College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 16

Electric Forces and Fields - all with Video Answers

Educators

Chapter Questions

Problem 1

Find the total positive charge of all the protons in $1.0 \mathrm{~mol}$ of water.

Zulfiqar Ali

Zulfiqar Ali

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Problem 2

Suppose a 1.0-g nugget of pure gold has zero net charge. What would be its net charge after it has $1.0 \%$ of its electrons removed?

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Problem 3

A balloon, initially neutral, is rubbed with fur until it acquires a net charge of $-0.60 \mathrm{nC}$. (a) Assuming that only electrons are transferred, were electrons removed from the balloon or added to it? (b) How many electrons were transferred?

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Problem 4

A metallic sphere has a charge of $+4.0 \mathrm{nC}$. A negatively charged rod has a charge of $-6.0 \mathrm{nC}$. When the rod touches the sphere, $8.2 \times 10^{9}$ electrons are transferred. What are the charges of the sphere and the rod now?

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Problem 5

A positively charged rod is brought near two uncharged conducting spheres of the same size that are initially touching each other (diagram a). The spheres are moved apart and then the charged rod is removed (diagram b). (a) What is the sign of the net charge on sphere 1 in diagram b? (b) In comparison with the charge on sphere 1, how much and what sign of charge is on sphere 2 ?

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Problem 6

A metal sphere A has charge $Q$. Two other spheres, $\mathrm{B}$ and $\mathrm{C}$, are identical to $\mathrm{A}$ except they have zero net charge. A touches $\mathrm{B}$, then the two spheres are separated. B touches $C$, then those spheres are separated. Finally, C touches A and those two spheres are separated. How much charge is on each sphere?

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Problem 7

Repeat Problem 6 with a slight change. The difference this time is that sphere $\mathrm{C}$ is grounded when it is touching $\mathrm{B}$, but $\mathrm{C}$ is not grounded at any other time. What is the final charge on each sphere?

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Problem 8

Five conducting spheres are charged as shown. All have the same magnitude net charge except E, whose net charge is zero. Which pairs are attracted to each other and which are repelled by each other when they are brought near each other, but well away from the other spheres?

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Problem 9

If the electric force of repulsion between two $1-\mathrm{C}$ charges is $10 \mathrm{~N}$, how far apart are they?

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Problem 10

Two small metal spheres are $25.0 \mathrm{~cm}$ apart. The spheres have equal amounts of negative charge and repel each other with a force of $0.036 \mathrm{~N}$. What is the charge on each sphere?

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Problem 11

What is the ratio of the electric force to the gravitational force between a proton and an electron separated by $5.3 \times 10^{-11} \mathrm{~m}$ (the radius of a hydrogen atom)?

Narayan Hari

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Problem 12

How many electrons must be removed from each of two $5.0-\mathrm{kg}$ copper spheres to make the electric force of repulsion between them equal in magnitude to the gravitational attraction between them?

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Problem 13

$\mathrm{A}+2.0-\mathrm{nC}$ point charge is $3.0 \mathrm{~cm}$ away from a $-3.0-\mathrm{nC}$
point charge. (a) What are the magnitude and direction of the electric force acting on the $+2.0-\mathrm{n} \mathrm{C}$ charge?
(b) What are the magnitude and direction of the electric force acting on the $-3.0-\mathrm{nC}$ charge?

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Problem 14

Two metal spheres separated by a distance much greater than either sphere's radius have equal mass $m$ and equal electric charge $q$. What is the ratio of charge to mass $q / m$ in $\mathrm{C} / \mathrm{kg}$ if the electrical and gravitational forces balance?

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Problem 15

In the figure, a third point charge $-q$ is placed at point $P$. What is the electric force on $-q$ due to the other two point charges?

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Problem 16

Two point charges are separated by a distance $r$ and repel each other with a force $F$. If their separation is reduced to $0.25$ times the original value, what is the magnitude of the force of repulsion between them?

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Problem 17

A $\mathrm{K}^{+}$ ion and a $\mathrm{Cl}^{-}$ ion are directly across from each other on opposite sides of a membrane $9.0 \mathrm{~nm}$ thick. What is the electric force on the $\mathrm{K}^{+}$ ion due to the $\mathrm{Cl}^{-}$ ion? Ignore the presence of other charges.

