when a beam of ultraviolet light hits it, it starts accelerating downward. However, increasing the potential difference to 250 $\mathrm{V}$ reduces the downward acceleration of the speck to zero. Explain the phenomenon qualitatively and quantitatively using symbols.

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Problem 60

A 0.050 -kg cart has a spherical dome that is charged to $+1.0 \times 10^{-5} \mathrm{C} .$ The cart is at the top of a $2.0-\mathrm{m}$ -long inclined plane that makes a $30^{\circ}$ angle with the horizontal. The cart and the plane are placed in an upward-pointing vertical uniform $4000-\mathrm{N} / \mathrm{C} \vec{E}$ field. How long will it take the cart to reach the bottom of the plane? What assumptions did you make?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 61

Can a shark detect an axon $\vec{E}$ field? A nerve signal is transmitted along the long, thin axon of a neuron in a small fish. The transmission occurs as sodium ions Na' transfer like tipping dominos across the axon membrane from outside to inside. Each short section of axon gets an excess of about $6 \times 10^{8}$ sodium ions/mm. Determine the $\vec{E}$ field 4.0 $\mathrm{cm}$ from the axon produced by the excess sodium ions on the inside of the axon and an equal number of negative ions on the out-
side of a $1-\mathrm{mm}$ length of axon. The ions are separated by the $8 \times 10^{-9}$ -m-thick axon membrane. Will a shark that is able to detect fields as small as $10^{-6} \mathrm{N} / \mathrm{C}$ be able to detect that axon field? Explain.

Rashmi Sinha

Numerade Educator

Problem 62

Lightning warms water A lightning flash occurs when $-40 \mathrm{C}$ of charge moves from a cloud to Earth through a potential difference of $4.0 \times 10^{8} \mathrm{V}$ . Estimate how much water can boil as a result of energy released during the process. Describe the assumptions you made.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 63

In a hot water heater, water warms when electric potential energy is converted into thermal energy. (a) Determine the energy needed to warm 180 $\mathrm{kg}$ of water by $10^{\circ} \mathrm{C} .(\mathrm{b})$ If $-10.0 \mathrm{C}$ of electric charge passes through a $+120-\mathrm{V}$ potential difference in the heating coils each second, determine the time needed to warm the water by $10^{\circ} \mathrm{C} .$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 64

Electrophoresis Electrophoresis is used to separate biological molecules of different dimensions and electric charge (the molecules can have different charge depending on the pH of the solution). A particular molecule of radius $R$ with charge $q$ is in a viscous solution that has an $\vec{E}$ field across it. The field exerts an electric force on the molecule and the viscous solution exerts an opposing drag force $F_{\text { drag }}=D R v,$ where $D$ is a constant drag coefficient that depends on the shape and other features of the molecule and the solution, and $v$ is the molecule's speed in the solution. When the molecule gets up to speed, the electric force exerted on it by the field is equal in magnitude and opposite in direction to the drag force. Show that during time interval $\Delta t,$ the molecule will travel a distance $\Delta x=\frac{q E \Delta t}{D R} .$ Describe any assumptions you made.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 65

Energy stored in axon electric field An axon has a surface area of $5 \times 10^{-6} \mathrm{m}^{2}$ and the membrane is $8 \times 10^{-9} \mathrm{m}$ thick. The dielectric constant of the membrane is 6 . (a) Determine the capacitance of the axon considered as a parallel plate capacitor. (b) If the potential difference across the membrane wall is 0.080 $\mathrm{V}$ , determine the magnitude of the charge on each wall. (c) Determine the energy needed to charge that axon capacitor. (d) Determine the magnitude of the $\vec{E}$ field across the membrane due to the opposite sign charges across the membrane walls. (e) Calculate the energy density of that field. (f) Multiply the volume of space occupied by that field by the volume of the membrane to get the total energy stored in the field. How do the answers to (c) and (f) compare?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 66

Suppose you have a 1.0-F capacitor (very large capacitance) with a 0.10-V potential difference across the capacitor. What is the magnitude of the electric charge on each plate of the capacitor?
$$
\begin{array}{llllllll}{\text { (a) } 0.05 \mathrm{C}} & {\text { (b) } 0.10 \mathrm{C}} & {\text { (c) } 0.20 \mathrm{C}} & {\text { (d) } 1.0 \mathrm{C}} & {\text { (e) } 10 \mathrm{C}} & {\text { (e) } 10 \mathrm{C}}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 67

Suppose you place two of these $1.0-\mathrm{F}$ capacitors with the charge calculated in the previous question as shown in Figure $\mathrm{P} 15.67 \mathrm{a}$ . What is the net potential difference across the two capacitors (from one dot to the other)?
$$
\begin{array}{l}{\text { (a) } 0.05 \mathrm{V} \quad \text { (b) } 0.10 \mathrm{V} \quad \text { (c) } 0.20} \\ {\mathrm{V} \quad \text { (d) None of these }} \\ {\text { (e) Not enough information }}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 68

