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University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 23

Electric Potential - all with Video Answers

Educators

Chapter Questions

07:42

Problem 1

A point charge $q_{1}=+2.10 \mu \mathrm{C}$ is held stationary at the origin. A second point charge $q_{2}=-4.60 \mu \mathrm{C}$ moves from the point $x=0.150 \mathrm{~m}, y=0$ to the point $x=0.250 \mathrm{~m}, y=0.270 \mathrm{~m} .$ How much work is done by the electric force on $q_{2} ?$

Linda Winkler
Linda Winkler
Numerade Educator
01:36

Problem 2

A point charge $q_{1}$ is held stationary at the origin. A second charge $q_{2}$ is placed at point $a,$ and the electric potential energy of the pair of charges is $+5.4 \times 10^{-8} \mathrm{~J}$. When the second charge is moved to point $b,$ the electric force on the charge does $-1.9 \times 10^{-8} \mathrm{~J}$ of work. What is the electric potential energy of the pair of charges when the second charge is at point $b ?$

Nishant Kumar
Nishant Kumar
Numerade Educator
03:37

Problem 3

How much work is needed to assemble an atomic nucleus containing three protons (such as Li) if we model it as an equilateral triangle of side $2.00 \times 10^{-15} \mathrm{~m}$ with a proton at each vertex? Assume the protons started from very far away.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
10:40

Problem 4

(a) How much work would it take to push two protons very slowly from a separation of $2.00 \times 10^{-10} \mathrm{~m}$ (a typical atomic distance) to $3.00 \times 10^{-15} \mathrm{~m}$ (a typical nuclear distance)? (b) If the protons are both released from rest at the closer distance in part (a), how fast are they moving when they reach their original separation?

Mohit Khurana
Mohit Khurana
Texas A&M University
10:29

Problem 5

A small metal sphere, carrying a net charge of $q_{1}=-2.60 \mu \mathrm{C}$ is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of $q_{2}=-7.50 \mu \mathrm{C}$ and mass $1.50 \mathrm{~g}$ is projected toward $q_{1}$. When the two spheres are $0.800 \mathrm{~m}$ apart, $q_{2}$ is moving toward $q_{1}$ with speed $\begin{array}{llll}22.0 \mathrm{~m} / \mathrm{s} & \text { (Fig. } & \text { E23.5). } & \text { Assume }\end{array}$ that the two spheres can be treated as point charges. You can ignore the force of gravity. (a) What is the speed of $q_{2}$ when the spheres are $0.420 \mathrm{~m}$ apart? (b) How close does $q_{2}$ get to $q_{1}$ ?

Linda Winkler
Linda Winkler
Numerade Educator
09:52

Problem 6

BIO Energy of DNA Base Pairing. (See Exercise 21.18.) (a) Calculate the electric potential energy of the adenine-thymine bond, using the same combinations of molecules $(\mathrm{O}-\mathrm{H}-\mathrm{N}$ and $\mathrm{N}-\mathrm{H}-\mathrm{N})$ as in Exercise 21.18 . (b) Compare this energy with the potential energy of the proton-electron pair in the hydrogen atom.

Sophie S
Sophie S
Numerade Educator
05:31

Problem 7

Two protons, starting several meters apart, are aimed directly at each other with speeds of $2.00 \times 10^{5} \mathrm{~m} / \mathrm{s}$, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
08:44

Problem 8

Three equal $1.40 \mu \mathrm{C}$ point charges are placed at the corners of an equilateral triangle with sides $0.300 \mathrm{~m}$ long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Linda Winkler
Linda Winkler
Numerade Educator
06:02

Problem 9

Two protons are released from rest when they are $0.750 \mathrm{nm}$ apart.
(a) What is the maximum speed they will reach? When does this speed occur?
(b) What is the maximum acceleration they will achieve? When does this acceleration occur?

Kai Chen
Kai Chen
Princeton University
14:11

Problem 10

Four electrons are located at the corners of a square $10.0 \mathrm{nm}$ on a side, with an alpha particle at its midpoint. How much work is needed to move the alpha particle to the midpoint of one of the sides of the square?

Mohit Khurana
Mohit Khurana
Texas A&M University
09:21

Problem 11

Points $a$ and $b$ lie in a region where the $y$ -component of the electric field is $E_{y}=\alpha+\beta / y^{2}$. The constants in this expression have the values $\alpha=600 \mathrm{~N} / \mathrm{C}$ and $\beta=5.00 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$. Points $a$ and $b$ are on the $+y$ -axis. Point $a$ is at $y=2.00 \mathrm{~cm}$ and point $b$ is at $y=3.00 \mathrm{~cm}$. What is the potential difference $V_{a}-V_{b}$ between these two points and which point, $a$ or $b$, is at higher potential?

Linda Winkler
Linda Winkler
Numerade Educator
08:53

Problem 12

An object with charge $q=-6.00 \times 10^{-9} \mathrm{C}$ is placed in a region of uniform electric field and is released from rest at point $A .$ After the charge has moved to point $B, 0.500 \mathrm{~m}$ to the right, it has kinetic energy $3.00 \times 10^{-7} \mathrm{~J}$. (a) If the electric potential at point $A$ is $+30.0 \mathrm{~V}$, what is the electric potential at point $B ?$ (b) What are the magnitude and direction of the electric field?

Mohit Khurana
Mohit Khurana
Texas A&M University
07:46

Problem 13

A small particle has charge $-5.70 \mu \mathrm{C}$ and mass $2.70 \times 10^{-4} \mathrm{~kg} .$ It moves from point $A,$ where the electric potential is $V_{A}=+270 \mathrm{~V},$ to point $B,$ where the electric potential is $V_{B}=+830 \mathrm{~V}$ The electric force is the only force acting on the particle. The particle has speed $5.90 \mathrm{~m} / \mathrm{s}$ at point $A$. What is its speed at point $B ?$ Is it moving faster or slower at $B$ than at $A$ ? Explain.

Linda Winkler
Linda Winkler
Numerade Educator
09:20

Problem 14

A particle with charge $+4.20 \mathrm{nC}$ is in a uniform electric field $\overrightarrow{\boldsymbol{E}}$ directed to the left. The charge is released from rest and moves to the left; after it has moved $6.00 \mathrm{~cm},$ its kinetic energy is $+2.20 \times 10^{-6} \mathrm{~J}$. What are (a) the work done by the electric force, $(b)$ the potential of the starting point with respect to the end point, and (c) the magnitude of $\overrightarrow{\boldsymbol{E}}$ ?

Mohit Khurana
Mohit Khurana
Texas A&M University
09:32

Problem 15

A charge of $30.0 \mathrm{nC}$ is placed in a uniform electric field that is directed vertically upward and has a magnitude of $3.60 \times 10^{4} \mathrm{~V} / \mathrm{m} .$ What work is done by the electric force when the charge moves (a) $0.490 \mathrm{~m}$ to the right; (b) $0.700 \mathrm{~m}$ upward; (c) $2.80 \mathrm{~m}$ at an angle of $45.0^{\circ}$ downward from the horizontal?