Brian Francisco

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Problem 18

Three point charges are fixed in place in a right triangle. What is the electric force on the $-0.60-\mu \mathrm{C}$ charge due to the other two charges?

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Problem 19

Three point charges are fixed in place in a right triangle. What is the electric force on the $+1.0-\mu \mathrm{C}$ charge due to the other two charges?

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Problem 20

A tiny sphere with a charge of $7.0 \mu \mathrm{C}$ is attached to a spring. Two other tiny charged spheres, each with a charge of $-4.0 \mu \mathrm{C}$, are placed in the positions shown in the figure and the spring stretches $5.0 \mathrm{~cm}$ from its preyious equilibrium position toward the two spheres. Calculate the spring constant.

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Problem 21

A total charge of $7.50 \times 10^{-6} \mathrm{C}$ is distributed on two different small metal spheres. When the spheres are $6.00 \mathrm{~cm}$ apart, they each feel a repulsive force of $20.0 \mathrm{~N}$. How much charge is on each sphere?

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Problem 22

Two Styrofoam balls with the same mass $m=9.0 \times 10^{-8} \mathrm{~kg}$ and the same positive charge $Q$ are suspended from the same point by insulating threads of length $L=0.98 \mathrm{~m}$. The separation of the balls is $d=0.020 \mathrm{~m} .$ What is the charge $Q ?$

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Problem 23

Using the three point charges of Example $16.3$, find the magnitude of the force on $q_{2}$ due to the other two charges, $q_{1}$ and $q_{3} .$ [Hint: After finding the force on $q_{2}$ due to $q_{1}$, separate that force into $x$ - and $y$ -components.]

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Problem 24

An equilateral triangle has a point charge $+q$ at each of the three vertices $(A, B, C)$. Another point charge $Q$ is placed at $D$, the midpoint of the side $B C$. Solve for $Q$ if the total electric force on the charge at $A$ due to the charges at $B, C$, and $D$ is zero.

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Problem 25

A small sphere with a charge of $-0.60 \mu \mathrm{C}$ is placed in a uniform electric field of magnitude $1.2 \times 10^{6} \mathrm{~N} / \mathrm{C}$ pointing to the west. What is the magnitude and direction of the force on the sphere due to the electric field?

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Problem 26

The electric field across a cellular membrane is $1.0 \times 10^{7} \mathrm{~N} / \mathrm{C}$ directed into the cell. (a) If a pore opens, which way do sodium ions ( $\mathrm{Na}^{+}$ ) flow-into the cell or out of the cell? (b) What is the magnitude of the electric force on the sodium ion? The charge on the sodium ion is $+e$

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Problem 27

What are the magnitude and direction of the acceleration of a proton at a point where the electric field has magnitude $33 \mathrm{kN} / \mathrm{C}$ and is directed straight up?

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Problem 28

What are the magnitude and direction of the acceleration of an electron at a point where the electric field has magnitude $6100 \mathrm{~N} / \mathrm{C}$ and is directed due north?

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Problem 29

What are the magnitude and direction of the electric field midway between two point charges, $-15 \mu \mathrm{C}$ and $+12 \mu \mathrm{C}$, that are $8.0 \mathrm{~cm}$ apart?

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Problem 30

An electron traveling horizontally from west to east enters a region where a uniform electric field is directed upward. What is the direction of the electric force exerted on the electron once it has entered the field?

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Problem 31

A negative point charge $-Q$ is situated near a large metal plate that has a total charge of $+Q$. Sketch the electric field lines.

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Problem 32

What is the electric field at $x=d$ (point $P$ )?

Vishal Gupta

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Problem 33

What is the electric field at $x=2 d$ (point $S$ )?

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Problem 34

Are there any points $n o t$ on the $x$ -axis where $\overrightarrow{\mathbf{E}}=0$ ? Explain.

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Problem 35

On the $x$ -axis, in which of the three regions $x<0$, $0<x<3 d$, and $x>3 d$ is there a point where $\overrightarrow{\mathbf{E}}=0 ?$ Explain.

