Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 24

Electric Potential - all with Video Answers

Educators

Chapter Questions

Problem 1

A particular $12 \mathrm{~V}$ car battery can send a total charge of $84 \mathrm{~A} \cdot \mathrm{h}$ (ampere-hours) through a circuit, from one terminal to the other. (a) How many coulombs of charge does this represent? (Hint: See Eq. $21-3 .$ ) (b) If this entire charge undergoes a change in electric potential of $12 \mathrm{~V}$, how much energy is involved?

Ankur S

Ankur S

Numerade Educator

Problem 2

The electric potential difference between the ground and a cloud in a particular thunderstorm is $1.2 \times 10^{9} \mathrm{~V} .$ In the unit electron-volts, what is the magnitude of the change in the electric potential energy of an electron that moves between the ground and the cloud?

Ben Nicholson

Numerade Educator

Problem 3

Suppose that in a 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 change in energy of that transferred charge? (b) If all the energy released could be used to accelerate a $1000 \mathrm{~kg}$ car from rest, what would be its final speed?

Andrew Puyleart

Andrew Puyleart

Numerade Educator

Problem 4

Two large, parallel, conducting plates are $12 \mathrm{~cm}$ apart and have charges of equal magnitude and opposite sign on their facing surfaces. An electric 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?

Ben Nicholson

Numerade Educator

Problem 5

An infinite nonconducting 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 $50 \mathrm{~V}$ ?

Andrew Puyleart

Andrew Puyleart

Numerade Educator

Problem 6

When an electron moves from $A$ to $B$ along an electric field line in Fig. 24-34, 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 $(\mathrm{c})$ $V_{c}-V_{e} ?$

Supratim Pal

Numerade Educator

Problem 7

The electric field in a region of space has the components $E_{y}=$ $E_{z}=0$ and $E_{x}=(4.00 \mathrm{~N} / \mathrm{C}) x .$ Point
$A$ is on the $y$ axis at $y=3.00 \mathrm{~m}$, and
point $B$ is on the $x$ axis at $x=4.00 \mathrm{~m}$. What is the potential difference $V_{B}-V_{A} ?$

Salamat Ali

Numerade Educator

Problem 8

A graph of the $x$ component of the electric field as a function of $x$ in a region of space is shown in Fig. $24-35 .$ The scale of the vertical axis is set by $E_{x s}=20.0 \mathrm{~N} / \mathrm{C}$. The $y$ and $z$ components of the electric field are zero in this region. If the electric potential at the
origin is $10 \mathrm{~V},(\mathrm{a})$ what is the electric potential at $x=2.0 \mathrm{~m}$, (b) what the greatest positive value of the elec tric potential for points on the $x$ axis for which $0 \leq x \leq 6.0 \mathrm{~m}$, and (c) for what value of $x$ is the electric poten

Supratim Pal

Numerade Educator

Problem 9

An infinite nonconducting sheet has a surface charge density $\sigma=+5.80 \mathrm{pC} / \mathrm{m}^{2} .$ (a) How much work is done by the electric field due to the sheet if a particle of charge $q=+1.60 \times 10^{-19} \mathrm{C}$ is moved from the sheet to a point $P$ at distance $d=3.56 \mathrm{~cm}$ from the sheet? (b) If the electric potential $V$ is defined to be zero on the sheet, what is $V$ at $P ?$

Salamat Ali

Numerade Educator

Problem 10

Two uniformly charged, infinite, nonconducting planes are parallel to a $y z$ plane and positioned at $x=-50 \mathrm{~cm}$ and $x=+50$ $\mathrm{cm} .$ The charge densities on the planes are $-50 \mathrm{nC} / \mathrm{m}^{2}$ and $+25$ $\mathrm{nC} / \mathrm{m}^{2}$, respectively. What is the magnitude of the potential difference between the origin and the point on the $x$ axis at $x=+80 \mathrm{~cm} ?$ (Hint: Use Gauss' law.)

Ben Nicholson

Numerade Educator

Problem 11

A nonconducting sphere has radius $R=2.31 \mathrm{~cm}$ and uniformly distributed charge $q=+3.50 \mathrm{fC}$. Take the electric potential at the sphere's center to be $V_{0}=0 .$ What is $V$ at radial distance
(a) $r=1.45 \mathrm{~cm}$ and
(b) $r=R$. (Hint: See Module 23-6.)

Salamat Ali

Numerade Educator

Problem 12

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

Ben Nicholson

Numerade Educator

Problem 13

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

Numerade Educator

Problem 14

Consider a particle with charge $q=1.0 \mu \mathrm{C}$, point $A$ at distance $d_{1}=2.0 \mathrm{~m}$ from $q$, and point $B$ at distance $d_{2}=1.0 \mathrm{~m} .(\mathrm{a})$ If $A$ and $B$ are diametrically opposite each other, as in Fig. $24-36 a$, what is the electric potential difference $V_{A}-V_{B} ?$ (b) What is that electric potential difference if $A$ and $B$ are located as in Fig. $24-36 b ?$

Ben Nicholson

Numerade Educator

Problem 15

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

Numerade Educator

Problem 16

Figure $24-37$ shows a rectangular array of charged particles fixed in place, with distance $a=39.0$ $\mathrm{cm}$ and the charges shown as integer multiples of $q_{1}=3.40 \mathrm{pC}$ and $q_{2}=$ $6.00 \mathrm{pC}$. With $V=0$ at infinity, what
is the net electric potential at the rectangle's center? (Hint: Thoughtful examination of the arrangement can reduce the calculation.)