Linda Winkler
Linda Winkler
Numerade Educator
14:22

Problem 16

Two stationary point charges $+3.00 \mathrm{nC}$ and $+2.00 \mathrm{nC}$ are separated by a distance of $50.0 \mathrm{~cm} .$ An electron is released from rest at a point midway between the two charges and moves along the line connecting the two charges. What is the speed of the electron when it is $10.0 \mathrm{~cm}$ from the $+3.00 \mathrm{nC}$ charge?

Mohit Khurana
Mohit Khurana
Texas A&M University
09:06

Problem 17

Point charges $q_{1}=+2.00 \mu \mathrm{C}$ and $q_{2}=-2.00 \mu \mathrm{C}$ are placed at adjacent corners of a square for which the length of each side is $5.00 \mathrm{~cm}$. Point $a$ is at the center of the square, and point $b$ is at the empty corner closest to $q_{2}$. Take the electric potential to be zero at a distance far from both charges. (a) What is the electric potential at point $a$ due to $q_{1}$ and $q_{2} ?$ (b) What is the electric potential at point $b$ ? (c) A point charge $q_{3}=-6.00 \mu \mathrm{C}$ moves from point $a$ to point $b$. How much work is done on $q_{3}$ by the electric forces exerted by $q_{1}$ and $q_{2}$ ? Is this work positive or negative?

Linda Winkler
Linda Winkler
Numerade Educator
05:24

Problem 18

Two point charges of equal magnitude $Q$ are held a distance $d$ apart. Consider only points on the line passing through both charges; take $V=0$ at infinity. (a) If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?). (b) Repeat part (a) for two point charges having opposite signs.

Sophie S
Sophie S
Numerade Educator
08:33

Problem 19

Two point charges $q_{1}=$$+2.00 \mathrm{nC}$ and $q_{2}=-6.10 \mathrm{nC}$ are $0.100 \mathrm{~m}$ apart. Point $A$ is midway between them; point $B$ is $0.080 \mathrm{~m}$ from $q_{1}$ and $0.060 \mathrm{~m}$ from $q_{2}$ (Fig. E23.19). Take the electric potential to be zero at infinity. Find (a) the potential at point $A ;$ (b) the potential at point $B ;(\mathrm{c})$ the work done by the electric field on a charge of $3.00 \mathrm{nC}$ that travels from point $B$ to point $A$.

Linda Winkler
Linda Winkler
Numerade Educator
10:50

Problem 20

(a) An electron is to be accelerated from $2.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$ to $8.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$. Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from $8.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$ to a halt?

Linda Winkler
Linda Winkler
Numerade Educator
04:43

Problem 21

Points $A$ and $B$ lie within a region of space where there is a uniform electric field that has no $x$ - or $z$ -component; only the $y$ -component $E_{y}$ is nonzero. Point $A$ is at $y=8.00 \mathrm{~cm}$ and point $B$ is at $y=15.0 \mathrm{~cm} .$ The potential difference between $B$ and $A$ is $V_{B}-V_{A}=+12.0 \mathrm{~V},$ so point $B$ is at higher potential than point $A$. (a) Is $E_{y}$ positive or negative? (b) What is the magnitude of the electric field? (c) Point $C$ has coordinates $x=5.00 \mathrm{~cm}, y=5.00 \mathrm{~cm} .$ What is the potential difference between points $B$ and $C$ ?

Sophie S
Sophie S
Numerade Educator
08:29

Problem 22

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98 \mathrm{~V}$ and $16.2 \mathrm{~V} / \mathrm{m}$ respectively. (Take $V=0$ at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric field directed toward or away from the point charge?

Mohit Khurana
Mohit Khurana
Texas A&M University
05:12

Problem 23

A uniform electric field has magnitude $E$ and is directed in the negative $x$ -direction. The potential difference between point $a$ (at $x=0.60 \mathrm{~m})$ and point $b$ (at $x=0.90 \mathrm{~m})$ is $240 \mathrm{~V}$.
(a) Which point, $a$ or $b$, is at the higher potential? (b) Calculate the value of $E$. (c) A negative point charge $q=-0.200 \mu \mathrm{C}$ is moved from $b$ to $a$. Calculate the work done on the point charge by the electric field.

John-Paul Mann
John-Paul Mann
Numerade Educator
05:27

Problem 24

A small sphere with charge $q=-5.00 \mu \mathrm{C}$ is moving in a uniform electric field that has no $y$ - or $z$ -component. The only force on the sphere is the force exerted by the electric field. Point $A$ is on the $x$ -axis at $x=-0.400 \mathrm{~m}$, and point $B$ is at the origin. At point $A$ the sphere has kinetic energy $K_{A}=8.00 \times 10^{-4} \mathrm{~J},$ and at point $B$ its kinetic energy is $K_{B}=3.00 \times 10^{-4} \mathrm{~J}$. (a) What is the potential difference $V_{A B}=V_{A}-V_{B} ?$ Which point, $A$ or $B,$ is at higher potential? (b) What are the magnitude and direction of the electric field?

Sophie S
Sophie S
Numerade Educator
01:47

Problem 25

Identical point charges $q_{1}$ and $q_{2}$ each have positive charge $+6.00 \mu \mathrm{C}$. Charge $q_{1}$ is held fixed on the $x$ -axis at $x=+0.400 \mathrm{~m}$, and $q_{2}$ is held fixed on the $x$ -axis at $x=-0.400 \mathrm{~m}$. A small sphere has charge $Q=-0.200 \mu \mathrm{C}$ and mass $12.0 \mathrm{~g} .$ The sphere is initially very far from the origin. It is released from rest and moves along the $y$ -axis toward the origin. (a) As the sphere moves from very large $y$ to $y=0$, how much work is done on it by the resultant force exerted by $q_{1}$ and $q_{2}$ ? (b) If the only force acting on the sphere is the force exerted by the point charges, what is its speed when it reaches the origin?

Anand Jangid
Anand Jangid
Numerade Educator
03:04

Problem 26

A solid conducting sphere of radius $5.00 \mathrm{~cm}$ carries a net charge. To find the value of the charge, you measure the potential difference $V_{A B}=V_{A}-V_{B}$ between point $A,$ which is $8.00 \mathrm{~cm}$ from the center of the sphere, and point $B$, which is a distance $r$ from the center of the sphere. You repeat these measurements for several values of $r>8.00 \mathrm{~cm} .$ When you plot your data as $V_{A B}$ versus $1 / r,$ the values lie close to a straight line with slope $-18.0 \mathrm{~V} \cdot \mathrm{m}$. What does your data give for the net charge on the sphere? Is the net charge positive or negative?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
17:57

Problem 27

A thin spherical shell with radius $R_{1}=4.00 \mathrm{~cm}$ is concentric with a larger thin spherical shell with radius $R_{2}=8.00 \mathrm{~cm} .$ Both shells are made of insulating material. The smaller shell has charge $q_{1}=+6.00 \mathrm{nC}$ distributed uniformly over its surface, and the larger shell has charge $q_{2}=-9.00 \mathrm{nC}$ distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells. (a) What is the electric potential due to the two shells at the following distance from their common center: (i) $r=0 ;$ (ii) $r=5.00 \mathrm{~cm} ;$ (iii) $r=9.00 \mathrm{~cm} ?$ (b) What is the magnitude of the potential difference between the surfaces of the two shells? Which shell is at higher potential: the inner shell or the outer shell?

