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Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 22

Electric Fields - all with Video Answers

Educators

Chapter Questions

16:09

Problem 1

Sketch qualitatively the electric field lines both between and outside two concentric conducting spherical shells when a uniform positive charge $q_{1}$ is on the inner shell and a uniform negative charge $-q_{2}$ is on the outer. Consider the cases $q_{1}>q_{2}, q_{1}=q_{2}$, and $q_{1}<q_{2}$

Jack Hou
Jack Hou
Numerade Educator
04:37

Problem 2

In Fig. $22-34$ the electric field lines on the left have twice the separation of those on the right. (a) If the magnitude of the field at $A$ is $40 \mathrm{~N} / \mathrm{C}$, what is the magnitude of the force on a proton at $A ?(\mathrm{~b})$ What is the magnitude of the field at $B$ ?

Abhishek Jana
Abhishek Jana
Numerade Educator
View

Problem 3

The nucleus of a plutonium- 239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the (a) magnitude and (b) direction (radially inward or outward) of the electric field produced by the protons?

Jack Hou
Jack Hou
Numerade Educator
06:37

Problem 4

Two charged particles are attached to an $x$ axis: Particle 1 of charge $-2.00 \times 10^{-7} \mathrm{C}$ is at position $x=6.00 \mathrm{~cm}$ and particle 2 of charge $+2.00 \times 10^{-7} \mathrm{C}$ is at position $x=21.0 \mathrm{~cm} .$ Midway between the particles, what is their net electric field in unit-vector notation?

Abhishek Jana
Abhishek Jana
Numerade Educator
01:41

Problem 5

A charged particle produces an electric field with a magnitude of $2.0 \mathrm{~N} / \mathrm{C}$ at a point that is $50 \mathrm{~cm}$ away from the particle. What is the magnitude of the particle's charge?

Sachin Rao
Sachin Rao
Numerade Educator
01:52

Problem 6

What is the magnitude of a point charge that would create an electric field of $1.00 \mathrm{~N} / \mathrm{C}$ at points $1.00$ $\mathrm{m}$ away?

Abhishek Jana
Abhishek Jana
Numerade Educator
07:48

Problem 7

In Fig. 22-35, the four particles form a square of edge length $a=5.00 \mathrm{~cm}$ and have charges $q_{1}=+10.0 \mathrm{nC}, \quad q_{2}=-20.0 \mathrm{nC}, \quad q_{3}=$ $+20.0 \mathrm{nC}$, and $q_{4}=-10.0 \mathrm{nC}$. In unitvector notation, what net electric field do the particles produce at the square's center?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
05:53

Problem 8

In Fig. $22-36$, the four particles are fixed in place and have charges $q_{1}=q_{2}=+5 e, \quad q_{3}=+3 e, \quad$ and $\quad q_{4}=$
$-12 e$. Distance $d=5.0 \mu \mathrm{m}$. What is the magnitude of the net electric field at point $P$ due to the particles?

Abhishek Jana
Abhishek Jana
Numerade Educator
03:39

Problem 9

Figure 22-37 shows two charged particles on an $x$ axis: $-\bar{q}=$ $-3.20 \times 10^{-19} \mathrm{C}$ at $x=-3.00 \mathrm{~m}$ and
$q=3.20 \times 10^{-19} \mathrm{C}$ at $x=+3.00 \mathrm{~m}$ What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the net electric field produced at point $P$ at $y=4.00 \mathrm{~m} ?$

Dading Chen
Dading Chen
Numerade Educator
07:24

Problem 10

Figure $22-38 a$ shows two charged particles fixed in place on an $x$ axis with separation $L$. The ratio $q_{1} / q_{2}$ of their charge magnitudes is 4.00. Figure $22-38 b$ shows the $x$ component $E_{\text {neL }, x}$ of their net electric field along the $x$ axis just to the right of particle 2 . The $x$ axis scale is set by $x_{s}=30.0 \mathrm{~cm} .$ (a) At what value of $x>0$ is $E_{\mathrm{nct}, \mathrm{x}}$ maximum? (b) If particle 2 has charge $-q_{2}=-3 e$, what is the value of that maximum?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:27

Problem 11

Two charged particles are fixed to an $x$ axis: Particle 1 of charge $q_{1}=2.1 \times 10^{-8} \mathrm{C}$ is at position $x=20 \mathrm{~cm}$ and particle 2 of charge $q_{2}=-4.00 q_{1}$ is at position $x=70 \mathrm{~cm}$. At what coordinate on the axis (other than at infinity) is the net electric field produced by the two particles equal to zero?

