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Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 25

Electric Potential - all with Video Answers

Educators

Chapter Questions

00:49

Problem 1

A particular $12 \mathrm{~V}$ car battery can send a total charge of $3.0 \times 10^{5} \mathrm{C}$ through a circuit, from one terminal to the other. (a) How many coulombs of charge does this represent? (b) If this entire charge undergoes a potential difference of $12 \mathrm{~V}$, how much energy is involved?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:11

Problem 2

The electric potential difference between the ground and a cloud in a particular thunderstorm is $1.2 \times 10^{9} \mathrm{~V}$. What is the magnitude of the change in the electric potential energy (in multiples of the electron-volt) of an electron that moves between the ground and the cloud?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:04

Problem 3

In a given lightning flash, the potential difference between a cloud and the ground is $1.0 \times 10^{9} \mathrm{~V}$ and the quantity of charge transferred is $30 \mathrm{C}$. (a) What is the decrease in energy of that transferred charge. (b) If all that energy could be used to accelerate a $1000 \mathrm{~kg}$ automobile from rest, what would be the automobile's final speed? (c) If the energy could be used to melt ice, how much ice would it melt at $0^{\circ} \mathrm{C} ?$ The heat of fusion of ice is $3.33 \times 10^{5} \mathrm{~J} / \mathrm{kg}$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:55

Problem 4

When an electron moves from $A$ to $B$ along an electric field line in Fig. $25-26$, the electric field does $3.94 \times$ $10^{-19} \mathrm{~J}$ of work on it. What are the electric potential differences (a) $V_{B}-V_{A}$, (b) $V_{C}-V_{A}$, and (c) $V_{C}-V_{B}$ ?

Ben Nicholson
Ben Nicholson
Numerade Educator
01:11

Problem 5

An infinite non-conducting sheet has a surface charge density $\sigma=0.10 \mu \mathrm{C} / \mathrm{m}^{2}$ on one side. How far apart are equipotential surfaces whose potentials differ by $50 \mathrm{~V}$ ?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:53

Problem 6

Wwo large, parallel, conducting plates are $12 \mathrm{~cm}$ apart and have charges of equal magnitude and opposite sign on their facing surfaces. An electrostatic force of $3.9 \times 10^{-15} \mathrm{~N}$ acts on an electron placed anywhere between the two plates. (Neglect fringing.) (a) Find the electric field at the position of the electron. (b) What is the potential difference between the plates?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
04:31

Problem 7

A Geiger counter has a metal cylinder $2.00 \mathrm{~cm}$ in diameter along whose axis is stretched a wire $1.30 \times 10^{-4} \mathrm{~cm}$ in diameter. If the potential difference between the wire and the cylinder is $850 \mathrm{~V}$, what is the electric field at the surface of (a) the wire and (b) the cylinder? (Hint: Use the result of Problem 30 of Chapter 24.)

Salamat Ali
Salamat Ali
Numerade Educator
06:52

Problem 8

The electric field inside a nonconducting sphere of radius $R$, with charge spread uniformly throughout its volume, is radially directed and has magnitude
$$E(r)=|\vec{E}(r)|=\frac{|q| r}{4 \pi \varepsilon_{0} R^{3}}.$$ Here $q$ (positive or negative) is the total charge within the sphere, and $r$ is the distance from the sphere's center. (a) Taking $V=0$ at the center of the sphere, find the electric potential $V(r)$ inside the sphere. (b) What is the difference in electric potential between a point on the surface and the sphere's center? (c) If $q$ is positive, which of those two points is at the higher potential?

Keshav Singh
Keshav Singh
Numerade Educator
05:01

Problem 9

A charge $q$ is distributed uniformly throughout a spherical volume of radius $R$. (a) Setting $V=0$ at infinity, show that the potential at a distance $r$ from the center, where $r<R$, is given by
$$
V=\frac{q\left(3 R^{2}-r^{2}\right)}{8 \pi \varepsilon_{0} R^{3}}
$$

Suhas Katkar
Suhas Katkar
Numerade Educator
03:56

Problem 10

Two Figure $25-27$ shows, edge-on, an infinite nonconducting sheet with positive surface charge density $\sigma$ on one side. (a) Use Eq. $25-16$ and Eq. $24-16$ to show that the electric potential of an infinite sheet of charge can be written $V=V_{0}-\left(\sigma / 2 \varepsilon_{0}\right) z$, where $V_{0}$ is the electric potential at the surface of the sheet and $z$ is the perpendicular distance from the sheet. (b) How much work is done by the electric field of the sheet as a small positive test charge $q_{0}$ is moved from an initial position on the sheet to a final position located a distance $z$ from the sheet?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
15:46

Problem 11

A thick spherical shell of charge $Q$ and uniform volume charge density $\rho$ is bounded by radii $r_{1}$ and $r_{2}$, where $r_{2}>r_{1}$. With $V=0$ at infinity, find the electric potential $\Delta V$ as a function of the distance $r$ from the center of the distribution, considering the regions (a) $r>r_{2}$, (b) $r_{2}>r>r_{1}$, and (c) $r<r_{1}$. (d) Do these solutions agree at $r=r_{2}$ and $r=r_{1} ?$ (Hint: See Section 24-6.)