Ben Nicholson

Numerade Educator

Problem 17

In Fig. $24-38$, what is the net electric potential at point $P$ due to the four particles if $\hat{V}=0$ at infinity, $q=5.00 \mathrm{fC}$, and $d=4.00 \mathrm{~cm} ?$

Supratim Pal

Numerade Educator

Problem 18

Two charged particles are shown in Fig. $24-39 a$. Particle 1 . with
charge $q_{1}$, is fixed in place at distance $d$. Particle 2 , with charge $\bar{q}_{2}$; can be moved along the $x$ axis. Figure $24-39 b$ gives the net electric potential $V$ at the origin due to the two particles as a function of the $x$ coordinate of particle $2 .$ The scale of the $x$ axis is set by $x_{s}=$ $16.0 \mathrm{~cm} .$ The plot has an asymptote of $V=5.76 \times 10^{-7} \mathrm{~V}$ as $x \rightarrow \infty .$
What is $q_{2}$ in terms of $e$ ?

Ben Nicholson

Numerade Educator

Problem 19

In Fig. $24-40$, particles with the charges $q_{1}=+5 e$ and $q_{2}=-15 e$ are fixed in place with a separation of $d=24.0 \mathrm{~cm} .$ With electric potential defined to be $V=0$ at infinity, what are the finite (a) positive and (b) negative values of $x$ at which the net electric potential on the $x$ axis is zero?

Salamat Ali

Numerade Educator

Problem 20

Two particles, of charges $q_{1}$ and $q_{2}$, are separated by distance $d$ in Fig. $24-40$. The net electric field due to 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

Numerade Educator

Problem 21

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

Numerade Educator

Problem 22

In Fig. $24-41 a$, a particle of elementary charge $+e$ is initially at coordinate $z=20 \mathrm{~nm}$ on the dipole axis (here a $z$ axis) through
an electric dipole, on the positive side of the dipole. (The origin of $z$ is at the center of the dipole.) The particle is then moved along a circular path around the dipole center until it is at coordinate $z=$ $-20 \mathrm{~nm}$, on the negative side of the dipole axis. Figure $24-41 b$ gives the work $W_{a}$ done by the force moving the particle versus the angle $\theta$ that locates the particle relative to the positive direction of the $z$ axis The scale of the vertical axis is set by $W_{a x}=4.0 \times 10^{-30} \mathrm{~J}$. What is the magnitude of the dipole moment?

Supratim Pal

Numerade Educator

Problem 23

(a) Figure 24-42a shows a nonconducting rod of length $L=$ $6.00 \mathrm{~cm}$ and uniform linear charge density $\lambda=+3.68 \mathrm{pC} / \mathrm{m}$. Assume that the electric potential is defined to be $V=0$ at infinity. What is $V$ at point $P$ at distance $d=8.00 \mathrm{~cm}$ along the rod's perpendicular bisector? (b) Figure $24-42 b$ shows an identical rod except that one half is now negatively charged. Both halves have a linear charge density of magnitude $3.68 \mathrm{pC} / \mathrm{m}$. With $V=0$ at infinity, what is $V$ at $P ?$

Supratim Pal

Numerade Educator

Problem 24

In Fig. $24-43$, a plastic rod having a uniformly distributed charge $Q=-25.6 \mathrm{pC}$ has been bent into a circular arc of radius $R=3.71 \mathrm{~cm}$ and central angle $\phi=120^{\circ} .$ With $V=0$ at infinity, what is the electric potential at $P$, the center of curvature of the rod?

Ben Nicholson

Numerade Educator

Problem 25

A plastic rod has been bent into a circle of radius $R=8.20 \mathrm{~cm} .$ It has a charge $Q_{1}=$ $+4.20 \mathrm{pC}$ uniformly distributed along onequarter of its circumference and a charge $Q_{2}=-6 Q_{1}$ uniformly distributed along the rest of the circumference (Fig. $24-44) .$ With
$V=0$ at infinity, what is the electric potential at (a) the center $C$ of the circle and (b) point $P$, on the central axis of the circle at distance $D=6.71 \mathrm{~cm}$ from the center?

Supratim Pal

Numerade Educator

Problem 26

Figure $24-45$ shows a thin rod with a uniform charge density of $2.00 \mu \mathrm{C} / \mathrm{m} .$ Evaluate the electric potential at point $P$ if $d=D=$ $L / 4.00$. Assume that the potential is zero at infinity.

Ben Nicholson

Numerade Educator

Problem 27

In Fig. $24-46$, three thin plastic rods form quarter-circles with a common center of curvature at the origin. The uniform charges on the three rods are $Q_{1}=+30 \mathrm{nC}, Q_{2}=$ $+3.0 Q_{1}$, and $Q_{3}=-8.0 Q_{1} .$ What is the net electric potential at the origin due to the rods?

Supratim Pal

Numerade Educator

Problem 28

Figure $24-47$ shows a thin plastic rod of length $L=12.0 \mathrm{~cm}$ and uniform positive charge $Q=$ $56.1 \mathrm{fClying}$ on an $x$ axis. With $V=0$ at infinity, find the electric potential at point $P_{1}$ on the axis, at distance $d=2.50 \mathrm{~cm}$ from the rod.