Sachin Rao
Sachin Rao
Numerade Educator
08:46

Problem 12

Figure $22-39$ shows an uneven arrangement of electrons (e) and protons (p) on a circular arc of radius $r=2.00 \mathrm{~cm}$, with angles $\theta_{1}=30.0^{\circ}, \theta_{2}=50.0^{\circ}, \theta_{3}=30.0^{\circ}$, and $\theta_{4}=20.0^{\circ} .$ What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the net electric field produced at the center of the arc?

Abhishek Jana
Abhishek Jana
Numerade Educator
10:48

Problem 13

Figure $22-40$ shows a proton (p) on the central axis through a disk with a uniform charge density due to excess electrons. The disk is seen from an edge-on view. Three of those electrons are shown: electron $\mathrm{e}_{c}$ at the disk center and electrons $\mathrm{e}_{s}$ at opposite sides of the disk, at radius $R$ from the center. The proton is initially at distance $z=R=2.00 \mathrm{~cm}$ from the disk. At that location, what are the magnitudes of (a) the electric field $\vec{E}_{c}$ due to electron $\mathrm{e}_{c}$ and $(\mathrm{b})$ the $n e t$ electric field $\vec{E}_{s, \text { net }}$ due to electrons $\mathrm{e}_{s} ?$ The proton is then moved to $z=R / 10.0 .$ What then are the magnitudes of (c) $\vec{E}_{c}$ and (d) $\vec{E}_{s, \text { net }}$ at the proton's location? (e) From (a) and (c) we see that as the proton gets nearer to the disk, the magnitude of $\vec{E}_{c}$ increases, as expected. Why does the magnitude of $\vec{E}_{s, \text { net }}$ from the two side electrons decrease, as we see from (b) and (d)?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
07:00

Problem 14

In Fig. 22-41, particle 1 of charge $q_{1}=-5.00 q$ and particle 2 of charge $q_{2}$ $=+2.00 q$ are fixed to an $x$ axis. (a) As a multiple of distance $L$, at what coordinate on the axis is the net electric field of the particles zero? (b) Sketch the net electric field lines between and around the particles.

Abhishek Jana
Abhishek Jana
Numerade Educator
02:53

Problem 15

In Fig. $22-42$, the three particles are fixed in place and have charges $q_{1}=q_{2}=$ $+e$ and $q_{3}=+2 e$. Distance $a=6.00 \mu \mathrm{m}$. What are the (a) magnitude and (b) direction of the net electric field at point $P$ due to the particles?

Sachin Rao
Sachin Rao
Numerade Educator
08:14

Problem 16

Figure $22-43$ shows a plastic ring of radius $R=50.0 \mathrm{~cm} .$ Two small charged beads are on the ring: Bead 1 of charge $+2.00 \mu \mathrm{C}$ is fixed in place at the left side; bead 2 of charge $+6.00 \mu \mathrm{C}$ can be moved along the ring. The two beads produce a net electric field of magnitude $E$ at the center of the ring. At what (a) positive and (b) negative value of angle $\theta$ should bead 2 be positioned such that $E=$ $2.00 \times 10^{5} \mathrm{~N} / \mathrm{C} ?$