Linda Winkler
Linda Winkler
Numerade Educator
10:41

Problem 28

A total electric charge of $1.50 \mathrm{nC}$ is distributed uniformly over the surface of a metal sphere with a radius of $24.0 \mathrm{~cm}$. If the potential is zero at a point at infinity, find the value of the potential at the following distances from the center of the sphere: (a) $55.0 \mathrm{~cm} ;$ (b) $24.0 \mathrm{~cm} ;$ (c) $13.0 \mathrm{~cm} .$

Linda Winkler
Linda Winkler
Numerade Educator
View

Problem 29

A uniformly charged, thin ring has radius $15.0 \mathrm{~cm}$ and total charge $+21.5 \mathrm{nC}$. An electron is placed on the ring's axis a distance $32.0 \mathrm{~cm}$ from the center of the ring and is constrained to stay on the axis of the ring. The electron is then released from rest. (a) Describe the subsequent motion of the electron. (b) Find the speed of the electron when it reaches the center of the ring.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
06:25

Problem 30

A solid conducting sphere has net positive charge and radius $R=0.400 \mathrm{~m}$. At a point $1.20 \mathrm{~m}$ from the center of the sphere, the electric potential due to the charge on the sphere is $24.0 \mathrm{~V}$. Assume that $V=0$ at an infinite distance from the sphere. What is the electric potential at the center of the sphere?

Mohit Khurana
Mohit Khurana
Texas A&M University
08:34

Problem 31

Charge $Q=8.00 \mu \mathrm{C}$ is distributed uniformly over the volume of an insulating sphere that has radius $R=12.0 \mathrm{~cm}$. A small sphere with charge $q=+1.00 \mu \mathrm{C}$ and mass $6.00 \times 10^{-5} \mathrm{~kg}$ is projected toward the center of the large sphere from an initial large distance. The large sphere is held at a fixed position and the small sphere can be treated as a point charge. What minimum speed must the small sphere have in order to come within $7.00 \mathrm{~cm}$ of the surface of the large sphere?

Linda Winkler
Linda Winkler
Numerade Educator
18:26

Problem 32

An infinitely long line of charge has linear charge density $5.50 \times 10^{-12} \mathrm{C} / \mathrm{m} . \mathrm{A}$ proton $\left(\right.$ mass $1.67 \times 10^{-27} \mathrm{~kg},$ charge $+1.60 \times 10^{-19} \mathrm{C}$ ) is $16.0 \mathrm{~cm}$ from the line and moving directly toward the line at $2.50 \times 10^{3} \mathrm{~m} / \mathrm{s}$. (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge?

Linda Winkler
Linda Winkler
Numerade Educator
07:05

Problem 33

A very long insulating cylindrical shell of radius $6.70 \mathrm{~cm}$ carries charge of linear density $9.00 \mu \mathrm{C} / \mathrm{m}$ spread uniformly over its outer surface. What would a voltmeter read if it were connected between (a) the surface of the cylinder and a point $4.70 \mathrm{~cm}$ above the surface, and (b) the surface and a point $1.00 \mathrm{~cm}$ from the central axis of the cylinder?

Linda Winkler
Linda Winkler
Numerade Educator
07:03

Problem 34

A very long insulating cylinder of charge of radius $2.50 \mathrm{~cm}$ carries a uniform linear density of $18.0 \mathrm{nC} / \mathrm{m} .$ If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads $190 \mathrm{~V} ?$

Linda Winkler
Linda Winkler
Numerade Educator
06:17

Problem 35

A very small sphere with positive charge $q=+7.00 \mu \mathrm{C}$ is released from rest at a point $1.70 \mathrm{~cm}$ from a very long line of uniform linear charge density $\lambda=+1.00 \mu \mathrm{C} / \mathrm{m} .$ What is the kinetic energy of the sphere when it is $3.50 \mathrm{~cm}$ from the line of charge if the only force on it is the force exerted by the line of charge?

Linda Winkler
Linda Winkler
Numerade Educator
08:34

Problem 36

Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by $2.20 \mathrm{~cm} .$ (a) If the surface charge density for each plate has magnitude $47.0 \mathrm{nC} / \mathrm{m}^{2},$ what is the magnitude of $\overrightarrow{\boldsymbol{E}}$ in the region between the plates? (b) What is the potential difference between the two plates? (c) If the separation between the plates is doubled while the surface charge density is kept constant at the value in part (a), what happens to the magnitude of the electric field and to the potential difference?

Mohit Khurana
Mohit Khurana
Texas A&M University
06:51

Problem 37

Two large, parallel, metal plates carry opposite charges of equal magnitude. They are separated by $50.0 \mathrm{~mm},$ and the potential difference between them is $365 \mathrm{~V}$. (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the force this field exerts on a particle with charge $+2.50 \mathrm{nC} ?(\mathrm{c})$ Use the results of part $(\mathrm{b})$ to compute the work done by the field on the particle as it moves from the higher-potential plate to the lower. (d) Compare the result of part (c) to the change of potential energy of the same charge, computed from the electric potential.

Linda Winkler
Linda Winkler
Numerade Educator
03:24

Problem 38

Certain sharks can detect an electric field as weak as $1.0 \mu \mathrm{V} / \mathrm{m} .$ To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary $1.5 \mathrm{~V}$ AA battery across these plates, how far apart would the plates have to be?

Mohit Khurana
Mohit Khurana
Texas A&M University
01:16

Problem 39

The electric field at the surface of a charged, solid, copper sphere with radius $0.200 \mathrm{~m}$ is $3800 \mathrm{~N} / \mathrm{C},$ directed toward the center of the sphere. What is the potential at the center of the sphere, if we take the potential to be zero infinitely far from the sphere?

Salamat Ali
Salamat Ali
Numerade Educator
View

Problem 40

(a) How much excess charge must be placed on a copper sphere $25.0 \mathrm{~cm}$ in diameter so that the potential of its center is $3.75 \mathrm{kV}$ ? Take the point where $V=0$ to be infinitely far from the sphere. (b) What is the potential of the sphere's surface?