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Problem 36

Find the $x$ -coordinates of the point(s) on the $x$ -axis where $\overrightarrow{\mathbf{E}}=0$.

Anas Venkitta

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Problem 37

Sketch the electric field lines in the
plane of the page due to the charges shown in the diagram.

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Problem 38

Sketch the electric field lines near two isolated and equal
(a) positive point charges and (b) negative point charges. Include arrowheads to show the field directions.

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Problem 39

Problems 39-42. Two tiny objects with equal charges of $7.00 \mu \mathrm{C}$ are placed at the two lower corners of a square with sides of $0.300 \mathrm{~m}$, as shown.
Find the electric field at point $B$, midway between the upper left and right corners.

Christopher Dzorkpata

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Problem 40

Two tiny objects with equal charges of $7.00 \mu \mathrm{C}$ are placed at the two lower corners of a square with sides of $0.300 \mathrm{~m}$, as shown.
Find the electric field at point $C$, the center of the square.

Zulfiqar Ali

Numerade Educator

Problem 41

Two tiny objects with equal charges of $7.00 \mu \mathrm{C}$ are placed at the two lower corners of a square with sides of $0.300 \mathrm{~m}$, as shown.
Find the electric field at point $A$, the upper left corner

Zulfiqar Ali

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Problem 42

Two tiny objects with equal charges of $7.00 \mu \mathrm{C}$ are placed at the two lower corners of a square with sides of $0.300 \mathrm{~m}$, as shown.
Where would you place a third small object with the same charge so that the electric field is zero at the corner of the square labeled $\bar{A}$ ?

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Problem 43

Three point charges are placed on the $x$ -axis. A charge of $3.00 \mu \mathrm{C}$ is at the origin. A charge of $-5.00 \mu \mathrm{C}$ is at $20.0 \mathrm{~cm}$, and a charge of $8.00 \mu \mathrm{C}$ is at $35.0 \mathrm{~cm}$. What is the force on the charge at the origin?

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Problem 44

Two equal charges $\quad(Q=$ $+1.00 \mathrm{nC}$ ) are situated at the diagonal corners $A$ and $B$ of a square of side $1.0 \mathrm{~m}$. What is the magnitude of the electric field at point $D$ ?

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Problem 45

Suppose a charge $q$ is placed at point $x=0, y=0 .$ A second charge $q$ is placed at point $x=8.0 \mathrm{~m}, y=0 .$ What charge must be placed at the point $x=4.0 \mathrm{~m}, y=0$ in order that the field at the point $x=4.0 \mathrm{~m}, y=3.0 \mathrm{~m}$ be zero?

Vishal Gupta

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Problem 46

Two point charges, $q_{1}=+20.0 \mathrm{nC}$ and $q_{2}=+10.0 \mathrm{nC}$, are located on the $x$ -axis at $x=0$ and $x=1.00 \mathrm{~m}$, respectively. Where on the $x$ -axis is the electric field equal to zero?

Zulfiqar Ali

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Problem 47

Two electric charges, $q_{1}=+20.0 \mathrm{nC}$ and $q_{2}=+10.0 \mathrm{nC}$, are located on the $x$ -axis at $x=0 \mathrm{~m}$ and $x=1.00 \mathrm{~m}$, respectively. What is the magnitude of the electric field at the point $x=0.50 \mathrm{~m}, y=0.50 \mathrm{~m}$ ?

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Problem 48

An electron is placed in a uniform electric field of strength $232 \mathrm{~N} / \mathrm{C}$. If the electron is at rest at the origin of a coordinate system at $t=0$ and the electric field is in the positive $x$ -direction, what are the $x$ - and $y$ coordinates of the electron at $t=2.30 \mathrm{~ns}$ ?

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Problem 49

An electron is projected horizontally into the space between two oppositely charged metal plates. The electric field between the plates is $500.0 \mathrm{~N} / \mathrm{C}$, directed up.
(a) While in the field, what is the force on the electron?
(b) If the vertical deflection of the electron as it leaves the plates is $3.00 \mathrm{~mm}$, how much has its kinetic energy increased due to the electric field?