Abhishek Jana
Abhishek Jana
Numerade Educator
06:46

Problem 17

Two charged beads are on the plastic ring in Fig. $22-44 a$. Bead 2, which is not shown, is fixed in place on the ring, which has radius $R=60.0 \mathrm{~cm} .$ Bead 1, which is not fixed in place, is initially on the $x$ axis at angle $\theta=0^{\circ} .$ It is then moved to the opposite side, at angle $\theta=180^{\circ}$, through the first and second quadrants of the $x y$ coordinate system. Figure $22-44 b$ gives the $x$ component of the net electric field produced at the origin by the two beads as a function of $\theta$, and Fig. $22-44 c$ gives the $y$ component of that net electric field. The vertical axis scales are set by $E_{x x}=5.0 \times 10^{4} \mathrm{~N} / \mathrm{C}$ and $E_{y s}=$ $-9.0 \times 10^{4} \mathrm{~N} / \mathrm{C}$. (a) At what angle $\theta$ is bead 2 located? What are the charges of (b) bead 1 and (c) bead $2 ?$

Dading Chen
Dading Chen
Numerade Educator
04:23

Problem 18

The electric field of an electric dipole along the dipole axis is approximated by Eqs. $22-8$ and $22-9 .$ If a binomial expansion is made of Eq. $22-7$, what is the next term in the expression for the dipole's electric field along the dipole axis? That is, what is $E_{\text {next }}$ in the expression
$$
E=\frac{1}{2 \pi \varepsilon_{0}} \frac{q d}{z^{3}}+E_{\text {neut }} ?
$$

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

Problem 19

Figure $22-45$ shows an electric dipole. What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the dipole's electric field at point $P$, located at distance $r$ ? $d$ ?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
05:30

Problem 20

Equations $22-8$ and $22-9$ are approximations of the magnitude of the electric field of an electric dipole, at points along the dipole axis. Consider a point $P$ on that axis at distance $z=5.00 d$ from the dipole center $(d$ is the separation distance between the particles of the dipole). Let $E_{\text {appe }}$ be the magnitude of the field at point $P$ as approximated by Eqs $22-8$ and 22-9. Let $E_{\text {act }}$ be the actual magnitude. What is the ratio $E_{\text {appr }} / E_{\text {act }} ?$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
08:19

Problem 21

Electric quadrupole. Figure $22-46$ shows a generic electric quadrupole. It consists of two dipoles with dipole moments that are equal in magnitude but opposite in direction. Show that the value of $E$ on the axis of the quadrupole for a point $P$ a distance $z$ from its center (assume $z$ ? $d$ ) is given by
$$
E=\frac{3 Q}{4 \pi \varepsilon_{0} z^{4}}
$$
in which $Q\left(=2 q d^{2}\right)$ is known as the quadrupole moment of the charge distribution.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:30

Problem 22

Density, density, density. (a) A charge $-300 e$ is uniformly distributed along a circular arc of radius $4.00 \mathrm{~cm}$, which subtends an angle of $40^{\circ}$. What is the linear charge density along the arc? (b) $\mathrm{A}$ charge $-300 e$ is uniformly distributed over one face of a circular disk of radius $2.00 \mathrm{~cm}$. What is the surface charge density over that face? (c) A charge $-300 e$ is uniformly distributed over the surface of a sphere of radius $2.00 \mathrm{~cm}$. What is the surface charge density over that surface? (d) A charge $-300 e$ is uniformly spread through the volume of a sphere of radius $2.00 \mathrm{~cm} .$ What is the volume charge density in that sphere?

Abhishek Jana
Abhishek Jana
Numerade Educator
03:05

Problem 23

Figure $22-47$ shows two parallel nonconducting rings with their central axes along a common line. Ring 1 has uniform charge $q_{1}$ and radius $R$; ring 2 has uniform charge $q_{2}$ and the same radius $R$. The rings are separated by distance $d=3.00 R$. The net electric field at point $P$ on the common line, at distance $\bar{R}$ from ring 1 , is zero. What is the ratio $q_{1} / q_{2}$ ?