Sophie S
Sophie S
Numerade Educator
View

Problem 41

A metal sphere with radius $r_{a}$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_{b}$ There is charge $+q$ on the inner sphere and charge $-q$ on the outer spherical shell. (a) Calculate the potential $V(r)$ for (i) $r<r_{a}$; (ii) $r_{a}<r<r_{b} ;$ (iii) $r>r_{b}$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite. (b) Show that the potential of the inner sphere with respect to the outer is
$$
V_{a b}=\frac{q}{4 \pi \epsilon_{0}}\left(\frac{1}{r_{a}}-\frac{1}{r_{b}}\right)
$$
(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude
$$
E(r)=\frac{V_{a b}}{\left(1 / r_{a}-1 / r_{b}\right)} \frac{1}{r^{2}}
$$
(d) Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distance $r$ from the center, where $r>r_{b} .$ (e) Suppose the charge on the outer sphere is not $-q$ but a negative charge of different magnitude, say $-Q .$ Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:49

Problem 42

A very large plastic sheet carries a uniform charge density of $-6.00 \mathrm{nC} / \mathrm{m}^{2}$ on one face. (a) As you move away from the sheet along a line perpendicular to it, does the potential increase or decrease? How do you know, without doing any calculations? Does your answer depend on where you choose the reference point for potential? (b) Find the spacing between equipotential surfaces that differ from each other by $1.00 \mathrm{~V}$. What type of surfaces are these?

Mohit Khurana
Mohit Khurana
Texas A&M University
05:48

Problem 43

In a certain region of space, the electric potential is $V(x, y, z)=A x y-B x^{2}+C y,$ where $A, B,$ and $C$ are positive constants. (a) Calculate the $x-, y-$, and $z$ -components of the electric field. (b) At which points is the electric field equal to zero?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:44

Problem 44

In a certain region of space the electric potential is given by $V=+A x^{2} y-B x y^{2},$ where $A=5.00 \mathrm{~V} / \mathrm{m}^{3}$ and $B=8.00 \mathrm{~V} / \mathrm{m}^{3}$. Calculate the magnitude and direction of the electric field at the point in the region that has coordinates $x=2.00 \mathrm{~m}, y=0.400 \mathrm{~m},$ and $z=0$.

Athiru Pathiraja
Athiru Pathiraja
Numerade Educator
03:53

Problem 45

A metal sphere with radius $r_{a}=1.20 \mathrm{~cm}$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_{b}=9.60 \mathrm{~cm}$. Charge $+q$ is put on the inner sphere and charge $-q$ on the outer spherical shell. The magnitude of $q$ is chosen to make the potential difference between the spheres $500 \mathrm{~V}$, with the inner sphere at higher potential.
(a) Use the result of Exercise $23.41(\mathrm{~b})$ to calculate q. (b) With the help of the result of Exercise $23.41($ a) , sketch the equipotential surfaces that correspond to $500,400,300,200,100,$ and $0 \mathrm{~V}$.
(c) In your sketch, show the electric field lines. Are the electric field lines and equipotential surfaces mutually perpendicular? Are the equipotential surfaces closer together when the magnitude of $\overrightarrow{\boldsymbol{E}}$ is largest?

Athiru Pathiraja
Athiru Pathiraja
Numerade Educator
10:04

Problem 46

A point charge $q_{1}=+15.00 \mu \mathrm{C}$ is held fixed in space. From a horizontal distance of $4.00 \mathrm{~cm},$ a small sphere with mass $4.00 \times 10^{-3} \mathrm{~kg}$ and charge $q_{2}=+2.00 \mu \mathrm{C}$ is fired toward the fixed charge with an initial speed of $44.0 \mathrm{~m} / \mathrm{s}$. Gravity can be neglected. What is the acceleration of the sphere at the instant when its speed is $20.0 \mathrm{~m} / \mathrm{s} ?$

Linda Winkler
Linda Winkler
Numerade Educator
17:05

Problem 47

A point charge $q_{1}=4.05 \mathrm{nC}$ is placed at the origin, and a second point charge $q_{2}=-2.95 \mathrm{nC}$ is placed on the $x$ -axis at $x=+21.0 \mathrm{~cm} .$ A third point charge $q_{3}=2.05 \mathrm{nC}$ is to be placed on the $x$ -axis between $q_{1}$ and $q_{2}$. (Take as zero the potential energy of the three charges when they are infinitely far apart.) (a) What is the potential energy of the system of the three charges if $q_{3}$ is placed at $x=+11.0 \mathrm{~cm} ?$ (b) Where should $q_{3}$ be placed to make the potential energy of the system equal to zero?

Linda Winkler
Linda Winkler
Numerade Educator
09:55

Problem 48

A point charge $+8.00 \mathrm{nC}$ is on the $-x$ -axis at $x=-0.200 \mathrm{~m}$ and a point charge $-4.00 \mathrm{nC}$ is on the $+x$ -axis at $x=0.200 \mathrm{~m}$. (a) In addition to $x=\pm \infty$, at what point on the $x$ -axis is the resultant field of the two charges equal to zero? (b) Let $V=0$ at $x=\pm \infty$. At what two other points on the $x$ -axis is the total electric potential due to the two charges equal to zero? (c) Is $E=0$ at either of the points in part (b) where $V=0 ?$ Explain.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
05:19

Problem 49

A very long uniform line of charge with charge per unit length $\lambda=+5.00 \mu \mathrm{C} / \mathrm{m}$ lies along the $x$ -axis, with its midpoint at the origin. A very large uniform sheet of charge is parallel to the $x y$ -plane; the center of the sheet is at $z=+0.600 \mathrm{~m}$. The sheet has charge per unit area $\sigma=+8.00 \mu \mathrm{C} / \mathrm{m}^{2},$ and the center of the sheet is at $x=0$, $y=0 .$ Point $A$ is on the $z$ -axis at $z=+0.300 \mathrm{~m},$ and point $B$ is on the $z$ -axis at $z=-0.200 \mathrm{~m}$. What is the potential difference $V_{A B}=V_{A}-V_{B}$ between points $A$ and $B$ ? Which point, $A$ or $B$, is at higher potential?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:08

Problem 50

A small sphere with mass $5.00 \times 10^{-7} \mathrm{~kg}$ and charge $+3.00 \mu \mathrm{C}$ is released from rest a distance of $0.500 \mathrm{~m}$ above a large horizontal insulating sheet of charge that has uniform surface charge density $\sigma=+8.00 \mathrm{pC} / \mathrm{m}^{2}$. Using energy methods, calculate the speed of the sphere when it is $0.200 \mathrm{~m}$ above the sheet.

Narayan Hari
Narayan Hari
Numerade Educator
06:11

Problem 51

A gold nucleus has a radius of $7.3 \times 10^{-15} \mathrm{~m}$ and a charge of $+79 e$. Through what voltage must an alpha particle, with charge $+2 e,$ be accelerated so that it has just enough energy to reach a distance of $2.0 \times 10^{-14} \mathrm{~m}$ from the surface of a gold nucleus? (Assume that the gold nucleus remains stationary and can be treated as a point charge.)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
09:54

Problem 52

A proton and an alpha particle are released from rest when they are $0.225 \mathrm{nm}$ apart. The alpha particle (a helium nucleus) has essentially four times the mass and two times the charge of a proton. Find the maximum speed and maximum acceleration of each of these particles. When do these maxima occur: just following the release of the particles or after a very long time?