Supratim Pal
Supratim Pal
Numerade Educator
01:30

Problem 12

As a space shuttle moves through the dilute ionized gas of Earth's ionosphere, its potential is typically changed by $-1.0 \mathrm{~V}$ during one revolution. By assuming that the shuttle is a sphere of radius $10 \mathrm{~m}$, estimate the amount of charge it collects.

Ben Nicholson
Ben Nicholson
Numerade Educator
02:39

Problem 13

Consider a point charge $q=1.0 \mu \mathrm{C}$, point $A$ at distance $d_{1}=2.0 \mathrm{~m}$ from $q$, and point $B$ at distance $d_{2}=$ $1.0 \mathrm{~m} .$ (a) If these points are diametrically opposite each other, as in Fig. $25-28 a$, what is the electric potential difference $V_{A}-V_{B} ?$
(b) What is that electric potential difference if points $A$ and $B$ are located as in Fig. $25-28 b ?$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:03

Problem 14

Figure $25-29$ shows two charged particles on an axis. Sketch the electric field lines and the equipotential surfaces in the plane of the page for (a) $q_{1}=+q$ and $q_{2}=+2 q$ and $(\mathrm{b})$ $q_{1}=+q$ and $q_{2}=-3 q$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:49

Problem 15

In Fig. $25-29$, set $V=0$ at infinity and let the particles have charges $q_{1}=+q$ and $q_{2}$ $=-3 q .$ Then locate (in terms of the separation distance $d$ ) any point on the $x$ axis (other than at infinity) at which the net potential due to the particles is zero.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:29

Problem 16

Two particles, of charges $q_{1}$ and $q_{2}$, are separated by distance $d$ in Fig. 25-29. The net electric field of the particles is zero at $x=d / 4$. With $V=0$ at infinity, locate (in terms of $d$ ) any point on the $x$ axis (other than at infinity) at which the electric potential due to the two particles is zero.

Ben Nicholson
Ben Nicholson
Numerade Educator
02:00

Problem 17

A spherical drop of water carrying a charge of $30 \mathrm{pC}$ has a potential of $500 \mathrm{~V}$ at its surface (with $V=0$ at infinity). (a) What is the radius of the drop? (b) If two such drops of the same charge and radius combine to form a single spherical drop, what is the potential at the surface of the new drop?

Salamat Ali
Salamat Ali
Numerade Educator
01:10

Problem 18

What are (a) the charge and (b) the charge density on the surface of a conducting sphere of radius $0.15 \mathrm{~m}$ whose potential is $200 \mathrm{~V}$ (with $V=0$ at infinity)?

Salamat Ali
Salamat Ali
Numerade Educator
00:32

Problem 19

An electric field of approximately $100 \mathrm{~V} / \mathrm{m}$ is often observed near the surface of Earth. If this were the field over the entire surface, what would be the electric potential of a point on the surface? (Set $V=0$ at infinity.)

Salamat Ali
Salamat Ali
Numerade Educator
02:56

Problem 20

In Fig. $25-30$, point $P$ is at the center of the rectangle. With $V=0$ at infinity, what is the net electric potential at $P$ due to the six charged particles?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:36

Problem 21

In Fig. $25-31$, what is the net potential at point $P$ due to the four point charges, if $V=0$ at infinity?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:39

Problem 22

(a) What is the electric potential energy of two electrons separated by $2.00 \mathrm{~nm} ?$ (b) If the separation increases, does the potential energy increase or decrease?

Ben Nicholson
Ben Nicholson
Numerade Educator
02:11

Problem 23

Derive an expression for the work required to set up the four-charge configuration of Fig. $25-32$, assuming the charges are initially infinitely far apart.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:31

Problem 24

What is the electric potential energy of the charge configuration of Fig. 25-13a? Use the numerical values provided in Touchstone Example 25-4.