Zulfiqar Ali

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Problem 50

A horizontal beam of electrons initially moving at $4.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$ is deflected vertically by the vertical electric field between oppositely charged parallel plates. The magnitude of the field is $2.00 \times 10^{4} \mathrm{~N} / \mathrm{C}$. (a) What is the direction of the field between the plates? (b) What is the charge per unit area on the plates? (c) What is the vertical deflection $d$ of the electrons as they leave the plates?

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Problem 51

A particle with mass $2.30 \mathrm{~g}$ and charge $+10.0 \mu \mathrm{C}$ enters through a small hole in a metal plate with a speed of $8.50 \mathrm{~m} / \mathrm{s}$ at an angle of $55.0^{\circ} .$ The uniform $\overrightarrow{\mathbf{E}}$ field in the region above the plate has magnitude $6.50 \times 10^{3} \mathrm{~N} / \mathrm{C}$ and is directed downward. The region above the metal plate is essentially a vacuum, so there is no air resistance. (a) Can you neglect the force of gravity when solving for the horizontal distance traveled by the particle? Why or why not?
(b) How far will the particle travel, $\Delta x$, before it hits the metal plate?

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Problem 52

Consider the same situation as in Problem 51 , but with a proton entering through the small hole at the same angle with a speed of $v=8.50 \times 10^{5} \mathrm{~m} / \mathrm{s}$. (a) Can you ignore the force of gravity when solving this problem for the horizontal distance traveled by the proton? Why or why not? (b) How far will the proton travel, $\Delta x$, before it hits the metal plate?

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Problem 53

Some forms of cancer can be treated using proton therapy in which proton beams are accelerated to high energies, then directed to collide into a tumor, killing the malignant cells. Suppose a proton accelerator is $4.0 \mathrm{~m}$ long and must accelerate protons from rest to a speed of $1.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$. Ignore any relativistic effects (Chapter
26) and determine the magnitude of the average electric field that could accelerate these protons.

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Problem 54

After the electrons in Example $16.9$ pass through the anode, they are moving at a speed of $8.4 \times 10^{6} \mathrm{~m} / \mathrm{s}$. They next pass between a pair of parallel plates $[(\mathrm{A})$ in Fig. 16.35]. The plates each have an area of $2.50 \mathrm{~cm}$ by $2.50 \mathrm{~cm}$ and they are separated by a distance of $0.50 \mathrm{~cm}$. The uniform electric field between them is $1.0 \times 10^{3} \mathrm{~N} / \mathrm{C}$ and the plates are charged as shown.
(a) In what direction are the electrons deflected? (b) By how much are the electrons deflected after passing through these plates?

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Problem 55

After the electrons pass through the parallel plates in Problem 54 , they pass between another set of parallel plates [(B) in Fig. 16.35]. These plates also have an area of $2.50 \mathrm{~cm}$ by $2.50 \mathrm{~cm}$ and are separated by a distance of $0.50 \mathrm{~cm}$. (a) In what direction must the field be oriented so that the electrons are deflected vertically upward? (b) If we neglect the gravitational force, how strong must the field be between these plates in order for the electrons to be deflected by $2.0 \mathrm{~mm} ?$ (c) How much less will the electrons be deflected if we do include the gravitational force?

Zulfiqar Ali

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Problem 56

A conducting sphere that carries a total charge of $-6 \mu \mathrm{C}$ is placed at the center of a conducting spherical shell that carries a total charge of $+1 \mu \mathrm{C}$. The conductors are in electrostatic equilibrium. Determine the charge on the outer surface of the shell. [Hint: Sketch a field line diagram.]

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Problem 57

A conducting sphere that carries a total charge of $+6 \mu \mathrm{C}$ is placed at the center of a conducting spherical shell that also carries a total charge of $+6 \mu \mathrm{C}$. The conductors are in electrostatic equilibrium.
(a) Determine the charge on the inner surface of the shell. (b) Determine the total charge on the outer surface of the shell.

Mayukh Banik

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Problem 58

A hollow conducting sphere of radius $R$ carries a negative charge $-q$. (a) Write expressions for the electric field $\overrightarrow{\mathbf{E}}$ inside $(r<R)$ and outside $(r>R)$ the sphere. Also indicate the direction of the field. (b) Sketch a graph of the field strength as a function of $r$ : [Hint: See Conceptual Example 16.8.]