Supratim Pal

Numerade Educator

Problem 29

In Fig. $24-48$, what is the net electric potential at the origin due, to the circular arc of charge $Q_{1}=38$, and $40 .$ $+7.21 \mathrm{pC}$ and the two particles of charges $Q_{2}=4.00 Q_{1}$ and $Q_{3}=-2.00 Q_{1} ?$ The arc's center of curvature is at the origin and its radius is $R=2.00 \mathrm{~m} ;$ the angle indicated is $\theta=20.0^{\circ}$

Supratim Pal

Numerade Educator

Problem 30

The smiling face of Fig. 24 49 consists of three items:
1. a thin rod of charge $-3.0 \mu \mathrm{C}$ that forms a full circle of radius $6.0 \mathrm{~cm}$;
2. a second thin rod of charge $2.0 \mu \mathrm{C}$ that forms a circular arc of radius $4.0 \mathrm{~cm}$, subtending an angle of $90^{\circ}$ about the center of the full circle;
3. an electric dipole with a dipole moment that is perpendicular to a radial line and has a magnitude of $1.28 \times 10^{-21} \mathrm{C} \cdot \mathrm{m}$
What is the net electric potential at the center?

Ben Nicholson

Numerade Educator

Problem 31

A plastic disk of radius $R=64.0 \mathrm{~cm}$ is charged on one side with a uniform surface charge density $\sigma=7.73 \mathrm{fC} / \mathrm{m}^{2}$,
and then three quadrants of the disk
are removed. The remaining quadrant is shown in Fig. $24-50 .$ With $V=0$ at infinity, what is the potential due to the remaining quadrant at point $P$, which is on the central axis of the original disk at distance $D=25.9 \mathrm{~cm}$ from the original center?

Supratim Pal

Numerade Educator

Problem 32

A nonuniform linear charge distribution given by $\lambda=$ $b x$, where $b$ is a constant, is located along an $x$ axis from $x=0$ to $x=0.20 \mathrm{~m}$. If $b=20 \mathrm{nC} / \mathrm{m}^{2}$ and $V=0$ at infinity, what is the electric potential at (a) the origin and (b) the point $y=0.15 \mathrm{~m}$ on the $y$ axis?

Ben Nicholson

Numerade Educator

Problem 33

The thin plastic rod shown in Fig. $24-47$ has length $L=$ $12.0 \mathrm{~cm}$ and a nonuniform linear charge density $\lambda=c x$, where $c=28.9 \mathrm{pC} / \mathrm{m}^{2} .$ With $V=0$ at infinity, find the electric potential at point $P_{1}$ on the axis, at distance $d=3.00 \mathrm{~cm}$ from one end.

Salamat Ali

Numerade Educator

Problem 34

Two large parallel metal plates are $1.5 \mathrm{~cm}$ apart and have charges of equal magnitudes but opposite signs 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?

Ben Nicholson

Numerade Educator

Problem 35

The electric potential at points in an $x y$ plane is given by $V=\left(2.0 \mathrm{~V} / \mathrm{m}^{2}\right) x^{2}-\left(3.0 \mathrm{~V} / \mathrm{m}^{2}\right) y^{2} .$ In unit-vector notation, what is
the electric field at the point $(3.0 \mathrm{~m}, 2.0 \mathrm{~m})$ ?

Salamat Ali

Numerade Educator

Problem 36

The electric potential $V$ in the space between two flat parallel plates 1 and 2 is given (in volts) by $V=1500 x^{2}$, where $x$ (in meters) is the perpendicular distance from plate $1 .$ At $x=1.3 \mathrm{~cm},(\mathrm{a})$ what is the magnitude of the electric field and (b) is the field directed toward or away from plate $1 ?$

Ben Nicholson

Numerade Educator

Problem 37

What is the magnitude of the electric field at the point $(3.00 \hat{i}-2.00 \hat{j}+4.00 \hat{k}) \mathrm{m}$ if the electric potential in the region is given by $V=2.00 x y z^{2}$, where $V$ is in volts and coordinates $x, y$, and $z$ are in meters?

Salamat Ali

Numerade Educator

Problem 38

Figure $24-47$ shows a thin plastic rod of length $L=13.5 \mathrm{~cm}$ and uniform charge $43.6 \mathrm{fC}$. (a) In terms of distance $d$, find an expression for the electric potential at point $P_{1} .$ (b) Next, substitute variable $x$ for $d$ and find an expression for the magnitude of the component $E_{x}$ of the electric field at $P_{1} .$ (c) What is the direction of $E_{x}$ relative to the positive direction of the $x$ axis? (d) What is the value of $E_{x}$ at $P_{1}$ for $x=d=6.20 \mathrm{~cm} ?$ (e) From the symmetry in Fig. $24-47$, determine $E_{\mathrm{y}}$ at $P_{1}$.

Ben Nicholson

Numerade Educator

Problem 39

An electron is placed in an $x y$ plane where the electric potential depends on $x$ and $y$ as shown, for the coordinate axes, in Fig. 24-51 (the potential does not depend on $z$ ). The scale of the vertical axis is set by $V_{s}=500 \mathrm{~V} .$ In unit-vector notation, what is the electric force on the electron?

Salamat Ali

Numerade Educator

Problem 40

The thin plastic rod of length $L=10.0 \mathrm{~cm}$ in Fig. $24-47$ has a nonuniform linear charge density $\lambda=c x$, where $c=$ $49.9 \mathrm{pC} / \mathrm{m}^{2} .$ (a) With $V=0$ at infinity, find the electric potential at point $P_{2}$ on the $y$ axis at $y=D=3.56 \mathrm{~cm} .$ (b) Find the electric field component $E_{y}$ at $P_{2} .(\mathrm{c})$ Why cannot the field component $E_{x}$ at $P_{2}$ be found using the result of (a)?

