Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 25

Electric Potential - all with Video Answers

Educators

Chapter Questions

Problem 1

How much work is done (by a battery, generator, or some other source of electrical energy) in moving Avogadro's number of electrons from an initial point where the electric potential is $9.00 \mathrm{~V}$ to a point where the potential is $-5.00 \mathrm{~V} ?$ (The potential in each case is measured relative to a common reference point.)

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

An ion accelerated through a potential difference of 115 V experiences an increase in kinetic energy of $7.37 \times 10^{-17} \mathrm{~J} .$ Calculate the charge on the ion.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 3

(a) Calculate the speed of a proton that is accelerated from rest through a potential difference of $120 \mathrm{~V}$.
(b) Calculate the speed of an electron that is accelerated through the same potential difference.

Paul Gabriel

Numerade Educator

Problem 4

Review Problem. Through what potential difference would an electron need to be accelerated for it to achieve a speed of $40.0 \%$ of the speed of light, starting from rest? The speed of light is $c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$ review Section $7.7 .$

Keshav Singh

Numerade Educator

Problem 5

What potential difference is needed to stop an electron having an initial speed of $4.20 \times 10^{5} \mathrm{~m} / \mathrm{s} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

A uniform electric field of magnitude $250 \mathrm{~V} / \mathrm{m}$ is directed in the positive $x$ direction. $\mathrm{A}+12.0-\mu \mathrm{C}$ charge moves from the origin to the point $(x, y)=$ $(20.0 \mathrm{~cm}, 50.0 \mathrm{~cm}) \cdot$ (a) What was the change in the potential energy of this charge? (b) Through what potential difference did the charge move?

Keshav Singh

Numerade Educator

Problem 7

The difference in potential between the accelerating plates of a TV set is about $25000 \mathrm{~V}$. If the distance between these plates is $1.50 \mathrm{~cm}$, find the magnitude of the uniform electric field in this region.

Keshav Singh

Numerade Educator

Problem 8

Suppose an electron is released from rest in a uniform electric field whose magnitude is $5.90 \times 10^{3} \mathrm{~V} / \mathrm{m}$.
(a) Through what potential difference will it have passed after moving $1.00 \mathrm{~cm} ?$ (b) How fast will the electron be moving after it has traveled $1.00 \mathrm{~cm}$ ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

An electron moving parallel to the $x$ axis has an initial speed of $8.70 \times 10^{6} \mathrm{~m} / \mathrm{s}$ at the origin. Its speed is reduced to $1.40 \times 10^{5} \mathrm{~m} / \mathrm{s}$ at the point $x=2.00 \mathrm{~cm}$. Calculate the potential difference between the origin and that point. Which point is at the higher potential?

Keshav Singh

Numerade Educator

Problem 10

A uniform electric field of magnitude $825 \mathrm{~V} / \mathrm{m}$ is directed in the negative $y$ direction as shown in Figure P25.10. The coordinates of point $A$ are $(-0.200,-0.300) \mathrm{m}$, and those of point $B$ are $(0.400,0.500) \mathrm{m} .$ Calculate the potential difference $V_{B}-V_{A}$, using the blue path.

Keshav Singh

Numerade Educator

Problem 11

A $4.00-\mathrm{kg}$ block carrying a charge $Q=50.0 \mu \mathrm{C}$ is connected to a spring for which $k=100 \mathrm{~N} / \mathrm{m}$. The block lies on a frictionless horizontal track, and the system is immersed in a uniform electric field of magnitude $E=$ $5.00 \times 10^{5} \mathrm{~V} / \mathrm{m}$, directed as shown in Figure P25.11. If the block is released from rest when the spring is unstretched (at $x=0$ ), (a) by what maximum amount does the spring expand? (b) What is the equilibrium position of the block? (c) Show that the block's motion is simple harmonic, and determine its period.
(d) Repeat part (a) if the coefficient of kinetic friction between block and surface is $0.200$.

Keshav Singh

Numerade Educator

Problem 12

A block having mass $m$ and charge $Q$ is connected to a spring having constant $k .$ The block lies on a frictionless horizontal track, and the system is immersed in a uniform electric field of magnitude $E$, directed as shown in Figure P25.11. If the block is released from rest when the spring is unstretched $($ at $x=0)$, (a) by what maximum amount does the spring expand? (b) What is the equilibrium position of the block? (c) Show that the block's motion is simple harmonic, and determine its period. (d) Repeat part (a) if the coefficient of kinetic friction between block and surface is $\mu_{k}$.