Mayukh Banik

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Problem 59

A conducting sphere is placed within a conducting spherical shell. The conductors are in electrostatic equilibrium. The inner sphere has a radius of $1.50 \mathrm{~cm}$, the inner radius of the spherical shell is $2.25 \mathrm{~cm}$, and the outer radius of the shell is $2.75 \mathrm{~cm}$. If the inner sphere has a charge of $230 \mathrm{nC}$, and the spherical shell has zero net charge, (a) what is the magnitude of the electric field at a point $1.75 \mathrm{~cm}$ from the center? (b) What is the electric field at a point $2.50 \mathrm{~cm}$ from the center? (c) What is the electric field at a point $3.00 \mathrm{~cm}$ from the center? [Hint: What must be true about the electric field inside a conductor in electrostatic equilibrium?]

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Problem 60

A conductor in electrostatic equilibrium contains a cavity in which there are two point charges: $q_{1}=+5 \mu \mathrm{C}$ and $q_{2}=-12 \mu \mathrm{C}$. The conductor itself carries a net charge $-4 \mu \mathrm{C}$. How much charge is on (a) the inner surface of the conductor? (b) the outer surface of the conductor?

Vishal Gupta

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Problem 61

In fair weather, over flat ground, there is a downward electric field of about $150 \mathrm{~N} / \mathrm{C}$. (a) Assume that the Earth is a conducting sphere with charge on its surface. If the electric field just outside is $150 \mathrm{~N} / \mathrm{C}$ pointing radially inward, calculate the total charge on the Earth and the charge per unit area. (b) At an altitude of $250 \mathrm{~m}$ above Earth's surface, the field is only $120 \mathrm{~N} / \mathrm{C}$. Calculate the charge density (charge per unit volume) of the air (assumed constant). [Hint: See Conceptual Example 16.8.]

Zulfiqar Ali

Numerade Educator

Problem 62

(a) Find the electric flux through each side of a cube of edge length $a$ in a uniform electric field of magnitude $E$. The field direction is perpendicular to two of the faces.
(b) What is the total flux through the cube?

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Problem 63

In a uniform electric field of magnitude $E$, the field lines cross through a rectangle of area $A$ at an angle of $60.0^{\circ}$ with respect to the plane of the rectangle. What is the flux through the rectangle?

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Problem 64

An object with a charge of $0.890 \mu \mathrm{C}$ is placed at the center of a cube. What is the electric flux through one surface of the cube?

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Problem 65

In this problem, you can show from Coulomb's law that the constant of proportionality in Gauss's law must be $1 / \epsilon_{0}$. Imagine a sphere with its center at a point charge
q. (a) Write an expression for the electric flux in terms of the field strength $E$ and the radius $r$ of the sphere. [Hint: The field strength $E$ is the same everywhere on the sphere and the field lines cross the sphere perpendicular to its surface.] (b) Use Gauss's law in the form $\Phi_{\mathrm{E}}=c q$ (where $c$ is the constant of proportionality) and the electric field strength given by Coulomb's law to show that $c=1 / \epsilon_{0}$.

Mayukh Banik

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Problem 66

(a) Use Gauss's law to prove that the electric field outside any spherically symmetric charge distribution is the same as if all of the charge were concentrated into a point charge. (b) Now use Gauss's law to prove that the electric field inside a spherically symmetric charge distribution is zero if none of the charge is at a distance from the center less than that of the point where we determine the field.

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Problem 67

Using the results of Problem 66 , we can find the electric field at any radius for any spherically symmetrical charge distribution. A solid sphere of charge of radius $R$ has a total charge of $q$ uniformly spread throughout the sphere. (a) Find the magnitude of the electric field for $r \geq R$. (b) Find the magnitude of the electric field for $r \leq R$. (c) Sketch a graph of $E(r)$ for $0 \leq r \leq 3 R$.

Mayukh Banik

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Problem 68

An electron is suspended at a distance of $1.20 \mathrm{~cm}$ above a uniform line of charge. What is the linear charge density of the line of charge? Ignore end effects.