Dading Chen
Dading Chen
Numerade Educator
08:01

Problem 24

A thin nonconducting rod with a uniform distribution of positive charge $Q$ is bent into a complete circle of radius $R$ (Fig. $22-48$ ). The central perpendicular axis through the ring is a $z$ axis, with the origin at the center of the ring. What is the magnitude of the electric field due to the rod at (a) $z=$ 0 and (b) $z=\infty$ ? (c) In terms of $R$, at what positive value of $z$ is that magnitude maximum? (d) If $R=2.00 \mathrm{~cm}$ and $Q=4.00 \mu \mathrm{C}$, what is the maximum magnitude?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
09:58

Problem 25

Figure $22-49$ shows three circuJar arcs centered on the origin of a coordinate system. On each arc, the uniformly distributed charge is given in terms of $Q=2.00 \mu \mathrm{C}$. The radii are given in terms of $R=10.0 \mathrm{~cm}$. What are the (a) magnitude and (b) direction (relative to the positive $x$ direction) of the net electric field at the origin due to the arcs?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:44

Problem 26

In Fig. $22-50$, a thin glass rod forms a semicircle of radius $r=5.00 \mathrm{~cm} .$ Charge is uniformly distributed along the rod, with $+q=4.50 \mathrm{pC}$ in the upper half and $-q$ $=-4.50 \mathrm{pC}$ in the lower half. What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the electric field $\vec{E}$ at $P$, the center of the semicircle?

Abhishek Jana
Abhishek Jana
Numerade Educator
03:33

Problem 27

In Fig. $22-51$, two curved plastic rods, one of charge $+q$ and the other of charge $-q$, form a circle of radius $R=$ $8.50 \mathrm{~cm}$ in an $x y$ plane. The $x$ axis passes through both of the connecting points, and the charge is distributed uniformly on both rods. If $q=15.0 \mathrm{pC}$, what are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the electric field $\vec{E}$ produced at $P$, the center of the circle?

Sachin Rao
Sachin Rao
Numerade Educator
01:10

Problem 28

Charge is uniformly distributed around a ring of radius $R=2.40 \mathrm{~cm}$, and the resulting electric field magnitude $E$ is measured along the ring's central axis (perpendicular to the plane of the ring). At what distance from the ring's center is E maximum?

Abhishek Jana
Abhishek Jana
Numerade Educator
06:05

Problem 29

Figure $22-52 a$ shows a nonconducting rod with a uniformly distributed charge $+Q .$ The rod forms a half-circle with radius $R$ and produces an electric field of magnitude $E_{a r c}$ at its center of curvature $P$. If the arc is collapsed to a point at distance $R$ from $P$ (Fig. $22-52 b$ ), by what factor is the magnitude of the electric field at $P$ multiplied?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:24

Problem 30

Figure $22-53$ shows two concentric rings, of radii $R$ and $R^{\prime}=3.00 R$, that lie on the same plane. Point $P$ lies on the central $z$ axis, at distance $D=2.00 R$ from the center of the rings. The smaller ring has uniformly distributed charge $+Q .$ In terms of $Q$, what is the uniformly distributed charge on the larger ring if the net electric field at $P$ is zero?

Abhishek Jana
Abhishek Jana
Numerade Educator
16:09

Problem 31

In Fig. 22-54, a nonconducting rod of length $L=$ $8.15 \mathrm{~cm}$ has a charge $-q=-4.23 \mathrm{fC}$ uniformly distributed along its length. (a) What is the linear charge density of the rod? What are the (b) magnitude and (c) direction (relative to the positive direction of the $x$ axis) of the electric field produced at point $P$, at distance $a=12.0 \mathrm{~cm}$ from the rod? What is the electric field magnitude produced at distance $a=50 \mathrm{~m}$ by $(\mathrm{d})$ the rod and (e) a particle of charge $-q=-4.23 \mathrm{fC}$ that we use to replace the rod? (At that distance, the rod "looks" like a particle.)