Keshav Singh

Numerade Educator

Problem 13

On planet Tehar, the acceleration due to gravity is the same as that on Earth but there is also a strong downward electric field with the field being uniform close to the planet's surface. A $2.00-\mathrm{kg}$ ball having a charge of $5.00 \mu \mathrm{C}$ is thrown upward at a speed of $20.1 \mathrm{~m} / \mathrm{s}$ and it hits the ground after an interval of $4.10 \mathrm{~s}$. What is the potential difference between the starting point and the top point of the trajectory?

Keshav Singh

Numerade Educator

Problem 14

An insulating rod having linear charge density $\lambda=$ $40.0 \mu \mathrm{C} / \mathrm{m}$ and linear mass density $\mu=0.100 \mathrm{~kg} / \mathrm{m}$ is
released from rest in a uniform electric field $E=$ $100 \mathrm{~V} / \mathrm{m}$ directed perpendicular to the rod (Fig. P25.14). (a) Determine the speed of the rod after it has traveled $2.00 \mathrm{~m} .$ (b) How does your answer to part (a) change if the electric field is not perpendicular to the rod? Explain.

Mayukh Banik

Numerade Educator

Problem 15

A particle having charge $q=+2.00 \mu \mathrm{C}$ and mass $m=$ $0.0100 \mathrm{~kg}$ is connected to a string that is $L=1.50 \mathrm{~m}$ long and is tied to the pivot point $P$ in Figure $\mathrm{P} 25.15$. The particle, string, and pivot point all lie on a horizontal table. The particle is released from rest when the string makes an angle $\theta=60.0^{\circ}$ with a uniform electric field of magnitude $E=300 \mathrm{~V} / \mathrm{m} .$ Determine the speed of the particle when the string is parallel to the electric field (point $a$ in Fig. $\mathrm{P} 25.15$ ).

Keshav Singh

Numerade Educator

Problem 16

(a) Find the potential at a distance of $1.00 \mathrm{~cm}$ from a proton. (b) What is the potential difference between two points that are $1.00 \mathrm{~cm}$ and $2.00 \mathrm{~cm}$ from a proton?
(c) Repeat parts (a) and (b) for an electron.

Manish Kumar

Numerade Educator

Problem 17

Given two $2.00-\mu \mathrm{C}$ charges, as shown in Figure $\mathrm{P} 25.17$, and a positive test charge $q=1.28 \times 10^{-18} \mathrm{C}$ at the origin, (a) what is the net force exerted on $q$ by the two $2.00-\mu \mathrm{C}$ charges? (b) What is the electric field at the origin due to the two $2.00-\mu \mathrm{C}$ charges? (c) What is the electric potential at the origin due to the two $2.00-\mu \mathrm{C}$ charges?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

A charge $+q$ is at the origin. A charge $-2 q$ is at $x=$ $2.00 \mathrm{~m}$ on the $x$ axis. For what finite value(s) of $x$ is
(a) the electric field zero? (b) the electric potential zero?

Rashmi Sinha

Numerade Educator

Problem 19

The Bohr model of the hydrogen atom states that the single electron can exist only in certain allowed orbits around the proton. The radius of each Bohr orbit is $r=$ $n^{2}(0.0529 \mathrm{~nm})$ where $n=1,2,8, \ldots \ldots$ Calculate
the electric potential energy of a hydrogen atom when the electron is in the (a) first allowed orbit, $n=1$;
(b) second allowed orbit, $n=2$; and (c) when the electron has escaped from the atom $(r=\infty)$. Express your answers in electron volts.

Keshav Singh

Numerade Educator

Problem 20

Two point charges $Q_{1}=+5.00 \mathrm{nC}$ and $Q_{2}=-3.00 \mathrm{nC}$ are separated by $95.0 \mathrm{~cm}$. (a) What is the potential energy of the pair? What is the significance of the algebraic sign of your answer? (b) What is the electric potential at a point midway between the charges?