Ben Nicholson

Numerade Educator

Problem 41

A particle of charge $+7.5 \mu \mathrm{C}$ is released from rest at the point $x=60 \mathrm{~cm}$ on an $x$ axis. The particle begins to move due to the presence of a charge $Q$ that remains fixed at the origin. What is the kinetic energy of the particle at the instant it has moved $40 \mathrm{~cm}$ if (a) $Q=+20 \mu \mathrm{C}$ and (b) $Q=-20 \mu \mathrm{C}$ ?

Salamat Ali

Numerade Educator

Problem 42

(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

Numerade Educator

Problem 43

How much work is required to set up the arrangement of Fig. $24-52$ if $q=2.30 \mathrm{pC}, a=64.0 \mathrm{~cm}$, and the particles are initially infinitely far apart and at rest?

Salamat Ali

Numerade Educator

Problem 44

In Fig. 24-53, seven charged particles are fixed in place to form a square with an edge length of $4.0 \mathrm{~cm} .$ How much work must we do to bring a particle of charge $+6 e$ initially at rest from an infinite distance to the center of the square?

Ben Nicholson

Numerade Educator

Problem 45

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 a 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

Numerade Educator

Problem 46

A charge of $-9.0 \mathrm{nC}$ is uniformly distributed around a thin plastic ring lying in a $y z$ plane with the ring center at the origin. A $-6.0 \mathrm{pC}$ particle is located on the $x$ axis at $x=3.0 \mathrm{~m}$. For a ring radius of $1.5 \mathrm{~m}$, how much work must an external force do on the particle to move it to the origin?

Ben Nicholson

Numerade Educator

Problem 47

What is the escape speed for an electron initially at rest on the surface of a sphere with a radius of $1.0 \mathrm{~cm}$ and a uniformly distributed charge of $1.6 \times 10^{-15} \mathrm{C} ?$ That is, what initial speed must the electron have in order to reach an infinite distance from the sphere and have zero kinetic energy when it gets there?

Salamat Ali

Numerade Educator

Problem 48

A thin, spherical, conducting shell of radius $R$ is mounted on an isolating support and charged to a potential of $-125 \mathrm{~V}$. An electron is then fired directly toward the center of the shell, from point $P$ at distance $r$ from the center of the shell $(r>R)$. What initial speed $v_{0}$ is needed for the electron to just reach the shell before reversing direction?

Supratim Pal

Numerade Educator

Problem 49

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

Numerade Educator

Problem 50

In Fig. $24-54$, how much work must we do to bring a particle, of charge $Q=+16 e$ and initially at rest, along the dashed line from
infinity to the indicated point near two fixed particles of charges $q_{1}=$ $+4 e$ and $q_{2}=-q_{1} / 2 ?$ Distance $d=$
$1.40 \mathrm{~cm}, \theta_{1}=43^{\circ}$, and $\theta_{2}=60^{\circ}$.

Ben Nicholson

Numerade Educator

Problem 51

In the rectangle of Fig. 24 55 , 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 is the electric potential at (a) corner $A$ and
(b) corner $B ?$ (c) How much work is required to move a charge $q_{3}=+3.0$ $\mu \mathrm{C}$ from $B$ to $A$ along a diagonal of the rectangle? (d) Does this work increase or decrease the electric potential energy of the three-charge system? Is more, less, or the same work required if $q_{3}$ is moved along a path that is (e) inside the rectangle but not on a diagonal and outside the rectangle?

Supratim Pal

Numerade Educator

Problem 52

Figure 24-56a shows an electron moving along an electric dipole axis toward the negative side of the dipole. The dipole is fixed in place. The electron was initially very far from the dipole, with kinetic energy $100 \mathrm{eV}$. Figure 24- $56 b$ gives the kinetic energy $K$ of the electron versus its distance $r$ from the dipole center. The scale of the horizontal axis is set by $r_{s}=0.10 \mathrm{~m}$. What is the magnitude of the dipole moment?

Supratim Pal

Numerade Educator

Problem 53

Two tiny metal spheres $A$ and $B$, mass $m_{A}=5.00 \mathrm{~g}$ and $m_{B}=$ $10.0 \mathrm{~g}$, have equal positive charge $q=5.00 \mu \mathrm{C}$. 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?

Salamat Ali

Numerade Educator

Problem 54

A positron (charge $+e$, mass equal to the electron mass) is moving at $1.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$ in the positive direction of an $x$ axis when, at $x=0$, it encounters an electric field directed along the $x$ axis. The electric potential $V$ associated with the field is given in Fig. $24-57$. The scale of the vertical axis is set by $V_{s}=500.0 \mathrm{~V}$.
(a) Does the positron emerge from the field at $x=0$ (which means its motion is reversed) or at $x=0.50$ $\mathrm{m}$ (which means its motion is not reversed)? (b) What is its speed when it emerges?