Suzanne W.
Suzanne W.
Numerade Educator
08:45

Problem 32

In Fig. $22-55$, positive charge $q=7.81 \mathrm{pC}$ is spread uniformly along a thin nonconducting rod of length $L=14.5 \mathrm{~cm} .$ What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the electric field produced at point $P$, at distance $R=$ $6.00 \mathrm{~cm}$ from the rod along its perpendicular bisector?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
07:09

Problem 33

In Fig. 22-56, a "semiinfinite" nonconducting rod (that is, infinite in one direction only) has uniform linear charge density $\lambda$. Show that the electric field $\vec{E}_{p}$ at point $P$ makes an angle of $45^{\circ}$ with the rod and that this result is independent of the distance $R$. (Hint: Separately find the component of $\vec{E}_{p}$ parallel to the rod and the component perpendicular to the rod.)

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

Problem 34

A disk of radius $2.5 \mathrm{~cm}$ has a surface charge density of $5.3 \mu \mathrm{C} / \mathrm{m}^{2}$ on its upper face. What is the magnitude of the electric field produced by the disk at a point on its central axis at distance $z=$ $12 \mathrm{~cm}$ from the disk?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:32

Problem 35

At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk?

Sachin Rao
Sachin Rao
Numerade Educator
03:52

Problem 36

A circular plastic disk with radius $R=2.00 \mathrm{~cm}$ has a uniformly distributed charge $Q=+\left(2.00 \times 10^{\circ}\right) e$ on one face. A circular ring of width $30 \mu \mathrm{m}$ is centered on that face, with the center of that width at radius $r=0.50 \mathrm{~cm} .$ In coulombs, what charge is contained within the width of the ring?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:39

Problem 37

Suppose you design an apparatus in which a uniformly charged disk of radius $R$ is to produce an electric field. The field magnitude is most important along the central perpendicular axis of the disk, at a point $P$ at distance $2.00 R$ from the disk (Fig. $22-57 a$ ). Cost analysis suggests that you switch to a ring of the same outer radius $R$ but with inner radius $R / 2.00$ (Fig. $22-57 b$ ). Assume that the ring will have the same surface charge density as the original disk. If you switch to the ring, by what percentage will you decrease the electric field magnitude at $P$ ?

Sachin Rao
Sachin Rao
Numerade Educator
01:04

Problem 38

Figure $22-58 a$ shows a circular disk that is uniformly charged. The central $z$ axis is perpendicular to the disk face, with the origin at the disk. Figure $22-58 b$ gives the magnitude of the electric field along that axis in terms of the maximum magnitude $E_{m}$ at the disk surface. The $z$ axis scale is set by $z_{s}=8.0 \mathrm{~cm} .$ What is the radius of the disk?

Abhishek Jana
Abhishek Jana
Numerade Educator
02:50

Problem 39

In Millikan's experiment, an oil drop of radius $1.64 \mu \mathrm{m}$ and density $0.851 \mathrm{~g} / \mathrm{cm}^{3}$ is suspended in chamber $\mathrm{C}$ (Fig. $22-16$ ) when a downward electric field of $1.92 \times 10^{5} \mathrm{~N} / \mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
07:19

Problem 40

An electron with a speed of $5.00 \times 10^{8} \mathrm{~cm} / \mathrm{s}$ enters an electric field of magnitude $1.00 \times 10^{3} \mathrm{~N} / \mathrm{C}$, traveling along a field line in the direction that retards its motion. (a) How far will the electron travel in the field before stopping momentarily, and (b) how much time will have elapsed? (c) If the region containing the electric field is $8.00 \mathrm{~mm}$ long (too short for the electron to stop within it), what fraction of the electron's initial kinetic energy will be lost in that region?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:51

Problem 41

A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge $-2.0 \times 10^{-9} \mathrm{C}$ is acted on by a downward electrostatic force of $3.0 \times 10^{-6} \mathrm{~N}$ when placed in this field. (a) What is the magnitude of the electric field? What are the (b) magnitude and (c) direction of the electrostatic force $\vec{F}_{d}$ on the proton placed in this field? (d) What is the magnitude of the gravitational force $\vec{F}_{g}$ on the proton? (e) What is the ratio $F_{d^{\prime}} / F_{g}$ in this case?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:25

Problem 42

Humid air breaks down (its molecules become ionized) in an electric field of $3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}$. In that field, what is the magnitude of the electrostatic force on (a) an electron and (b) an ion with a single electron missing?