Farhanul Hasan
Farhanul Hasan
Numerade Educator
08:38

Problem 25

In the rectangle of Fig. 25-33, the sides have lengths $5.0 \mathrm{~cm}$ and 15 $\mathrm{cm}, q_{1}=-5.0 \mu \mathrm{C}$, and $q_{2}=+2.0 \mu \mathrm{C}$. With
$V=0$ at infinity, what are the electric potentials (a) at corner $A$ and (b) at corner $B ?$ (c) How much work is required to move a third charge $q_{3}=+3.0 \mu \mathrm{C}$ from $B$ to $A$ along $\mathrm{a}$ diagonal of the rectangle? (d) Does this work increase or decrease the electric energy of the three-charge system? Is more, less, or the same work required if $q_{3}$ is moved along paths that are (e) inside the rectangle but not on a diagonal and (f) outside the rectangle?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:43

Problem 26

In Fig. 25-34, how much work is required to bring the charge of $+5 q$ in from infinity along the dashed line and place it as shown near the two fixed charges $+4 q$ and $-2 q ?$ Take distance $d=$ 1.40 $\mathrm{cm}$ and charge $q=1.6 \times$ $10^{-19} \mathrm{C}$.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
05:16

Problem 27

A particle of positive charge $Q$ is fixed at point $P$. A second particle of mass $m$ and negative charge $-q$ moves at constant speed in a circle of radius $r_{1}$, centered at $P .$ Derive an expression for the work $W$ that must be done by an external agent on the second particle to increase the radius of the circle of motion to $r_{2}$.

Ben Nicholson
Ben Nicholson
Numerade Educator
03:19

Problem 28

Calculate (a) the electric potential established by the nucleus of a hydrogen atom at the average distance $\left(r=5.29 \times 10^{-11} \mathrm{~m}\right)$ of the atom's electron (take $V=0$ at infinite distance), (b) the electric potential energy of the atom when the electron is at this radius, and (c) the kinetic energy of the electron, assuming it to be moving in a circular orbit of this radius centered on the nucleus. (d) How much energy is required to ionize the hydrogen atom (that is, to remove the electron from the nucleus so that the separation is effectively infinite)? Express all energies in electron-volts.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:19

Problem 29

A particle of charge $q$ is fixed at point $P$, and a second particle of mass $m$ and the same charge $q$ is initially held a distance $r_{1}$ from $P$. The second particle is then released. Determine its speed when it is distance $r_{2}$ from $P$. Let $q=3.1 \mu \mathrm{C}, m=20 \mathrm{mg}$, $r_{1}=0.90 \mathrm{~mm}$, and $r_{2}=2.5 \mathrm{~mm}$

Salamat Ali
Salamat Ali
Numerade Educator
04:06

Problem 30

A charge of $-9.0 \mathrm{nC}$ is uniformly distributed around a thin plastic ring of radius $1.5 \mathrm{~m}$ that lies in the $y z$ plane with its center at the origin. A point charge of $-6.0 \mathrm{pC}$ is located on the $x$ axis at $x=3.0 \mathrm{~m} .$ Calculate the work done on the point charge by an external force to move the point charge to the origin.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
06:07

Problem 31

Two tiny metal spheres $A$ and $B$ of mass $m_{A}=5.00 \mathrm{~g}$ and $m_{B}=10.0 \mathrm{~g}$ have equal positive charges $q=5.00 \mu \mathrm{C}$. The spheres are connected by a massless nonconducting string of length $d=1.00 \mathrm{~m}$, which is much greater than the radii of the spheres. (a) What is the electric potential energy of the system? (b) Suppose you cut the string. At that instant, what is the acceleration of each sphere? (c) A long time after you cut the string, what is the speed of each sphere?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:53

Problem 32

A thin, spherical, conducting shell of radius $R$ is mounted on an isolating support and charged to a potential of $-V$. An electron is then fired from point $P$ at distance $r$ from the center of the shell $(r \geqslant R)$ with initial speed $v_{1}$ and directly toward the shell's center. What value of $v_{1}$ is needed for the electron to just reach the shell before reversing direction?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:31

Problem 33

Two electrons are fixed $2.0 \mathrm{~cm}$ apart. Another electron is shot from infinity and stops midway between the two. What is its initial speed?

Salamat Ali
Salamat Ali
Numerade Educator
01:45

Problem 34

Two charged, parallel, flat conducting surfaces are spaced $d=1.00 \mathrm{~cm}$ apart and produce a potential difference $\Delta V=625 \mathrm{~V}$ between them. An electron is projected from one surface directly toward the second. What is the initial speed of the electron if it stops just at the second surface?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:53

Problem 35

An electron is projected with an initial speed of $3.2 \times 10^{5} \mathrm{~m} / \mathrm{s}$ directly toward a proton that is fixed in place. If the electron is initially a great distance from the proton, at what distance from the proton is the speed of the electron instantaneously equal to twice the initial value?