Ben Nicholson

Numerade Educator

Problem 55

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

Numerade Educator

Problem 56

Particle 1 (with a charge of $+5.0 \mu \mathrm{C}$ ) and particle 2 (with a charge of $+3.0 \mu \mathrm{C}$ ) are fixed in place with separation $d=4.0 \mathrm{~cm}$
on the $x$ axis shown in Fig. $24-58 a$. Particle 3 can be moved along the $x$ axis to the right of particle 2 . Figure $24-58 b$ gives the electric potential energy $U$ of the three-particle system as a function of the $x$ coordinate of particle $3 .$ The scale of the vertical axis is set by $U_{s}=5.0 \mathrm{~J}$. What is the charge of particle $3 ?$

Ben Nicholson

Numerade Educator

Problem 57

Identical $50 \mu \mathrm{C}$ charges are fixed on an $x$ axis at $x=\pm 3.0 \mathrm{~m} .$ A particle of charge $q=-15 \mu \mathrm{C}$ is then released from rest at a point on the positive part of the $y$ axis. Due to the symmetry of the situation, the particle moves along the $y$ axis and has kinetic energy $1.2 \mathrm{~J}$ as it passes through the point $x=0, y=4.0 \mathrm{~m}$.
(a) What is the kinetic energy of the particle as it passes through the origin? (b) At what negative value of $y$ will the particle momentarily stop?

Supratim Pal

Numerade Educator

Problem 58

Proton in a well. Figure $24-59$ shows electric potential $V$ along $\operatorname{an} x$ axis. The scale of the vertical axis is set by $V_{s}=10.0 \mathrm{~V} .$ A pro-
ton is to be released at $x=3.5 \mathrm{~cm}$ with
initial kinetic energy $4.00 \mathrm{eV} .(\mathrm{a})$ If it is initially moving in the negative direction of the axis, does it reach a turning point (if so, what is the $x$ coordinate of that point) or does it escape from the plotted region (if so, what is its speed at $x=0) ?$ (b) If it is initially moving in the positive direction of the axis, does it reach a turning point (if so, what is the $x$ coordinate of that point) or does it escape from the plotted region (if so, what is its speed at $x=6.0 \mathrm{~cm})$ ? What are the
(c) magnitude $F$ and (d) direction (positive or negative direction of the $x$ axis) of the electric force on the proton if the proton moves just to the left of $x=3.0 \mathrm{~cm}$ ? What are (e) $F$ and (f) the direction if the proton moves just to the right of $x=5.0 \mathrm{~cm} ?$

Supratim Pal

Numerade Educator

Problem 59

In Fig. 24-60, a charged particle (either an electron or a proton) is moving rightward between two parallel charged plates separated by distance $d=2.00 \mathrm{~mm}$. The plate potentials are $V_{1}=-70.0 \mathrm{~V}$ and $V_{2}=-50.0 \mathrm{~V}$. The particle is slowing from an initial speed of $90.0 \mathrm{~km} / \mathrm{s}$ at the left plate. (a) Is the particle an electron or a proton? (b) What is its speed just as it reaches plate $2 ?$

Narayan Hari

Numerade Educator

Problem 60

In Fig. $24-61 a$, we move an electron from an infinite distance to a point at distance $R=8.00 \mathrm{~cm}$ from a tiny charged ball. The move requires work $W=2.16 \times 10^{-13} \mathrm{~J}$ by us. (a) What is the charge $Q$ on the ball? In Fig. $24-61 b$, the ball has been sliced up and the slices spread out so that an equal amount of charge is at the hour positions on a circular clock face of radius $R=8.00 \mathrm{~cm}$. Now the electron is brought from an infinite distance to the center of
the circle. (b) With that addition of the electron to the system of 12 charged particles, what is the change in the electric potential energy of the system?

Penny Riley

Numerade Educator

Problem 61

Suppose $N$ electrons can be placed in either of two configurations. In configuration 1 , they are all placed on the circumference of a narrow ring of radius $R$ and are uniformly distributed so that the distance between adjacent electrons is the same everywhere. In configuration $2, N-1$ electrons are uniformly distributed on the ring and one electron is placed in the center of the ring. (a) What is the smallest value of $N$ for which the second configuration is less energetic than the first? (b) For that value of $N$, consider any one circumference electron - call it $\mathrm{e}_{0}$. How many other circumference electrons are closer to $e_{0}$ than the central electron is?

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 62

Sphere 1 with radius $R_{1}$ has positive charge $q .$ Sphere 2 with radius $2.00 R_{1}$ is far from sphere 1 and initially uncharged. After the separated spheres are connected with a wire thin enough to retain only negligible charge, (a) is potential $V_{1}$ of sphere 1 greater than, less than, or equal to potential $V_{2}$ of sphere $2 ?$ What fraction of $q$ ends up on (b) sphere 1 and (c) sphere 2 ? (d) What is the ratio $\sigma_{1} / \sigma_{2}$ of the surface charge densities of the spheres?

Ben Nicholson

Numerade Educator

Problem 63

Two metal spheres, each of radius $3.0 \mathrm{~cm}$, have a center-to-center separation of $2.0 \mathrm{~m} .$ Sphere 1 has charge $+1.0 \times$ $10^{-8} \mathrm{C}$; sphere 2 has charge $-3.0 \times 10^{-8} \mathrm{C}$. Assume that the separation is large enough for us to say that the charge on each sphere is uniformly distributed (the spheres do not affect each other). With $V=0$ at infinity, calculate (a) the potential at the point halfway between the centers and the potential on the surface of
(b) sphere 1 and $(\mathrm{c})$ sphere $2 .$

Supratim Pal

Numerade Educator

Problem 64

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

Ben Nicholson

Numerade Educator

Problem 65

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

Numerade Educator

Problem 66

Two isolated, concentric, conducting spherical shells have radii $R_{1}=0.500 \mathrm{~m}$ and $R_{2}=1.00 \mathrm{~m}$, uniform charges $q_{1}=+2.00 \mu \mathrm{C}$
and $q_{2}=+1.00 \mu \mathrm{C}$, and negligible thicknesses. What is the magnitude of the electric field $E$ at radial distance (a) $r=4.00 \mathrm{~m},(\mathrm{~b}) r=$ $0.700 \mathrm{~m}$, and (c) $r=0.200 \mathrm{~m}$ ? With $V=0$ at infinity, what is $V$ at
(d) $r=4.00 \mathrm{~m}$,
(e) $r=1.00 \mathrm{~m}$,
(f) $r=0.700 \mathrm{~m}$,
(g) $r=0.500 \mathrm{~m}$,
(h) $r=0.200 \mathrm{~m}$, and (i) $r=0 ?$ (j) $\operatorname{Sketch} E(r)$ and $V(r)$.