Keshav Singh

Numerade Educator

Problem 21

The three charges in Figure $\mathrm{P} 25.21$ are at the vertices of an isosceles triangle. Calculate the electric potential at the midpoint of the base, taking $q=7.00 \mu \mathrm{C}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

Compare this problem with Problem 55 in Chapter $23 .$ Four identical point charges $(q=+10.0 \mu \mathrm{C})$ are located on the corners of a rectangle, as shown in Figure $\mathrm{P} 28.55 .$ The dimensions of the rectangle are $L=60.0 \mathrm{~cm}$ and $W=15.0 \mathrm{~cm} .$ Calculate the electric potential energy of the charge at the lower left corner due to the other three charges.

Keshav Singh

Numerade Educator

Problem 23

Show that the amount of work required to assemble four identical point charges of magnitude $Q$ at the corners of a square of side $s$ is $5.41 k_{e} Q^{2} / s$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

Compare this problem with Problem 18 in Chapter 23. Two point charges each of magnitude $2.00 \mu \mathrm{C}$ are located on the $x$ axis. One is at $x=1.00 \mathrm{~m}$, and the other is at $x=-1.00 \mathrm{~m} .$ (a) Determine the electric potential on the $y$ axis at $y=0.500 \mathrm{~m}$. (b) Calculate the electric potential energy of a third charge, of $-3.00 \mu \mathrm{C}$, placed on the $y$ axis at $y=0.500 \mathrm{~m}$.

Keshav Singh

Numerade Educator

Problem 25

Compare this problem with Problem 22 in Chapter 23. Five equal negative point charges $-q$ are placed symmetrically around a circle of radius $R$. Calculate the electric potential at the center of the circle.

Keshav Singh

Numerade Educator

Problem 26

Compare this problem with Problem 17 in Chapter $23 .$ Three equal positive charges $q$ are at the corners of an equilateral triangle of side $a$, as shown in Figure $\mathrm{P} 23.17 .$
(a) At what point, if any, in the plane of the charges is the electric potential zero? (b) What is the electric potential at the point $P$ due to the two charges at the base of the triangle?

Keshav Singh

Numerade Educator

Problem 27

Review Problem. Two insulating spheres having radii $0.300 \mathrm{~cm}$ and $0.500 \mathrm{~cm}$, masses $0.100 \mathrm{~kg}$ and $0.700 \mathrm{~kg}$
and charges $-2.00 \mu \mathrm{C}$ and $3.00 \mu \mathrm{C}$ are released from rest when their centers are separated by $1.00 \mathrm{~m}$.
(a) How fast will each be moving when they collide? (Hint: Consider conservation of energy and linear momentum.) (b) If the spheres were conductors would the speeds be larger or smaller than those calculated in part
(a)? Explain.

Keshav Singh

Numerade Educator

Problem 28

Review Problem. Two insulating spheres having radii $r_{1}$ and $r_{2}$, masses $m_{1}$ and $m_{2}$, and charges $-q_{1}$ and $q_{2}$ are released from rest when their centers are separated by a distance $d$. (a) How fast is each moving when they collide? (Hint: Consider conservation of energy and conservation of linear momentum.) (b) If the spheres were conductors, would the speeds be greater or less than those calculated in part (a)?

Keshav Singh

Numerade Educator

Problem 29

A small spherical object carries a charge of $8.00 \mathrm{nC.}$ At what distance from the center of the object is the potential equal to $100 \mathrm{~V} ? 50.0 \mathrm{~V} ? 25.0 \mathrm{~V} ?$ Is the spacing of the equipotentials proportional to the change in potential?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

Two point charges of equal magnitude are located along the $y$ axis equal distances above and below the $x$ axis, as shown in Figure P25.80. (a) Plot a graph of the potential at points along the $x$ axis over the interval $-3 a<x<8 a$. You should plot the potential in units of $k_{e} Q / a .(\mathrm{b})$ Let the charge located at $-a$ be negative and plot the potential along the $y$ axis over the interval $-4 a<y<4 a$.

Keshav Singh

Numerade Educator

Problem 31

In Rutherford's famous scattering experiments that led to the planetary model of the atom, alpha particles (charge $+2 e$, mass $=6.64 \times 10^{-27} \mathrm{~kg}$ ) were fired at a gold nucleus (charge $+79 e$ ). An alpha particle, initially very far from the gold nucleus, is fired with a velocity of $2.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$ directly toward the center of the nucleus. How close does the alpha particle get to this center before turning around? Assume the gold nucleus remains stationary.