Kai Chen
Kai Chen
Princeton University
10:21

Problem 53

A particle with charge $+7.80 \mathrm{nC}$ is in a uniform electric field directed to the left. Another force, in addition to the electric force, acts on the particle so that when it is released from rest, it moves to the right. After it has moved $7.50 \mathrm{~cm}$, the additional force has done $7.10 \times 10^{-5} \mathrm{~J}$ of work and the particle has $3.55 \times 10^{-5} \mathrm{~J}$ of kinetic energy. (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of the electric field?

Linda Winkler
Linda Winkler
Numerade Educator
14:10

Problem 54

Identical charges $q=+5.00 \mu \mathrm{C}$ are placed at opposite corners of a square that has sides of length $8.00 \mathrm{~cm}$. Point $A$ is at one of the empty corners, and point $B$ is at the center of the square. A charge $q_{0}=-3.00 \mu \mathrm{C}$ is placed at point $A$ and moves along the diagonal of the square to point $B$. (a) What is the magnitude of the net electric force on $q_{0}$ when it is at point $A ?$ Sketch the placement of the charges and the direction of the net force. (b) What is the magnitude of the net electric force on $q_{0}$ when it is at point $B ?$ (c) How much work does the electric force do on $q_{0}$ during its motion from $A$ to $B ?$ Is this work positive or negative? When it goes from $A$ to $B,$ does $q_{0}$ move to higher potential or to lower potential?

Mohit Khurana
Mohit Khurana
Texas A&M University
06:28

Problem 55

A vacuum tube diode consists of concentric cylindrical electrodes, the negative cathode and the positive anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is given by $V(x)=C x^{4 / 3}$ where $x$ is the distance from the cathode and $C$ is a constant, characteristic of a particular diode and operating conditions. Assume that the distance between the cathode and anode is $13.0 \mathrm{~mm}$ and the potential difference between electrodes is $240 \mathrm{~V}$. (a) Determine the value of $C$. (b) Obtain a formula for the electric field between the electrodes as a function of $x$. (c) Determine the force on an electron when the electron is halfway between the electrodes.

Sophie S
Sophie S
Numerade Educator
02:37

Problem 56

When you scuff your feet on a carpet, you gain electrons and become negatively charged. If you then place your finger near a metallic surface, such as a doorknob, an electric field develops between your finger and the doorknob. As your finger gets closer to the surface, the magnitude of the electric field increases. When it exceeds a threshold of $3 \mathrm{MV} / \mathrm{m},$ the air breaks down, creating a small "lightning" strike, which you feel as a shock. (a) Estimate the distance between your finger and a doorknob at the point you feel the shock. (b) Using this estimate and the threshold field strength of $3 \mathrm{MV} / \mathrm{m},$ find the potential between your finger and the doorknob. (c) At this (small) distance, we can treat the tip of your finger and the end of the doorknob as infinite planar surfaces with opposite charge density. Estimate that density. (d) Estimate the effective area of your fingertip as presented to the doorknob. (e) Use these results to estimate the magnitude of charge on your finger. (f) Assuming that your net excess charge has built up on your finger, attracted there by the doorknob, estimate the number of electrons that you lost while scuffing your feet.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
01:54

Problem 57

An Ionic $\quad$ Crystal. Figure $\mathbf{P} 23.57$ shows eight point charges arranged at the corners of a cube with sides of length $d$. The values of the charges are $+q$ and $-q$, as shown. This is a model of one cell of a cubic ionic crystal. In sodium chloride $(\mathrm{NaCl}),$ for instance, the positive ions are $\mathrm{Na}^{+}$ and the negative ions are $\mathrm{Cl}^{-}$. (a) Calculate the potential energy $U$ of this arrangement. (Take as zero the potential energy of the eight charges when they are infinitely far apart.) (b) In part (a), you should have found that $U<0$. Explain the relationship between this result and the observation that such ionic crystals exist in nature.

Dading Chen
Dading Chen
Numerade Educator
02:06

Problem 58

Electrical power is transmitted to our homes by overhead wires strung between poles. A typical residential utility line has a maximum potential of $22 \mathrm{kV}$ relative to ground. We can treat this potential as constant and model it as generated by a net charge distributed on the wire. (a) Estimate the height of an electrical transmission line. (b) Treat the electrical wire as a long conducting cylinder with a radius of $2 \mathrm{~cm}$. If the potential between the surface of the wire and a position directly beneath the wire is $22 \mathrm{kV}$, what is the linear charge density on the wire? (c) Use your estimate of the linear charge density to estimate the strength of the electric field on the ground beneath the wire.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
02:25

Problem 59

A small sphere with mass $1.50 \mathrm{~g}$ hangs by a thread between two very large parallel vertical plates $5.00 \mathrm{~cm}$ apart (Fig. P23.59). The plates are insulating and have uniform surface charge densities $+\sigma$ and $-\sigma .$ The charge on the sphere is $q=8.90 \times 10^{-6} \mathrm{C}$. What potential difference between the plates will cause the thread to assume an angle of $30.0^{\circ}$ with the vertical?

Salamat Ali
Salamat Ali
Numerade Educator
02:51

Problem 60

Two spherical shells have a common center. The inner shell has radius $R_{1}=5.00 \mathrm{~cm}$ and charge $q_{1}=+3.00 \times 10^{-6} \mathrm{C} ;$ the outer shell has radius $R_{2}=15.0 \mathrm{~cm}$ and charge $q_{2}=-5.00 \times 10^{-6} \mathrm{C}$. Both charges are spread uniformly over the shell surface. What is the electric potential due to the two shells at the following distances from their common center:
(a) $r=2.50 \mathrm{~cm} ;$
(b) $r=10.0 \mathrm{~cm} ;$
(c) $r=20.0 \mathrm{~cm} ?$
Take $V=0$ at a large distance from the shells.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
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Problem 61

A long metal cylinder with radius $a$ is supported on an insulating stand on the axis of a long, hollow, metal tube with radius $b$. The positive charge per unit length on the inner cylinder is $\lambda,$ and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential $V(r)$ for (i) $r<a$;
(ii) $a<r<b$; (iii) $r>b$. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take $V=0$ at $r=b$. (b) Show that the potential of the inner cylinder with respect to the outer is
$$
V_{a b}=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{b}{a}
$$
(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude
$$
E(r)=\frac{V_{a b}}{\ln (b / a)} \frac{1}{r}
$$
(d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:13

Problem 62

A Geiger counter detects radiation such as alpha particles by using the fact that the radiation ionizes the air along its path. A thin wire lies on the axis of a hollow metal cylinder and is insulated from it (Fig. P23.62). A large potential difference is established between the wire and the outer cylinder, with the wire at higher potential; this sets up a strong electric field directed radially outward. When ionizing radiation enters the device, it ionizes a few air molecules. The free electrons produced are accelerated by the electric field toward the wire and, on the way there, ionize many more air molecules. Thus a current pulse is produced that can be detected by appropriate electronic circuitry and converted to an audible "click." Suppose the radius of the central wire is $145 \mu \mathrm{m}$ and the radius of the hollow cylinder is $1.80 \mathrm{~cm} .$ What potential difference between the wire and the cylinder produces an electric field of $2.00 \times 10^{4} \mathrm{~V} / \mathrm{m}$ at a distance of $1.20 \mathrm{~cm}$ from the axis of the wire? (The wire and cylinder are both very long in comparison to their radii, so the results of Problem 23.61 apply.)