Supratim Pal

Numerade Educator

Problem 67

A 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 $500 \mathrm{~V} ?$

Salamat Ali

Numerade Educator

Problem 68

Here are the charges and coordinates of two charged particles located in an $x y$ plane: $q_{1}=+3.00 \times 10^{-6} \mathrm{C}, x=+3.50 \mathrm{~cm}$
$y=+0.500 \mathrm{~cm}$ and $q_{2}=-4.00 \times 10^{-6} \mathrm{C}, x=-2.00 \mathrm{~cm}, \quad y=$
$+1.50 \mathrm{~cm}$. How much work must be done to locate these charges at their given positions, starting from infinite separation?

Supratim Pal

Numerade Educator

Problem 69

A long, solid, conducting cylinder has a radius of $2.0 \mathrm{~cm}$. The electric field at the surface of the cylinder is $160 \mathrm{~N} / \mathrm{C}$, directed radially outward. Let $A, B$, and $C$ be points that are $1.0 \mathrm{~cm}, 2.0 \mathrm{~cm}$, and $5.0 \mathrm{~cm}$, respectively, from the central axis of the cylinder. What are (a) the magnitude of the electric field at $C$ and the electric potential differences (b) $V_{B}-V_{C}$ and (c) $V_{A}-V_{B}$ ?

Salamat Ali

Numerade Educator

Problem 70

The chocolate crumb mystery. This story begins with Problem 60 in Chapter 23. (a) From the answer to part (a) of that problem, find an expression for the electric potential as a function of the radial distance $r$ from the center of the pipe. (The electric potential is zero on the grounded pipe wall.) (b) For the typical volume charge density $\rho=-1.1 \times 10^{-3} \mathrm{C} / \mathrm{m}^{3}$, what is the difference in the electric potential between the pipe's center and its inside wall? (The story continues with Problem 60 in Chapter 25.)

Supratim Pal

Numerade Educator

Problem 71

Starting from Eq. $24-30$, derive an expression for the electric field due to a dipole at a point on the dipole axis.

Salamat Ali

Numerade Educator

Problem 72

The magnitude $E$ of an electric field depends on the radial distance $r$ according to $E=A / r^{4}$, where $A$ is a constant with the unit volt-cubic meter. As a multiple of $\bar{A}$, what is the magnitude of the electric potential difference between $r=2.00 \mathrm{~m}$ and $r=3.00 \mathrm{~m} ?$

Ben Nicholson

Numerade Educator

Problem 73

(a) If an isolated conducting sphere $10 \mathrm{~cm}$ in radius has a net charge of $4.0 \mu \mathrm{C}$ and if $V=0$ at infinity, what is the potential on the surface of the sphere? (b) Can this situation actually occur, given that the air around the sphere undergoes electrical breakdown when the field exceeds $3.0 \mathrm{MV} / \mathrm{m}$ ?

Salamat Ali

Numerade Educator

Problem 74

Three particles, charge $q_{1}=+10 \mu \mathrm{C}$ $q_{2}=-20 \mu \mathrm{C}$, and $q_{3}=+30 \mu \mathrm{C}$, are posi-
tioned at the vertices of an isosceles triangle as shown in Fig. $24-62 .$ If $a=10 \mathrm{~cm}$ and $b=$ $6.0 \mathrm{~cm}$, how much work must an external agent do to exchange the positions of (a) $q_{1}$ and $q_{3}$ and, instead, (b) $q_{1}$ and $q_{2}$ ?

Supratim Pal

Numerade Educator

Problem 75

An electric field of approximately 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. $)$

Supratim Pal

Numerade Educator

Problem 76

A Gaussian sphere of radius $4.00 \mathrm{~cm}$ is centered on a ball that has a radius of $1.00 \mathrm{~cm}$ and a uniform charge distribution. The total (net) electric flux through the surface of the Gaussian sphere is $+5.60 \times 10^{4} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$
What is the electric potential $12.0 \mathrm{~cm}$ from the center of the ball?

Ben Nicholson

Numerade Educator

Problem 77

In a Millikan oil-drop experiment (Module 22-6), a uniform electric field of $1.92 \times 10^{5} \mathrm{~N} / \mathrm{C}$ is maintained in the
region between two plates separated by $1.50 \mathrm{~cm} .$ Find the potential difference between the plates.

Salamat Ali

Numerade Educator

Problem 78

Figure $24-63$ shows three circular, nonconducting arcs of radius $R=8.50$ $\mathrm{cm}$. The charges on the arcs are $q_{1}=4.52$
$\mathrm{pC}, q_{2}=-2.00 q_{1}, q_{3}=+3.00 q_{1}$. With $V=0$ at infinity, what is the net electric potential of the arcs at the common center of curvature?