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Problem 69

A thin, flat sheet of charge has a uniform surface charge density $\sigma(\sigma / 2$ on each side). (a) Sketch the field lines due to the sheet. (b) Sketch the field lines for an infinitely large sheet with the same charge density. (c) For the infinite sheet, how does the field strength depend on the distance from the sheet? [Hint: Refer to your field line sketch.] (d) For points close to the finite sheet and far from its edges, can the sheet be approximated by an infinitely large sheet? [Hint: Again, refer to the field line sketches.] (e) Use Gauss's law to show that the magnitude of the electric field near a sheet of uniform charge density $\sigma$ is $E=\sigma /\left(2 \epsilon_{0}\right)$.

Brian Francisco

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Problem 70

A flat conducting sheet of area $A$ has a charge $q$ on each surface. (a) What is the electric field inside the sheet?
(b) Use Gauss's law to show that the electric field just outside the sheet is $E=q /\left(\epsilon_{0} A\right)=\sigma / \epsilon_{0} .$ (c) Does this contradict the result of Problem 69 ? Compare the field line diagrams for the two situations.

Mayukh Banik

Numerade Educator

Problem 71

A parallel-plate capacitor consists of two flat metal plates of area $A$ separated by a small distance $d$. The plates are given equal and opposite net charges $\pm q .$
(a) Sketch the field lines and use your sketch to explain why almost all of the charge is on the inner surfaces of the plates. (b) Use Gauss's law to show that the electric field between the plates and away from the edges is $E=q /\left(\epsilon_{0} A\right)=\sigma / \epsilon_{0} .(\mathrm{c})$ Does this agree with or contra-
dict the result of Problem $70 ?$ Explain. (d) Use the principle of superposition and the result of Problem 69 to arrive at this same answer. [Hint: The inner surfaces of the two plates are thin, flat sheets of charge.]

Mayukh Banik

Numerade Educator

Problem 72

A coaxial cable consists of a wire of radius $a$ surrounded by a thin metal cylindrical shell of radius $b$. The wire has a uniform linear charge density $\lambda>0$ and the outer shell has a uniform linear charge density $-\lambda$. (a) Sketch the field lines for this cable. (b) Find expressions for the magnitude of the electric field in the regions $r \leq a, a<r<b$, and $b \leq r$

Mayukh Banik

Numerade Educator

Problem 73

Use Gauss's law to derive an expression for the electric field outside the thin spherical shell of Conceptual Example $16.8$.

Mayukh Banik

Numerade Educator

Problem 74

Consider two protons (charge $+e$ ), separated by a distance of $2.0 \times 10^{-15} \mathrm{~m}$ (as in a typical atomic nucleus). The electric force between these protons is equal in magnitude to the gravitational force on an object of what mass near Earth's surface?

Mayukh Banik

Numerade Educator

Problem 75

In lab tests it was found that rats can detect electric fields of about $5.0 \mathrm{kN} / \mathrm{C}$ or more. If a point charge of $1.0 \mu \mathrm{C}$ is sitting in a maze, how close must the rat come to the charge in order to detect it?

Zulfiqar Ali

Numerade Educator

Problem 76

A raindrop inside a thundercloud has charge $-8 e$. What is the electric force on the raindrop if the electric field at its location (due to other charges in the cloud) has magnitude $2.0 \times 10^{6} \mathrm{~N} / \mathrm{C}$ and is directed upward?

Zulfiqar Ali

Numerade Educator

Problem 77

An electron beam in an oscilloscope is deflected by the electric field produced by oppositely charged metal plates. If the electric field between the plates is $2.00 \times 10^{5} \mathrm{~N} / \mathrm{C}$ directed downward, what is the force on each electron when it passes between the plates?

Zulfiqar Ali

Numerade Educator

Problem 78

A point charge $q_{1}=+5.0 \mu \mathrm{C}$ is fixed in place at $x=0$ and a point charge $q_{2}=-3.0 \mu \mathrm{C}$ is fixed at $x=$ $-20.0 \mathrm{~cm}$. Where can we place a point charge $q_{3}=-8.0 \mu \mathrm{C}$ so that the net electric force on $q_{1}$ due to $q_{2}$ and $q_{3}$ is zero?