Keshav Singh

Numerade Educator

Problem 32

An electron starts from rest $3.00 \mathrm{~cm}$ from the center of a uniformly charged insulating sphere of radius $2.00 \mathrm{~cm}$ and total charge $1.00 \mathrm{nC.}$ What is the speed of the electron when it reaches the surface of the sphere?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

Calculate the energy required to assemble the array of charges shown in Figure $\mathrm{P} 25.83$, where $a=0.200 \mathrm{~m}$, $b=0.400 \mathrm{~m}$, and $q=6.00 \mu \mathrm{C}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

Four identical particles each have charge $q$ and mass $m$. They are released from rest at the vertices of a square of side $L$. How fast is each charge moving when their distance from the center of the square doubles?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 35

How much work is required to assemble eight identical point charges, each of magnitude $q$, at the corners of a cube of side s?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

The potential in a region between $x=0$ and $x=$ $6.00 \mathrm{~m}$ is $V=a+b x$ where $a=10.0 \mathrm{~V}$ and $b=$ $-7.00 \mathrm{~V} / \mathrm{m} .$ Determine (a) the potential at $x=$ $0,3.00 \mathrm{~m}$, and $6.00 \mathrm{~m}$ and $(\mathrm{b})$ the magnitude and direction of the electric field at $x=0,3.00 \mathrm{~m}$, and $6.00 \mathrm{~m}$.

Mayukh Banik

Numerade Educator

Problem 37

Over a certain region of space, the electric potential is $V=5 x-3 x^{2} y+2 y z^{2}$. Find the expressions for the $x, y$ and $z$ components of the electric field over this region. What is the magnitude of the field at the point $P$, which has coordinates $(1,0,-2) \mathrm{m}$ ?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 38

The electric potential inside a charged spherical conductor of radius $R$ is given by $V=k_{e} Q / R$ and outside the conductor is given by $V=k_{e} Q / r$. Using $E_{r}=-d V / d r$, derive the electric field (a) inside and
(b) outside this charge distribution.

Mayukh Banik

Numerade Educator

Problem 39

It is shown in Example $25.7$ that the potential at a point $P$ a distance $a$ above one end of a uniformly charged rod of length $\ell$ lying along the $x$ axis is
$$
V=\frac{k_{e} Q}{\ell} \ln \left(\frac{\ell+\sqrt{\ell^{2}+a^{2}}}{a}\right)
$$
Use this result to derive an expression for the $y$ component of the electric field at $P$. (Hint: Replace $a$ with $y$.)

Keshav Singh

Numerade Educator

Problem 40

When an uncharged conducting sphere of radius $a$ is placed at the origin of an $x y$ coordinate system that lies in an initially uniform electric field $\mathbf{E}=E_{0} \mathbf{k}$, the resulting electric potential is
$$
V(x, y, z)=V_{0}-E_{0} z+\frac{E_{0} a^{3} z}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}
$$
for points outside the sphere, where $V_{0}$ is the (constant) electric potential on the conductor. Use this equation to determine the $x, y$, and $z$ components of the resulting electric field.

Keshav Singh

Numerade Educator

Problem 41

Consider a ring of radius $R$ with the total charge $Q$ spread uniformly over its perimeter. What is the potential difference between the point at the center of the ring and a point on its axis a distance $2 R$ from the center?

Mayukh Banik

Numerade Educator

Problem 42

Compare this problem with Problem 33 in Chapter 23. A uniformly charged insulating rod of length $14.0 \mathrm{~cm}$ is bent into the shape of a semicircle, as shown in Figure P28.83. If the rod has a total charge of $-7.50 \mu \mathrm{C}$, find the electric potential at $O$, the center of the semicircle.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 43

A rod of length $L$ (Fig. P25.43) lies along the $x$ axis with its left end at the origin and has a nonuniform charge density $\lambda=\alpha x$ (where $\alpha$ is a positive constant).
(a) What are the units of $\alpha ?$ (b) Calculate the electric potential at $A$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

For the arrangement described in the previous problem, calculate the electric potential at point $B$ that lies on the perpendicular bisector of the rod a distance $b$ above the $x$ axis.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

Calculate the electric potential at point $P$ on the axis of the annulus shown in Figure $\mathrm{P} 25.45$, which has a uniform charge density $\sigma$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

A wire of finite length that has a uniform linear charge density $\lambda$ is bent into the shape shown in Figure $\mathrm{P} 25.46$. Find the electric potential at point $O$.