Ben Nicholson

Numerade Educator

Problem 79

An electron is released from rest on the axis of an electric dipole that has charge $e$ and charge separation $d=20 \mathrm{pm}$ and that is fixed in place. The release point is on the positive side of the dipole, at distance $7.0 d$ from the dipole center. What is the electron's speed when it reaches a point $5.0 d$ from the dipole center?

Supratim Pal

Numerade Educator

Problem 80

Figure $24-64$ shows a ring of outer radius $R=13.0 \mathrm{~cm}$, inner radius $r=0.200 R$, and uniform surface charge density $\sigma=6.20 \mathrm{pC} / \mathrm{m}^{2} .$ With $V=0$ at infinity, find the electric potential at point $P$ on the central axis of the ring, at distance $z=2.00 R$ from the center of the ring.

Ben Nicholson

Numerade Educator

Problem 81

Electron in a well. Figure 24-65 shows electric potential $V$ along an $x$ axis. The scale of the vertical axis is set by $V_{s}=8.0 \mathrm{~V}$. An electron
to be released at $x=4.5$ $\mathrm{cm}$ with initial kinetic energy $3.00 \mathrm{eV}$. (a) If it is initially moving in the nega- $\quad$ Figure 24-65 Problem 81 . tive direction of the axis, does it reach a turning point (if so, what is the $x$ coordinate of that point) or does it escape from the plotted region (if so, what is its speed at $x=0$ )? (b) If it is initially moving in the positive direction of the axis, does it reach a turning point (if so, what is the $x$ coordinate of that point) or does it escape from the plotted region (if so, what is its speed at $x=7.0 \mathrm{~cm}) ?$ What are the (c) magnitude $F$ and $(\mathrm{d})$ direction (positive or negative direction of the $x$ axis) of the electric force on the electron if the electron moves just to the left of $x-4.0 \mathrm{~cm}$ ? What are (e) $\underline{F}$ and (f) the direction if it moves just to the right of $x=5.0 \mathrm{~cm}$ ?

Supratim Pal

Numerade Educator

Problem 82

(a) If Earth had a uniform surface charge density of $1.0$ electron $/ \mathrm{m}^{2}$ (a very artificial assumption), what would its potential be? (Set $V=0$ at infinity.) What would be the (b) magnitude and (c) direction (radially inward or outward) of the electric field due to Earth just outside its surface?

Ben Nicholson

Numerade Educator

Problem 83

In Fig. $24-66$, point $P$ is at distance $d_{1}=4.00 \mathrm{~m}$ from particle $1\left(q_{1}=-2 e\right)$ and distance $d_{2}=2.00 \mathrm{~m}$ from particle $2\left(q_{2}=+2 e\right)$, with both particles fixed in place. (a) With $V=0$ at infinity, what $0 q_{z}$
is $V$ at $P ?$ If we bring a particle of charge $q_{3}=+2 e$ from infinity to $P$,
(b) how much work do we do and
(c) what is the potential energy of the three-particle system?

Supratim Pal

Numerade Educator

Problem 84

A solid conducting sphere of radius $3.0 \mathrm{~cm}$ has a charge of $30 \mathrm{nC}$ distributed uniformly over its surface. Let $A$ be a point $1.0 \mathrm{~cm}$ from the center of the sphere, $S$ be a point on the surface of the sphere, and $B$ be a point $5.0 \mathrm{~cm}$ from the center of the sphere. What are the electric potential differences (a) $V_{S}-V_{B}$ and
(b) $V_{A}-V_{B} ?$

Ben Nicholson

Numerade Educator

Problem 85

In Fig. $24-67$, we move a particle of charge $+2 e$ in from infinity to the $x$ axis. How much work do we do? Distance $D$ is $4.00 \mathrm{~m}$.

Salamat Ali

Numerade Educator

Problem 86

Figure 24-68 shows a hemisphere with a charge of $4.00 \mu \mathrm{C}$ distributed uniformly through its volume. The hemisphere lies on an $x y$ plane the way half a grapefruit might lie face down on a kitchen table. Point $P$ is located on the plane, along a radial line from the hemisphere's center of curvature, at radial distance $15 \mathrm{~cm} .$ What is the electric potential at point $P$ due to the hemisphere?

Ben Nicholson

Numerade Educator

Problem 87

Three $+0.12$ C charges form an equilateral triangle $1.7 \mathrm{~m}$ on a side. Using energy supplied at the rate of $0.83 \mathrm{~kW}$, how many days would be required to move one of the charges to the midpoint of the line joining the other two charges?

Salamat Ali

Numerade Educator

Problem 88

Two charges $q=+2.0 \mu \mathrm{C}$ are fixed a distance $d=2.0 \mathrm{~cm}$ apart (Fig. $24-69) .$ (a) With $V=0$ at infinity, what is the electric potential at point $C ?$ (b) You bring a third charge $q=+2.0 \mu \mathrm{C}$ from infinity to $C$. How much work must you do? (c) What is the potential energy $U$ of the three-charge configuration when the third charge is in place?

Supratim Pal

Numerade Educator

Problem 89

Initially two electrons are fixed in place with a separation of $2.00 \mu \mathrm{m} .$ How much work must we do to bring a third electron in from infinity to complete an equilateral triangle?

Salamat Ali

Numerade Educator

Problem 90

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

Numerade Educator

Problem 91

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 inst at the second surface?