Zulfiqar Ali

Numerade Educator

Problem 79

The Bohr model of the hydrogen atom proposes that the electron orbits around the proton in a circle of radius $5.3 \times 10^{-11} \mathrm{~m}$. The electric force is responsible for the radial acceleration of the electron. What is the speed of the electron in this model?

Zulfiqar Ali

Numerade Educator

Problem 80

In a thunderstorm, charge is separated through a complicated mechanism
that is ultimately powered by the Sun. A simplified model of the charge in a thundercloud represents the positive charge accumulated at the top and the negative charge at the bottom as a pair of point charges.
(a) What is the magnitude and direction of the electric field produced by the two point charges at point $P$, which is just above Earth's surface? (b) Thinking of Earth as a conductor, what sign of charge would accumulate on the surface near point $P ?$ (This accumulated charge increases the magnitude of the electric field near point $P$.)

Mayukh Banik

Numerade Educator

Problem 81

Two point charges are located on the $x$ -axis: a charge of $+6.0 \mathrm{nC}$ at $x=0$ and an unknown charge $q$ at $x=$ $0.50 \mathrm{~m}$. No other charges are nearby. If the electric field is zero at the point $x=1.0 \mathrm{~m}$, what is $q$ ?

Mayukh Banik

Numerade Educator

Problem 82

Three equal charges are placed on three corners of a square. If the force that $Q_{\mathrm{a}}$ exerts on $Q_{\mathrm{b}}$ has magnitude $F_{\mathrm{bs}}$ and the force that $Q_{\mathrm{a}}$ exerts on $Q_{\mathrm{c}}$ has magnitude $F_{\mathrm{ca}}$, what is the ratio of $F_{\mathrm{ca}}$ to $F_{\mathrm{ba}}$ ?

Mayukh Banik

Numerade Educator

Problem 83

Two otherwise identical conducting spheres carry charges of $+5.0 \mu \mathrm{C}$ and $-1.0 \mu \mathrm{C}$. They are initially a large distance $L$ apart. The spheres are brought together, touched together, and then returned to their original

Mayukh Banik

Numerade Educator

Problem 84

Two metal spheres of radius $5.0 \mathrm{~cm}$ carry net charges of $+1.0 \mu \mathrm{C}$ and $+0.2 \mu \mathrm{C} .$ (a) What (approximately) is the magnitude of the electrical repulsion on either sphere when their centers are $1.00 \mathrm{~m}$ apart? (b) Why cannot Coulomb's law be used to find the force of repulsion when their centers are $12 \mathrm{~cm}$ apart? (c) Would the actual force be larger or smaller than the result of using Coulomb's law with $r=12 \mathrm{~cm}$ ? Explain.

Mayukh Banik

Numerade Educator

Problem 85

A charge of $63.0 \mathrm{nC}$ is located at a distance of $3.40 \mathrm{~cm}$ from a charge of $-47.0 \mathrm{nC}$. What are the $x$ - and $y$ -components of the electric field at a point $P$ that is directly above the $63.0-\mathrm{n} \mathrm{C}$ charge at a distance of $1.40 \mathrm{~cm}$ ? Point $P$ and the two charges are on the vertices of a right triangle.

Mayukh Banik

Numerade Educator

Problem 86

Point charges are arranged on the vertices of a square with sides of $2.50 \mathrm{~cm}$. Starting at the upper left corner and going clockwise, we have charge A with a charge of $0.200 \mu \mathrm{C}, \mathrm{B}$ with a charge of $-0.150 \mu \mathrm{C}, \mathrm{C}$ with a charge
of $0.300 \mu \mathrm{C}$, and $\mathrm{D}$ with a mass of $2.00 \mathrm{~g}$, but with an unknown charge. Charges A, $\mathrm{B}$, and $\mathrm{C}$ are fixed in place, and $\mathrm{D}$ is free to move. Particle D's instantaneous acceleration at point $\mathrm{D}$ is $248 \mathrm{~m} / \mathrm{s}^{2}$ in a direction $30.8^{\circ}$ below the negative $x$ -axis. What is the charge on $\mathrm{D}$ ?