Salamat Ali
Salamat Ali
Numerade Educator
00:51

Problem 36

The ammonia molecule $\mathrm{NH}_{3}$ has a permanent electric dipole moment equal to $1.47 \mathrm{D}$, where $1 \mathrm{D}=1$ debye unit $=$ $3.34 \times 10^{-30} \mathrm{C} \cdot \mathrm{m} .$ Calculate the electric potential due to an ammonia molecule at a point $52.0 \mathrm{~nm}$ away along the axis of the dipole. (Set $V=0$ at infinity.)

Salamat Ali
Salamat Ali
Numerade Educator
02:27

Problem 37

Figure $25-35$ shows three charged particles located on a horizontal axis. For points (such as $P$ ) on the axis with $r \geqslant d$, show that the electric potential $V(r)$ is given by
$$
V(r)=\frac{k q}{r}\left(1+\frac{2 d}{r}\right) .
$$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
05:07

Problem 38

(a) Figure 25-36a shows a positively charged plastic rod of length $L$ and uniform linear charge density $\lambda$. Setting $V$ $=0$ at infinity and considering Fig. $25-17$ and Eq. $25-34$, find the electric potential at point $P$ without written calculation. (b) Figure $25-36 b$ shows an identical rod, except that it is split in half and the right half is negatively charged; the left and right halves have the same magnitude $\lambda$ of uniform linear charge density. With $V$ still zero at infinity, what is the electric potential at point $P$ in Fig. $25-36 b ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:05

Problem 39

The plastic rod shown in Fig. $25-37$ has length $L$ and a nonuniform linear charge density $\lambda=c x$, where $c$ is a positive constant. With $V=0$ at infinity, find the electric potential at point $P_{1}$ on the axis, at distance $d$ from one end.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:14

Problem 40

Figure $25-37$ shows a plastic rod of length $L$ and uniform positive charge $Q$ lying on an $x$ axis. With $V=0$ at infinity, find the electric potential at point $P_{1}$ on the axis, at distance $d$ from one end of the rod.

Supratim Pal
Supratim Pal
Numerade Educator
02:23

Problem 41

The electric potential at points in an $x y$ plane is given by $V=\left(2.0 \mathrm{~V} / \mathrm{m}^{2}\right) x^{2}-\left(3.0 \mathrm{~V} / \mathrm{m}^{2}\right) y^{2} .$ What are the magnitude and direction of the electric field at the point $(3.0 \mathrm{~m}$, $2.0 \mathrm{~m}) ?$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:15

Problem 42

Two large parallel metal plates are $1.5 \mathrm{~cm}$ apart and have equal but opposite charges on their facing surfaces. Take the potential of the negative plate to be zero. If the potential halfway between the plates is then $+5.0 \mathrm{~V}$, what is the electric field in the region between the plates?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:53

Problem 43

(a) Using Eq. $25-31$, show that the electric potential at a point on the central axis of a thin ring of charge of radius $R$ and a distance $z$ from the ring is
$$V=\frac{k q}{\sqrt{z^{2}+R^{2}}}$$
(b) From this result, derive an expression for the $E$ -field magnitude $|\vec{E}|=E$ at points on the ring's axis; compare your result with the calculation of $E$ in Section $23-7$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:38

Problem 44

The plastic rod of length $L$ in Fig. $25-37$ has the nonuniform linear charge density $\lambda=c x$, where $c$ is a positive constant. (a) With $V=0$ at infinity, find the electric potential at point $P_{2}$ on the $y$ axis, a distance $y$ from one end of the rod. (b) From that result, find the electric field component $E_{y}$ at $P_{2} .$ (c) Why cannot the field component $E_{x}$ at $P_{2}$ be found using the result of (a)?

Amit Srivastava
Amit Srivastava
Numerade Educator
06:08

Problem 45

(a) Use the result of Problem 39 to find the electric field component $E_{x}$ at point $P_{1}$ in Fig. $25-37$ ( Hint: First substitute the variable $x$ for the distance $d$ in the result.) (b) Use symmetry to determine the electric field component $E_{y}$ at $P_{1}$.