Supratim Pal

Numerade Educator

Problem 92

In Fig. $24-70$, point $P$ is at the center of the rectangle. With $\bar{V}=0$ at infinity, $\quad q_{1}=5.00 \mathrm{fC}, \quad q_{2}=2.00 \mathrm{fC}$
$q_{3}=3.00 \mathrm{fC}$, and $d=2.54 \mathrm{~cm}$, what is
the net electric potential at $P$ due to the six charged particles?

Ben Nicholson

Numerade Educator

Problem 93

A uniform charge of $+16.0$ Figure 24-70 Problem 92 . $\mu \mathrm{C}$ is on a thin circular ring lying in an $x y$ plane and centered on the origin. The ring's radius is $3.00 \mathrm{~cm}$. If point $A$ is at the origin and point $B$ is on the $z$ axis at $z=4.00$ $\mathrm{cm}$, what is $V_{B}-V_{A} ?$

Supratim Pal

Numerade Educator

Problem 94

Consider a particle with charge $q=1.50 \times 10^{-8} \mathrm{C}$, and take $V=0$ at infinity. (a) What are the shape and dimensions of an equipotential surface having a potential of $30.0 \mathrm{~V}$ due to $q$ alone?
(b) Are surfaces whose potentials differ by a constant amount (1.0 V, say) evenly spaced?

Ben Nicholson

Numerade Educator

Problem 95

A thick spherical shell of charge $Q$ and uniform volume charge density $\rho$ is bounded by radii $r_{1}$ and $r_{2}>r_{1} .$ With $V=0$ at infinity, find the electric potential $V$ as a function of distance $r$ from the center of the distribution, considering regions (a) $r>r_{2}$,
(b) $r_{2}>r>r_{1}$, and
(c) $r<r_{1^{\circ}}$
(d) Do these solutions agree with each other at $r=r_{2}$ and $r=r_{1}$ ? (Hint: See Module 23-6.)

Supratim Pal

Numerade Educator

Problem 96

A charge $q$ is distributed uniformly throughout a spherical volume of radius $R .$ Let $V=0$ at infinity. What are (a) $V$ at radial distance $r<R$ and (b) the potential difference between points at $r=R$ and the point at $r=0 ?$

Ben Nicholson

Numerade Educator

Problem 97

A solid copper sphere whose radius is $1.0 \mathrm{~cm}$ has a very thin surface coating of nickel. Some of the nickel atoms are radioactive, each atom emitting an electron as it decays. Half of these electrons enter the copper sphere, each depositing $100 \mathrm{keV}$ of energy there. The other half of the electrons escape, each carrying away a charge $-e$. The nickel coating has an activity of $3.70 \times 10^{8}$ radioactive decays per second. The sphere is hung from a long, nonconducting string and isolated from its surroundings. (a) How long will it take for the potential of the sphere to increase by $1000 \mathrm{~V} ?$ (b) How long will it take for the temperature of the sphere to increase by $5.0 \mathrm{~K}$ due to the energy deposited by the electrons? The heat capacity of the sphere is $14 . \mathrm{J} / \mathrm{K}$

Supratim Pal

Numerade Educator

Problem 98

In Fig. $24-71$, a metal sphere with charge $q=5.00 \mu \mathrm{C}$ and radius $r=3.00 \mathrm{~cm}$ is concentric with a larger metal sphere with charge $Q=$ $15.0 \mu \mathrm{C}$ and radius $R=6.00 \mathrm{~cm} .$ (a) What is the potential difference between the spheres? If we connect the spheres with a wire, what then is the charge on (b) the smaller sphere and
(c) the larger sphere?

Vishal Gupta

Numerade Educator

Problem 99

Using Eq. 24-32, show that the electric potential at a point on stand
the central axis of a thin ring (of charge $q$ and radius $R$ ) and at distance $z$ from the ring is
$$
V=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{\sqrt{z^{2}+R^{2}}}
$$
(b) From this result, derive an expression for the electric field magnitude $E$ at points on the ring's axis; compare your result with the calculation of $E$ in Module $22-4 .$

Supratim Pal

Numerade Educator

Problem 100

An alpha particle (which has two protons) is sent directly toward a target nucleus containing 92 protons. The alpha particle has an initial kinetic energy of $0.48 \mathrm{pJ}$. What is the least center-to-center distance the alpha particle will be from the target nucleus, assuming the nucleus does not move?

Ben Nicholson

Numerade Educator

Problem 101

In the quark model of fundamental particles, a proton is composed of three quarks: two "up" quarks, each having charge $+2 e / 3$, and one "down" quark, having charge $-e / 3$. Suppose that the three quarks are equidistant from one another. Take that separation distance to be $1.32 \times 10^{-15} \mathrm{~m}$ and calculate the electric potential energy of the system of (a) only the two up quarks and
(b) all three quarks.

Salamat Ali

Numerade Educator

Problem 102

A charge of $1.50 \times 10^{-8} \mathrm{C}$ lies on an isolated metal sphere of radius $16.0 \mathrm{~cm}$. With $V=0$ at infinity, what is the electric potential at points on the sphere's surface?

Supratim Pal

Numerade Educator

Problem 103

In Fig. $24-72$, two particles of charges $q_{1}$ and $q_{2}$ are fixed to an $x$ axis. If a third particle, of charge $+6.0 \mu \mathrm{C}$, is brought from an infinite $\quad$ Figure 24-72 Problem 103 . distance to point $P$, the three-particle system has the same electric potential energy as the original two-particle system. What is the charge ratio $q_{1} / q_{2} ?$

Supratim Pal

Numerade Educator