Mohit Khurana
Mohit Khurana
Texas A&M University
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Problem 63

Cathode-ray tubes (CRTs) were often found in oscilloscopes and computer monitors. In Fig. $\mathbf{P 2 3 . 6 3}$ an electron with an initial speed of $6.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$ is projected along the axis midway between the deflection plates of a cathode-ray tube. The potential difference between the two plates is $22.0 \mathrm{~V}$ and the lower plate is the one at higher potential.
(a) What is the force (magnitude and direction) on the electron when it is between the plates? (b) What is the acceleration of the electron (magnitude and direction) when acted on by the force in part (a)? (c) How far below the axis has the electron moved when it reaches the end of the plates? (d) At what angle with the axis is it moving as it leaves the plates? (e) How far below the axis will it strike the fluorescent screen $S$ ?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 64

The vertical deflecting plates of a typical classroom oscilloscope are a pair of parallel square metal plates carrying equal but opposite charges. Typical dimensions are about $3.3 \mathrm{~cm}$ on a side, with a separation of about $5.3 \mathrm{~mm}$. The potential difference between the plates is $25.0 \mathrm{~V}$. The plates are close enough that we can ignore fringing at the ends. Under these conditions: (a) how much charge is on each plate, and (b) how strong is the electric field between the plates? (c) If an electron is ejected at rest from the negative plate, how fast is it moving when it reaches the positive plate?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:51

Problem 65

Electrostatic precipitators use electric forces to remove pollutant particles from smoke, in particular in the smokestacks of coal-burning power plants. One form of precipitator consists of a vertical, hollow, metal cylinder with a thin wire, insulated from the cylinder, running along its axis (Fig. P23.65). A large potential difference is established between the wire and the outer cylinder, with the wire at lower potential. This sets up a strong radial electric field directed inward. The field produces a region of ionized air near the wire. Smoke enters the precipitator at the bottom, ash and dust in it pick up electrons, and the charged pollutants are accelerated toward the outer cylinder wall by the electric field. Suppose the radius of the central wire is $90.0 \mu \mathrm{m},$ the radius of the cylinder is $14.0 \mathrm{~cm},$ and a potential difference of $50.0 \mathrm{kV}$ is established between the wire and the cylinder. Also assume that the wire and cylinder are both very long in comparison to the cylinder radius, so the results of Problem 23.61 apply. (a) What is the magnitude of the electric field midway between the wire and the cylinder wall? (b) What magnitude of charge must a $30.0 \mu \mathrm{g}$ ash particle have if the electric field computed in part (a) is to exert a force ten times the weight of the particle?

Salamat Ali
Salamat Ali
Numerade Educator
06:16

Problem 66

A disk with radius $R$ has uniform surface charge density $\sigma$.
(a) By regarding the disk as a series of thin concentric rings, calculate the electric potential $V$ at a point on the disk's axis a distance $x$ from the center of the disk. Assume that the potential is zero at infinity. (Hint: Use the result of Example 23.11 in Section 23.3.)
(b) Calculate $-\partial V / \partial x .$ Show that the result agrees with the expression for $E_{x}$ calculated in Example 21.11 (Section 21.5).

Mohit Khurana
Mohit Khurana
Texas A&M University
19:43

Problem 67

A solid sphere of radius $R$ contains a total charge $Q$ distributed uniformly throughout its volume. Find the energy needed to assemble this charge by bringing infinitesimal charges from far away. This energy is called the "self-energy" of the charge distribution. (Hint: After you have assembled a charge $q$ in a sphere of radius $r$, how much energy would it take to add a spherical shell of thickness $d r$ having charge $d q$ ? Then integrate to get the total energy.)

Linda Winkler
Linda Winkler
Numerade Educator
05:29

Problem 68

A thin insulating rod is bent into a semicircular arc of radius $a$, and a total electric charge $Q$ is distributed uniformly along the rod. Calculate the potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.

Mohit Khurana
Mohit Khurana
Texas A&M University
10:48

Problem 69

Charge $Q=+6.00 \mu \mathrm{C}$ is distributed uniformly over the volume of an insulating sphere that has radius $R=3.00 \mathrm{~cm} .$ What is the potential difference between the center of the sphere and the surface of the sphere?

Linda Winkler
Linda Winkler
Numerade Educator
04:25

Problem 70

(a) If a spherical raindrop of radius $0.550 \mathrm{~mm}$ carries a charge of -3.80 pC uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infinite distance from the raindrop.) (b) Two identical raindrops, each with radius and charge specified in part (a), collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 71

Electric charge is distributed uniformly along a thin rod of length $a$, with total charge $Q$. Take the potential to be zero at infinity. Find the potential at the following points (Fig. $\mathbf{P 2 3 . 7 1}$ ): (a) point $P$, a distance $x$ to the right of the rod, and (b) point $R$, a distance $y$ above the right-hand end of the rod. (c) In parts (a) and (b), what does your result reduce to as $x$ or $y$ becomes much larger than $a$ ?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 72

A rigid bar withmass $M,$ length $L,$ and a uniformly distributed positive charge $Q$ is free to pivot about the origin in the presence of a spatially uniform electric field $\overrightarrow{\boldsymbol{E}}=\boldsymbol{E} \hat{\jmath}$ as shown in Fig. P23.72. Assume that the electric forces are large enough for gravity to be neglected. (a) Write the potential $V(y)$ due to the electric field as a function of $y,$ using the convention $V(0)=V_{0},$ where $V_{0}$ is a constant to be determined. (b) Determine the electric potential energy $U$ of the system as a function of $\theta$ and $V_{0} .$ (Hint: $U=\int V d q .$ ) (c) For what value of $V_{0}$ does the potential energy vanish when $\theta=0$ ? (d) If the bar is released from rest at the position $\theta=90^{\circ}$, what is the angular speed of the bar as it passes the position $\theta=0 ?$ (e) If the bar starts from rest at a small value of $\theta,$ with what frequency will the bar oscillate around the position $\theta=0 ?$

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 73

A helium nucleus, also known as an $\alpha$ (alpha) particle, consists of two protons and two neutrons and has a diameter of $10^{-15} \mathrm{~m}$ $=1 \mathrm{fm} .$ The protons, with a charge of $+e,$ are subject to a repulsive Coulomb force. Since the neutrons have zero charge, there must be an attractive force that counteracts the electric repulsion and keeps the protons from flying apart. This so-called strong force plays a central role in particle physics. (a) As a crude model, assume that an $\alpha$ particle consists of two pointlike protons attracted by a Hooke's-law spring with spring constant $k,$ and ignore the neutrons. Assume further that in the absence of other forces, the spring has an equilibrium separation of zero. Write an expression for the potential energy when the protons are separated by distance $d$. (b) Minimize this potential to find the equilibrium separation $d_{0}$ in terms of $e$ and $k$. (c) If $d_{0}=1.00 \mathrm{fm},$ what is the value of $k$ ? (d) How much energy is stored in this system, in terms of electron volts? (e) A proton has a mass of $1.67 \times 10^{-27} \mathrm{~kg}$. If the spring were to break, the $\alpha$ particle would disintegrate and the protons would fly off in opposite directions. What would be their ultimate speed?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
13:03

Problem 74

A metal sphere with radius $R_{1}$ has a charge $Q_{1}$. Take the electric potential to be zero at an infinite distance from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now connected by a long, thin conducting wire to another sphere of radius $R_{2}$ that is several meters from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere; (d) the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.