Ben Nicholson
Ben Nicholson
Numerade Educator
00:56

Problem 46

An empty hollow metal sphere has a potential of $+400 \mathrm{~V}$ with respect to ground (defined to be at $V=0$ ) and has a charge of $5.0 \times 10^{-9} \mathrm{C}$. Find the electric potential at the center of the sphere.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:03

Problem 47

What is the excess charge on a conducting sphere of radius $r=0.15 \mathrm{~m}$ if the potential of the sphere is $1500 \mathrm{~V}$ and $V=0$ at infinity?

Salamat Ali
Salamat Ali
Numerade Educator
02:49

Problem 48

Consider two widely separated conducting spheres, 1 and 2 , the second having twice the diameter of the first. The smaller sphere initially has a positive charge $q$, and the larger one is initially uncharged. You now connect the spheres with a long thin wire. (a) How are the final potentials $V_{1}$ and $V_{2}$ of the spheres related? (b) What are the final charges $q_{1}$ and $q_{2}$ on the spheres, in terms of $q ?(\mathrm{c})$ What is the ratio of the final surface charge density of sphere 1 to that of sphere $2 ?$

Supratim Pal
Supratim Pal
Numerade Educator
03:39

Problem 49

Two metal spheres, each of radius $3.0 \mathrm{~cm}$, have a center-to-center separation of $2.0 \mathrm{~m}$. One has a charge of $+1.0 \times 10^{-8} \mathrm{C} ;$ the other has a charge of $-3.0 \times 10^{-8} \mathrm{C}$. Assume that the separation is large enough relative to the size of the spheres to permit us to consider the charge on each to be uniformly distributed (the spheres do not affect each other). With $V=0$ at infinity, calculate (a) the potential at the point halfway between their centers and (b) the potential of each sphere.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:59

Problem 50

A charged metal sphere of radius $15 \mathrm{~cm}$ has a net charge of $3.0 \times 10^{-8} \mathrm{C}$. (a) What is the electric field at the sphere's surface? (b) If $V=0$ at infinity, what is the electric potential at the sphere's surface? (c) At what distance from the sphere's surface has the electric potential decreased by $500 \mathrm{~V} ?$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:15

Problem 51

(a) If Earth had a net surface charge density of $1.0$ electron per square meter (a very artificial assumption), what would its potential be? (Set $V=0$ at infinity.) (b) What would be the electric field due to the Earth just outside its surface?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
10:24

Problem 52

Two thin, isolated, concentric conducting spheres of radii $R_{1}$ and $R_{2}$ (with $R_{1}<R_{2}$ ) have charges $q_{1}$ and $q_{2}$. With $V=0$ at infinity, derive expressions for the electric field magnitude $E(r)$ and the electric potential $V(r)$, where $r$ is the distance from the center of the spheres. Plot $E(r)$ and $V(r)$ from $r=0$ to $r=4.0 \mathrm{~m}$ for $R_{1}=0.50 \mathrm{~m}, R_{2}=1.0 \mathrm{~m}, q_{1}=+2.0 \mu \mathrm{C}$, and $q_{2}=+1.0 \mu \mathrm{C}$

Supratim Pal
Supratim Pal
Numerade Educator
03:14

Problem 53

Consider a charge $q=$ $-2.0 \mu \mathrm{C}$ that moves from $A$ to $B$ or $C$ to $D$ along the paths shown in Fig. $25-38$. This charge is moving in the presence of a uniform electric field of magnitude $E=100 \mathrm{~N} / \mathrm{C}$.
(a) What is the total work done on the charge if the distance between $A$ and $B$ is $0.62 \mathrm{~m} ?$
(b) What is the total work done on the charge if the distance between $C$ and $D$ is $0.58 \mathrm{~m} ?$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:18

Problem 54

(a) Figure $25-39$ shows a contour plot of part of a range of hills in Virginia. The outer part of the figure is at sea level (marked 0). Each contour line from the region marked 0 shows a level $10 \mathrm{~m}$ higher than the previous line. The maximum height is $70 \mathrm{~m}$ and is shown by the number 70 . Answer the following questions by giving the pair of grid markers (a letter and a number) closest to the point being requested.
i. Where is there a steep cliff?
ii. Where is there a pass between two hills?
iii. Where is the easiest climb up the hill?
(b) Now suppose the figure represents a plot of the electric equipotentials for the surface of a glass plate, and the numbers now represent voltage. The maximum is $70 \mathrm{~V}$ and each contour line from the region marked 0 shows a level $10 \mathrm{~V}$ higher than the previous line.
i. Where would a test charge placed on the glass feel the strongest electric force? In what direction would the force point?
ii. Is there a place on the glass where a charge could be placed so it feels no electric force? Where?

Adriano Chikande
Adriano Chikande
Numerade Educator