If both capacitors are discharged simultaneously, how much electric charge goes through the wire shown in pink in Figure P15.67a?
$$
\begin{array}{llll}{\text { (a) } 0.05 \mathrm{C}} & {\text { (b) } 0.10 \mathrm{C}} & {\text { (c) } 0.20 \mathrm{C}} & {\text { (d) } 1.0 \mathrm{C} \text { (e) } 10 \mathrm{C}}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 69

Suppose you place the two capacitors with the 0.10-V potential difference across each capacitor as shown in Figure P15.67b. What is the net potential difference across the two capacitors?
$$
\begin{array}{l}{\text { (a) } 0.05 \mathrm{V} \quad \text { (b) } 0.10 \mathrm{V} \quad \text { (c) 0.20 } \mathrm{V} \quad \text { (d) None of these }} \\ {\text { (e) Not enough information }}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 70

Look at the electrocyte shown in Figure 15.31c. What causes the 0.10-V potential difference from the outer left to the outer right side of the cell?
(a) The membrane is thicker on the left than on the right.
(b) The ion distribution across the left membrane is different than across the right membrane.
(c) The left and right membranes have different capacitances.
(d) b and c
(e) a, b, and c

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Problem 71

Suppose that one cell of the electrocyte is regarded as a small capacitor with a 0.10-V potential difference across it. How should we arrange 10 cells to get a 1.0-V potential difference across them?
(a) In series , as in Figure P15.67a
(b) In parallel, as in Figure P15.67b
(c) Randomly so that they do not cancel each other
(d) Not enough information

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 72

An $\vec{E}$ field of approximately $3 \times 10^{6} \mathrm{V} / \mathrm{m}$ is needed to ionize
dry air molecules. If the potential difference between the thin wires and metal plates is $60,000 \mathrm{V},$ what minimum distance must the wires be from the plates in order to cause dry air to ionize?
$$
\begin{array}{llll}{\text { (a) } 1.0 \mathrm{cm}} & {\text { (b) } 2.0 \mathrm{cm}} & {\text { (c) } 4.0 \mathrm{cm}} & {\text { (d) } 50 \mathrm{cm}} & {\text { (e) } 50 \mathrm{m}}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 73

Why are the smoke particles attracted to the closely spaced plates?
(a) The particles are negatively charged.
(b) The particles are positively charged.
(c) The plates are positively charged.
(d) The plates are negatively charged.
(e) a and c are correct.
(f) b and c are correct

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 74

Suppose a $2.0 \times 10^{-6}-\mathrm{kg}$ dust particle with charge $-4.0 \times 10^{-9} \mathrm{C}$ is moving vertically up a chimney at speed 6.0 $\mathrm{m} / \mathrm{s}$ when it enters the $+2000-\mathrm{N} / \mathrm{C} \vec{E}$ field pointing away from a metal collection plate of an electrostatic precipitator. If the particle is 2.0 $\mathrm{cm}$ from the plate at that instant, which answer below is closest to the magnitude of its horizontal acceleration toward the plate?
$$(a) 2.0 \times 10^{-5} \mathrm{m} / \mathrm{s}^{2} \quad (b) 1.2 \times 10^{-4} \mathrm{m} / \mathrm{s}^{2}$$
$$(c) 4.0 \times 10^{-4} \mathrm{m} / \mathrm{s}^{2} \quad (d) 4.0 \mathrm{m} / \mathrm{s}^{2}$$
$$(e) 24 \mathrm{m} / \mathrm{s}^{2}$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 75

Suppose everything is the same as in the previous problem. Which answer below is closest to the vertical distance the dust particle will move before hitting the plate?
$$
\begin{array}{llll}{\text { (a) } 0.01 \mathrm{m}} & {\text { (b) } 0.06 \mathrm{m}} & {\text { (c) } 0.1 \mathrm{m}} & {\text { (d) } 0.6 \mathrm{m}} & {\text { (e) } 1.0 \mathrm{m}}\end{array}
$$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 76

What is the purpose of giving a negative charge to the particles collected by the precipitator?
(a) The particles are then attracted to the positively charged plates while moving upward.
(b) The particles are then repelled from the positively charged plates while moving upward.
(c) The extra mass slows the upward movement of the particles.
(d) The negative charge cancels the positive charge the particles have when entering the precipitator.
(e) a and c are correct

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 77

Suppose the average particle moving with air up a chimney has a mass of $4.0 \times 10^{-6} \mathrm{kg}$ and charge of $-6.0 \times 10^{-8} \mathrm{C}$ The particles move with the air at speed 6.0 $\mathrm{m} / \mathrm{s}$ and average distance of 3.0 $\mathrm{cm}$ from a vertical plate. The plate is 0.60 $\mathrm{m}$ tall. Which answer below is closest to the minimum horizontal electric field that the plate must have so that the particles will be collected before moving past the top of the plate?
$$\begin{array}{ll}{\text { (a) } 100 \mathrm{N} / \mathrm{C}} & {\text { (b) } 200 \mathrm{N/C}} \\ {\text { (c) } 400 \mathrm{N} / \mathrm{C}} & {\text { (d) } 1000 \mathrm{N/C}}\end{array}$$
$$(e) 4000 \mathrm{N} / \mathrm{C}$$

Guilherme Barros

Guilherme Barros

Numerade Educator