Mohit Khurana
Mohit Khurana
Texas A&M University
07:06

Problem 75

An alpha particle with kinetic energy $9.50 \mathrm{MeV}$ (when far away) collides head-on with a lead nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the lead nucleus remains stationary and may be treated as a point charge. The atomic number of lead is $82 .$ The alpha particle is a helium nucleus, with atomic number $2 .)$

Mohit Khurana
Mohit Khurana
Texas A&M University
04:12

Problem 76

The electric potential $V$ in a region of space is given by $V(x, y, z)=A\left(x^{2}-3 y^{2}+z^{2}\right)$ where $A$ is a constant. (a) Derive an expression for the electric field $\overrightarrow{\boldsymbol{E}}$ at any point in this region. (b) The work done by the field when a $1.50 \mu \mathrm{C}$ test charge moves from the point $(x, y, z)=(0,0,0.250 \mathrm{~m})$ to the origin is measured to be $6.00 \times 10^{-5} \mathrm{~J}$. Determine $A$. (c) Determine the electric field at the point $(0,0,0.250 \mathrm{~m})$. (d) Show that in every plane parallel to the $x z$ -plane the equipotential contours are circles. (e) What is the radius of the equipotential contour corresponding to $V=1280 \mathrm{~V}$ and $y=2.00 \mathrm{~m} ?$

Hubert Agamasu
Hubert Agamasu
Numerade Educator
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Problem 77

The electric potential in a region that is within $2.00 \mathrm{~m}$ of the origin of a rectangular coordinate system is given by $V=A x^{l}+B y^{m}+C z^{n}+D,$ where $A, B, C, D, l, m,$ and $n$ are constants. The units of $A, B, C,$ and $D$ are such that if $x, y,$ and $z$ are in meters, then $V$ is in volts. You measure $V$ and each component of the electric field at four points and obtain these results:
$$
\begin{array}{lccccc}
\text { Point } & (x, y, z)(\mathbf{m}) & V(\mathbf{V}) & E_{x}(\mathbf{V} / \mathbf{m}) & E_{y}(\mathbf{V} / \mathbf{m}) & E_{z}(\mathbf{V} / \mathbf{m}) \\
\hline 1 & (0,0,0) & 10.0 & 0 & 0 & 0 \\
2 & (1.00,0,0) & 4.0 & 12.0 & 0 & 0 \\
3 & (0,1.00,0) & 6.0 & 0 & 12.0 & 0 \\
4 & (0,0,1.00) & 8.0 & 0 & 0 & 12.0
\end{array}
$$
(a) Use the data in the table to calculate $A, B, C, D, l, m,$ and $n$.
(b) What are $V$ and the magnitude of $E$ at the points (0,0,0) , $(0.50 \mathrm{~m}, 0.50 \mathrm{~m}, 0.50 \mathrm{~m}),$ and $(1.00 \mathrm{~m}, 1.00 \mathrm{~m}, 1.00 \mathrm{~m}) ?$

Lainey Roebuck
Lainey Roebuck
Numerade Educator
12:22

Problem 78

A small, stationary sphere carries a net charge $Q$. You perform the following experiment to measure $Q:$ From a large distance you fire a small particle with mass $m=4.00 \times 10^{-4} \mathrm{~kg}$ and charge $q=5.00 \times 10^{-8} \mathrm{C}$ directly at the center of the sphere. The apparatus you are using measures the particle's speed $v$ as a function of the distance $x$ from the sphere. The sphere's mass is much greater than the mass of the projectile particle, so you assume that the sphere remains at rest. All of the measured values of $x$ are much larger than the radius of either object, so you treat both objects as point particles. You plot your data on a graph of $v^{2}$ versus $(1 / x)$ (Fig. P23.78). The straight line $v^{2}=400 \mathrm{~m}^{2} / \mathrm{s}^{2}-\left[\left(15.75 \mathrm{~m}^{3} / \mathrm{s}^{2}\right) / x\right]$ gives a good
fit to the data points. (a) Explain why the graph is a straight line. (b) What is the initial speed $v_{0}$ of the particle when it is very far from the sphere? (c) What is $Q ?$ (d) How close does the particle get to the sphere? Assume that this distance is much larger than the radii of the particle and sphere, so continue to treat them as point particles and to assume that the sphere remains at rest.

Mohit Khurana
Mohit Khurana
Texas A&M University
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Problem 79

The charge of an electron was first measured by the American physicist Robert Millikan during $1909-1913 .$ In his experiment, oil was sprayed in very fine drops (about $10^{-4} \mathrm{~mm}$ in diameter) into the space between two parallel horizontal plates separated by a distance $d$. A potential difference $V_{A B}$ was maintained between the plates, causing a downward electric field between them. Some of the oil drops acquired a negative charge because of frictional effects or because of ionization of the surrounding air by $x$ rays or radioactivity. The drops were observed through a microscope. (a) Show that an oil drop of radius $r$ at rest between the plates remained at rest if the magnitude of its charge was $q=\frac{4 \pi}{3} \frac{\rho r^{3} g d}{V_{A B}}$ where $\rho$ is oil's density. (Ignore the buoyant force of the air.) By adjusting $V_{A B}$ to keep a given drop at rest, Millikan determined the charge on that drop, provided its radius $r$ was known. (b) Millikan's oil drops were much too small to measure their radii directly. Instead, Millikan determined $r$ by cutting off the electric field and measuring the terminal speed $v_{\mathrm{t}}$ of the drop as it fell. (We discussed terminal speed in Section 5.3.) The viscous force $F$ on a sphere of radius $r$ moving at speed $v$ through a fluid with viscosity $\eta$ is given by Stokes's law: $F=6 \pi \eta r v$. When a drop fell at $v_{\mathrm{t}},$ the viscous force just balanced the drop's weight $w=m g$. Show that the magnitude of the charge on the drop was $q=18 \pi \frac{d}{V_{A B}} \sqrt{\frac{\eta^{3} v_{\mathrm{t}}^{3}}{2 \rho g}}$ (c) You repeat the Millikan oil-drop experiment. Four of your measured values of $V_{A B}$ and $v_{\mathrm{t}}$ are listed in the table:
$$
\begin{array}{lcccc}
\text { Drop } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\
\hline V_{A B}(\mathrm{~V}) & 9.16 & 4.57 & 12.32 & 6.28 \\
\mathrm{v}_{\mathrm{t}}\left(10^{-5} \mathrm{~m} / \mathrm{s}\right) & 2.54 & 0.767 & 4.39 & 1.52
\end{array}
$$
In your apparatus, the separation $d$ between the horizontal plates is $1.00 \mathrm{~mm}$. The density of the oil you use is $824 \mathrm{~kg} / \mathrm{m}^{3}$. For the viscosity $\eta$ of air, use the value $1.81 \times 10^{-5} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$. Assume that $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$.
Calculate the charge $q$ of each drop. (d) If electric charge is quantized (that is, exists in multiples of the magnitude of the charge of an electron), then the charge on each drop is $-n e,$ where $n$ is the number of excess electrons on each drop. (All four drops in your table have negative charge.) Drop 2 has the smallest magnitude of charge observed in the experiment, for all 300 drops on which measurements were made, so assume that its charge is due to an excess charge of one electron. Determine the number of excess electrons $n$ for each of the other three drops. (e) Use $q=-n e$ to calculate $e$ from the data for each of the four drops, and average these four values to get your best experimental value of $e$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
09:59