Mayukh Banik

Numerade Educator

Problem 87

In a cathode ray tube, electrons initially at rest are accelerated by a uniform electric field of magnitude $4.0 \times 10^{5} \mathrm{~N} / \mathrm{C}$ during the first $5.0 \mathrm{~cm}$ of the tube's length; then they move at essentially constant velocity another $45 \mathrm{~cm}$ before hitting the screen. (a) Find the speed of the electrons when they hit the screen. (b) How long does it take them to travel the length of the tube?

Mayukh Banik

Numerade Educator

Problem 88

In the diagram, regions $A$ and $C$ extend far to the left and right, respectively. The electric field due to the two point charges is zero at some point in which region or regions? Explain.

Supratim Pal

Numerade Educator

Problem 89

A thin wire with positive charge evenly spread along its length is shaped into a semicircle. What is the direction of the electric field at the center of curvature of the semicircle? Explain.

Mayukh Banik

Numerade Educator

Problem 90

A very small charged block with a mass of $2.35 \mathrm{~g}$ is placed on an insulated, frictionless plane inclined at an angle of $17.0^{\circ}$ with respect to the horizontal. The block does not slide down the plane because of a $465-\mathrm{N} / \mathrm{C}$ uniform electric field that
points parallel to the surface downward along the plane. What is the sign and magnitude of the charge on the block?

Mayukh Banik

Numerade Educator

Problem 91

(a) What would the net charges on the Sun and Earth have to be if the electric force instead of the gravitational force were responsible for keeping Earth in its orbit? There are many possible answers, so restrict yourself to the case where the magnitude of the charges is proportional to the masses. (b) If the magnitude of the charges of the proton and electron were not exactly equal, astronomical bodies would have net charges that are approximately proportional to their masses. Could this possibly be an explanation for the Earth's orbit?

Mayukh Banik

Numerade Educator

Problem 92

What is the electric force on the chloride ion in the lower righthand corner in the diagram? Since the ions are in water, the "effective charge" on the chloride ions (Cl) is $-2 \times 10^{-21} \mathrm{C}$ and that of the sodium ions $\left(\mathrm{Na}^{+}\right)$ is $+2 \times 10^{-21} \mathrm{C}$. The effective charge is a way to account for the partial shielding due to nearby water molecules. Assume that all four ions are coplanar.

Mayukh Banik

Numerade Educator

Problem 93

A dipole consists of two equal and opposite point charges $(\pm q)$ separated by a distance $\bar{d}$.
(a) Write an expression for the magnitude of the electric field at a point $(x, 0)$ a large distance $(x \gg d)$ from the midpoint of the charges on a line perpendicular to the dipole axis. [Hint: Use small angle approximations.] (b) Give the direction of the field for $x>0$ and for $x<0$.

Mayukh Banik

Numerade Educator

Problem 94

A dipole consists of two equal and opposite point charges $(\pm q)$ separated by a distance $d$. (a) Write an expression for the electric field at a point $(0, y)$ on the dipole axis. Specify the direction of the field in all four regions:
(b) At distant $y>\frac{1}{2} d, 0<y<\frac{1}{2} d,-\frac{1}{2} d<y<0$, and $y<-\frac{1}{2} d$.
points $(|y| \gg d)$, write a simpler, approximate expression for the field. To what power of $y$ is the field proportional? Does this conflict with Coulomb's law? [Hint: Use the binomial approximation $(1 \pm x)^{n}=1 \pm n x$ for $\left.x \ll 1 .\right]$

Mayukh Banik

Numerade Educator

Problem 95

A dipole consists of two opposite charges $(q$ and $-q)$ separated by a fixed distance $d .$ The dipole is placed in an electric field in the $+x$ -direction of magnitude $E$. The dipole axis makes an angle $\theta$ with the electric field as shown in the diagram. (a) Calculate the net electric force acting on the dipole. (b) Calculate the net torque acting on the dipole due to the electric forces as a function of $\theta$. Let counterclockwise torque be positive. (c) Evaluate the net torque for $\theta=0^{\circ}$, $\theta=36.9^{\circ}$, and $\theta=90.0^{\circ}$. Let $q=\pm 3.0 \mu \mathrm{C}, d=7.0 \mathrm{~cm}$,
and $E=2.0 \times 10^{4} \mathrm{~N} / \mathrm{C} .$

Mayukh Banik

Numerade Educator