Keshav Singh

Numerade Educator

Problem 47

How many electrons should be removed from an initially uncharged spherical conductor of radius $0.300 \mathrm{~m}$ to produce a potential of $7.50 \mathrm{kV}$ at the surface?

Mayukh Banik

Numerade Educator

Problem 48

Two charged spherical conductors are connected by a long conducting wire, and a charge of $20.0 \mu \mathrm{C}$ is placed on the combination. (a) If one sphere has a radius of $4.00 \mathrm{~cm}$ and the other has a radius of $6.00 \mathrm{~cm}$, what is the electric field near the surface of each sphere?
(b) What is the electric potential of each sphere?

Aja S

Numerade Educator

Problem 49

A spherical conductor has a radius of $14.0 \mathrm{~cm}$ and charge of $26.0 \mu \mathrm{C}$. Calculate the electric field and the electric potential at (a) $r=10.0 \mathrm{~cm}$, (b) $r=20.0 \mathrm{~cm}$, and (c) $r=14.0 \mathrm{~cm}$ from the center.

Mayukh Banik

Numerade Educator

Problem 50

Two concentric spherical conducting shells of radii $a=$ $0.400 \mathrm{~m}$ and $b=0.500 \mathrm{~m}$ are connected by a thin wire, as shown in Figure $\mathrm{P} 25.50 .$ If a total charge $Q=$ $10.0 \mu \mathrm{C}$ is placed on the system, how much charge settles on each sphere?

Supratim Pal

Numerade Educator

Problem 51

Consider a Van de Graaff generator with a $90.0-\mathrm{cm}$ diameter dome operating in dry air. (a) What is the maximum potential of the dome? (b) What is the maximum charge on the dome?

Ajay Singhal

Numerade Educator

Problem 52

The spherical dome of a Van de Graaff generator can be raised to a maximum potential of $600 \mathrm{kV}$; then additional charge leaks off in sparks, by producing breakdown of the surrounding dry air. Determine (a) the charge on the dome and (b) the radius of the dome.

Ajay Singhal

Numerade Educator

Problem 53

The liquid-drop model of the nucleus suggests that high-energy oscillations of certain nuclei can split the nucleus into two unequal fragments plus a few neutrons. The fragments acquire kinetic energy from their mutual Coulomb repulsion. Calculate the electric potential energy (in electron volts) of two spherical fragments from a uranium nucleus having the following charges and radii: $38 e$ and $5.50 \times 10^{-15} \mathrm{~m} ; 54 e$ and $6.20 \times 10^{-15} \mathrm{~m}$. Assume that the charge is distributed uniformly throughout the volume of each spherical fragment and that their surfaces are initially in contact at rest. (The electrons surrounding the nucleus can be neglected.)

Keshav Singh

Numerade Educator

Problem 54

On a dry winter day you scuff your leather-soled shoes across a carpet and get a shock when you extend the tip of one finger toward a metal doorknob. In a dark room you see a spark perhaps $5 \mathrm{~mm}$ long. Make order-ofmagnitude estimates of (a) your electric potential and
(b) the charge on your body before you touch the doorknob. Explain your reasoning.

Keshav Singh

Numerade Educator

Problem 55

The charge distribution shown in Figure $\mathrm{P} 25.55$ is referred to as a linear quadrupole. (a) Show that the potential at a point on the $x$ axis where $x>a$ is
$$
V=\frac{2 k_{e} Q a^{2}}{x^{3}-x a^{2}}
$$
(b) Show that the expression obtained in part (a) when $x \gg a$ reduces to
$$
V=\frac{2 k_{e} Q a^{2}}{x^{3}}
$$

Mayukh Banik

Numerade Educator

Problem 56

(a) Use the exact result from Problem 55 to find the electric field at any point along the axis of the linear quadrupole for $x>a .$ (b) Evaluate $E$ at $x=3 a$ if $a=$ $2.00 \mathrm{~mm}$ and $Q=3.00 \mu \mathrm{C}$.