Problem 80

An annulus with an inner radius of $a$ and an outer radius of $b$ has charge density $\sigma$ and lies in the $x y$ -plane with its center at the origin, as shown in Fig. $\mathbf{P 2 3 . 8 0}$.
(a) Using the convention that the potential vanishes at infinity, determine the potential at all points on the $z$ -axis.
(b) Determine the electric field at all points on the $z$ -axis by differentiating the potential. (c) Show that in the limit $a \rightarrow 0, b \rightarrow \infty$ the electric field reproduces the result obtained in Example 22.7 for an infinite plane sheet of charge. (d) If $a=5.00 \mathrm{~cm}, b=10.0 \mathrm{~cm}$ and the total charge on the annulus is $1.00 \mu \mathrm{C},$ what is the potential at the origin?
(e) If a particle with mass $1.00 \mathrm{~g}$ (much less than the mass of the annulus) and charge $1.00 \mu \mathrm{C}$ is placed at the origin and given the slightest nudge, it will be projected along the $z$ -axis. In this case, what will be its ultimate speed?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
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Problem 81

A heart cell can be modeled as a cylindrical shell that is $100 \mu \mathrm{m}$ long, with an outer diameter of $20.0 \mu \mathrm{m}$ and a cell wall thickness of $1.00 \mu \mathrm{m}$, as shown in Fig. P23.81. Potassium ions move across the cell wall, depositing positive charge on the outer surface and leaving a net negative charge on the inner surface. During the so-called resting phase, the inside of the cell has a potential that is $90.0 \mathrm{mV}$ lower than the potential on its outer surface.
(a) If the net charge of the cell is zero, what is the magnitude of the total charge on either cell wall membrane? Ignore edge effects and treat the cell as a very long cylinder.
(b) What is the magnitude of the electric field just inside the cell wall?
(c) In a subsequent depolarization event, sodium ions move through channels in the cell wall, so that the inner membrane becomes positively charged. At the end of this event, the inside of the cell has a potential that is $20.0 \mathrm{mV}$ higher than the potential outside the cell. If we model this event by charge moving from the outer membrane to the inner membrane, what magnitude of charge moves across the cell wall during this event? (d) If this were done entirely by the motion of sodium ions, $\mathrm{Na}^{+}$, how many ions have moved?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
16:08

Problem 82

A hollow, thin-walled insulating cylinder of radius $R$ and length $L$ (like the cardboard tube in a roll of toilet paper) has charge $Q$ uniformly distributed over its surface. (a) Calculate the electric potential at all points along the axis of the tube. Take the origin to be at the center of the tube, and take the potential to be zero at infinity. (b) Show that if $L \ll R,$ the result of part (a) reduces to the potential on the axis of a ring of charge of radius $R$. (See Example 23.11 in Section 23.3.) (c) Use the result of part (a) to find the electric field at all points along the axis of the tube.

Mohit Khurana
Mohit Khurana
Texas A&M University
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Problem 83

In experiments in which atomic nuclei collide, head-on collisions like that described in Problem 23.75 do happen, but "near misses" are more common. Suppose the alpha particle in that problem is not "aimed" at the center of the lead nucleus but has an initial nonzero angular momentum (with respect to the stationary lead nucleus) of magnitude $L=p_{0} b$, where $p_{0}$ is the magnitude of the particle's initial momentum and $b=1.00 \times 10^{-12} \mathrm{~m} .$ What is the distance of closest approach? Repeat for $b=1.00 \times 10^{-13} \mathrm{~m}$ and $b=1.00 \times 10^{-14} \mathrm{~m}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:04

Problem 84

For a particular experiment, helium ions are to be given a kinetic energy of $3.0 \mathrm{MeV}$. What should the voltage at the center of the accelerator be, assuming that the ions start essentially at rest? (a) $-3.0 \mathrm{MV}$ (b) $+3.0 \mathrm{MV} ;(\mathrm{c})+1.5 \mathrm{MV}$ $;(\mathrm{d})+1.0 \mathrm{MV}$

Mohit Khurana
Mohit Khurana
Texas A&M University
06:23

Problem 85

A helium ion $\left(\mathrm{He}^{++}\right)$ that comes within about $10 \mathrm{fm}$ of the center of the nucleus of an atom in the sample may induce a nuclear reaction instead of simply scattering. Imagine a helium ion with a kinetic energy of $3.0 \mathrm{MeV}$ heading straight toward an atom at rest in the sample. Assume that the atom stays fixed. What minimum charge can the nucleus of the atom have such that the helium ion gets no closer than $10 \mathrm{fm}$ from the center of the atomic nucleus? (1 $\mathrm{fm}=1 \times 10^{-15} \mathrm{~m},$ and $e$ is the magnitude (a) $2 e ;$ (b) $11 e ;$ (c) $20 e$; of the charge of an electron or a proton.) (d) $22 e$.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
01:58

Problem 86

The maximum voltage at the center of a typical tandem electrostatic accelerator is $6.0 \mathrm{MV}$. If the distance from one end of the acceleration tube to the midpoint is $12 \mathrm{~m}$, what is the magnitude of the average electric field in the tube under these conditions?
(a) $41,000 \mathrm{~V} / \mathrm{m} ;$
(b) $250,000 \mathrm{~V} / \mathrm{m} ;$
(c) $500,000 \mathrm{~V} / \mathrm{m} ;$
(d) $6,000,000 \mathrm{~V} / \mathrm{m}$.

Mohit Khurana
Mohit Khurana
Texas A&M University