Surjit Tewari

Numerade Educator

Problem 57

At a certain distance from a point charge, the magnitude of the electric field is $500 \mathrm{~V} / \mathrm{m}$ and the electric potential is $-3.00 \mathrm{kV}$. (a) What is the distance to the charge? (b) What is the magnitude of the charge?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

An electron is released from rest on the axis of a uniform positively charged ring, $0.100 \mathrm{~m}$ from the ring's center. If the linear charge density of the ring is $+0.100 \mu \mathrm{C} / \mathrm{m}$ and the radius of the ring is $0.200 \mathrm{~m}$, how fast will the electron be moving when it reaches the center of the ring?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

(a) Consider a uniformly charged cylindrical shell having total charge $Q$, radius $R$, and height $h .$ Determine the electrostatic potential at a point a distance $d$ from the right side of the cylinder, as shown in Figure $\mathrm{P} 25.59 .$ ( Hint: Use the result of Example $25.5$ by treating the cylinder as a collection of ring charges.) (b) Use the result of Example $25.6$ to solve the same problem for a solid cylinder.

Keshav Singh

Numerade Educator

Problem 60

Two parallel plates having charges of equal magnitude but opposite sign are separated by $12.0 \mathrm{~cm}$. Each plate has a surface charge density of $36.0 \mathrm{nC} / \mathrm{m}^{2} . \mathrm{A}$ proton is released from rest at the positive plate. Determine
(a) the potential difference between the plates, $(\mathrm{b})$ the energy of the proton when it reaches the negative plate,
(c) the speed of the proton just before it strikes the negative plate, (d) the acceleration of the proton, and
(e) the force on the proton. (f) From the force, find the magnitude of the electric field and show that it is equal to that found from the charge densities on the plates.

Keshav Singh

Numerade Educator

Problem 61

Calculate the work that must be done to charge a spherical shell of radius $R$ to a total charge $Q$.

Rodger Claar

Numerade Educator

Problem 62

A Geiger-Müller counter is a radiation detector that essentially consists of a hollow cylinder (the cathode) of inner radius $r_{a}$ and a coaxial cylindrical wire (the anode) of radius $n_{b}$ (Fig. P25.62). The charge per unit length on the anode is $\lambda$, while the charge per unit length on the cathode is $-\lambda$. (a) Show that the magnitude of the potential difference between the wire and the cylinder in the sensitive region of the detector is
$$
\Delta V=2 k_{e} \lambda \ln \left(\frac{r_{a}}{r_{b}}\right)
$$
(b) Show that the magnitude of the electric field over that region is given by
$$
E=\frac{\Delta V}{\ln \left(r_{a} / r_{b}\right)}\left(\frac{1}{r}\right)
$$
where $r$ is the distance from the center of the anode to the point where the field is to be calculated.

Keshav Singh

Numerade Educator

Problem 63

From Gauss's law, the electric field set up by a uniform line of charge is
$$
\mathbf{E}=\left(\frac{\lambda}{2 \pi \epsilon_{0} r}\right) \hat{\mathbf{r}}
$$
where $\hat{\mathbf{r}}$ is a unit vector pointing radially away from the line and $\lambda$ is the charge per unit length along the line. Derive an expression for the potential difference between $r=r_{1}$ and $r=r_{2}$.

Keshav Singh

Numerade Educator

Problem 64

A point charge $q$ is located at $x=-R$, and a point charge $-2 q$ is located at the origin. Prove that the equipotential surface that has zero potential is a sphere centered at $(-4 R / 3,0,0)$ and having a radius $r=$ $2 R / 3$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

Consider two thin, conducting, spherical shells as shown in cross-section in Figure P25.65. The inner shell has a radius $r_{1}=15.0 \mathrm{~cm}$ and a charge of $10.0 \mathrm{nC}$. The outer shell has a radius $r_{2}=30.0 \mathrm{~cm}$ and a charge of $-15.0 \mathrm{nC}$. Find (a) the electric field $\mathbf{E}$ and $(\mathrm{b})$ the electric potential $V$ in regions $A, B$, and $C$, with $V=0 \mathrm{at}$ $r=\infty$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

The $x$ axis is the symmetry axis of a uniformly charged ring of radius $R$ and charge $Q$ (Fig. P25.66). A point charge $Q$ of mass $M$ is located at the center of the ring. When it is displaced slightly, the point charge accelerates along the $x$ axis to infinity. Show that the ultimate speed of the point charge is
$$
v=\left(\frac{2 k_{e} Q^{2}}{M R}\right)^{1 / 2}
$$

Keshav Singh

Numerade Educator

Problem 67

An infinite sheet of charge that has a surface charge density of $25.0 \mathrm{nC} / \mathrm{m}^{2}$ lies in the $y z$ plane, passes through the origin, and is at a potential of $1.00 \mathrm{kV}$ at the point $y=0, z=0 .$ A long wire having a linear charge density of $80.0 \mathrm{nC} / \mathrm{m}$ lies parallel to the $y$ axis and intersects the $x$ axis at $x=3.00 \mathrm{~m} .$ (a) Determine, as a function of $x$, the potential along the $x$ axis between wire and sheet. (b) What is the potential energy of a $2.00-\mathrm{n} \mathrm{C}$ charge placed at $x=0.800 \mathrm{~m} ?$

Dominador Tan

Numerade Educator

Problem 68

The thin, uniformly charged rod shown in Figure P25.68 has a linear charge density $\lambda$. Find an expression for the electric potential at $P$.

Mayukh Banik

Numerade Educator

Problem 69

A dipole is located along the $y$ axis as shown in Figure P25.69. (a) At a point $P$, which is far from the dipole $(r \gg a)$, the electric potential is
$$
V=k_{e} \frac{p \cos \theta}{r^{2}}
$$
where $p=2 q a$. Calculate the radial component $E_{r}$ and the perpendicular component $E_{\theta}$ of the associated electric field. Note that $E_{\theta}=-(1 / r)(\partial V / \partial \theta) .$ Do these results seem reasonable for $\theta=90^{\circ}$ and $0^{\circ} ?$ for $r=0$ ?
(b) For the dipole arrangement shown, express $V$ in terms of cartesian coordinates using $r=\left(x^{2}+y^{2}\right)^{1 / 2}$ and
$$
\cos \theta=\frac{y}{\left(x^{2}+y^{2}\right)^{1 / 2}}
$$
Using these results and taking $r \gg a$, calculate the field components $E_{x}$ and $E_{y}$.
$0 .$ Figure $\mathrm{P} 25.70$ shows several equipotential lines each labeled by its potential in volts. The distance between the lines of the square grid represents $1.00 \mathrm{~cm} .$ (a) Is the magnitude of the field bigger at $A$ or at $B$ ? Why?
(b) What is $\mathbf{E}$ at $B ?$ (c) Represent what the field looks like by drawing at least eight field lines.
(b) For the dipole arrangement shown, express $V$ in terms of cartesian coordinates using $r=\left(x^{2}+y^{2}\right)^{1 / 2}$ and
$$
\cos \theta=\frac{y}{\left(x^{2}+y^{2}\right)^{1 / 2}}
$$
Using these results and taking $r \gg a$, calculate the field components $E_{x}$ and $E_{y}$.

Keshav Singh

Numerade Educator

Problem 70

Figure $\mathrm{P} 25.70$ shows several equipotential lines each labeled by its potential in volts. The distance between the lines of the square grid represents $1.00 \mathrm{~cm}$. (a) Is the magnitude of the field bigger at $A$ or at $B$ ? Why?
(b) What is $\mathbf{E}$ at $B ?$ (c) Represent what the field looks like by drawing at least eight field lines.

Keshav Singh

Numerade Educator

Problem 71

A disk of radius $R$ has a nonuniform surface charge density $\sigma=C r$, where $C$ is a constant and $r$ is measured from the center of the disk (Fig. P25.71). Find (by direct integration) the potential at $P$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

A solid sphere of radius $R$ has a uniform charge density $\rho$ and total charge $Q$. Derive an expression for its total electric potential energy. (Hint: Imagine that the sphere is constructed by adding successive layers of concentric shells of charge $d q=\left(4 \pi r^{2} d r\right) \rho$ and use $\left.d U=V d q .\right)$

Keshav Singh

Numerade Educator

Problem 73

The results of Problem 62 apply also to an electrostatic precipitator (see Figs. 25.28a and P25.62). An applied voltage $\Delta V=V_{a}-V_{b}=50.0 \mathrm{kV}$ is to produce an electric field of magnitude $5.50 \mathrm{MV} / \mathrm{m}$ at the surface of the central wire. The outer cylindrical wall has uniform radius $r_{a}=0.850 \mathrm{~m} .$ (a) What should be the radius $r_{b}$ of the central wire? You will need to solve a transcendental equation. (b) What is the magnitude of the electric field at the outer wall?

Keshav Singh

Numerade Educator