Abhishek Jana
Abhishek Jana
Numerade Educator
02:03

Problem 43

An electron is released from rest in a uniform electric field of magnitude $2.00 \times 10^{4} \mathrm{~N} / \mathrm{C}$ Calculate the acceleration of the electron. (Ignore gravitation.)

Sachin Rao
Sachin Rao
Numerade Educator
01:44

Problem 44

An alpha particle (the nucleus of a helium atom) has a mass of $6.64 \times 10^{-27} \mathrm{~kg}$ and a charge of $+2 e$. What are the (a) magnitude and (b) direction of the electric field that will balance the gravitational force on the particle?

Nishant Kumar
Nishant Kumar
Numerade Educator
02:02

Problem 45

An electron on the axis of an electric dipole is $25 \mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \times 10^{-29} \mathrm{C} \cdot \mathrm{m} ?$ Assume that $25 \mathrm{~nm}$ is much larger than the separation of the charged particles that form the dipole.

Sachin Rao
Sachin Rao
Numerade Educator
02:08

Problem 46

An electron is accelerated eastward at $1.80 \times 10^{9} \mathrm{~m} / \mathrm{s}^{2}$ by an electric field. Determine the field (a) magnitude and (b) direction.

Abhishek Jana
Abhishek Jana
Numerade Educator
01:41

Problem 47

Beams of high-speed protons can be produced in "guns" using electric fields to accelerate the protons. (a) What acceleration would a proton experience if the gun's electric field were $2.00 \times 10^{4} \mathrm{~N} / \mathrm{C} ?$ (b) What speed would the proton attain if the field accelerated the proton through a distance of $1.00 \mathrm{~cm} ?$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
06:50

Problem 48

In Fig. $22-59$, an electron (e) is to be released from rest on the central axis of a uniformly charged disk of radius $R$. The surface charge density on the disk is $+4.00 \mu \mathrm{C} / \mathrm{m}^{2} .$ What is the magnitude of the electron's initial acceleration if it is released at a distance (a) $R$, (b) $R / 100$, and (c) $R / 1000$ from the center of the disk? (d) Why does the acceleration magnitude increase only slightly as the release point is moved closer to the disk?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:39

Problem 49

A $10.0 \mathrm{~g}$ block with a charge of $+8.00 \times 10^{-5} \mathrm{C}$ is placed in an electric field $\vec{E}=(3000 \hat{\mathrm{i}}-600 \mathrm{j}) \mathrm{N} / \mathrm{C}$. What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the electrostatic force on the block? If the block is released from rest at the origin at time $t=0$, what are its (c) $x$ and (d) $y$ coordinates at $t=3.00 \mathrm{~s}$ ?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
04:32

Problem 50

At some instant the velocity components of an electron moving between two charged parallel plates are $v_{x}=1.5 \times 10^{5} \mathrm{~m} / \mathrm{s}$ and $v_{y}=3.0 \times 10^{3} \mathrm{~m} / \mathrm{s}$. Suppose the electric field between the plates is uniform and given by $\vec{E}=(120 \mathrm{~N} / \mathrm{C}) \hat{j} .$ In unit-vector notation, what are (a) the electron's acceleration in that field and (b) the electron's velocity when its $x$ coordinate has changed by $2.0 \mathrm{~cm}$ ?

Abhishek Jana
Abhishek Jana
Numerade Educator
06:50

Problem 51

Assume that a honeybee is a sphere of diameter $1.000$ $\mathrm{cm}$ with a charge of $+45.0 \mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \mu \mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \mathrm{pC}$ on the near side of the grain and a charge of $+1.00 \mathrm{pC}$ on the far side. (a) What is the magnitude of the net electrostatic force on the grain due to the bee? Next, assume that the bee brings the grain to a distance of $1.000 \mathrm{~mm}$ from the tip of a flower's stigma and that the tip is a particle of charge $-45.0 \mathrm{pC}$ (b) What is the magnitude of the net electrostatic force on the grain due to the stigma? (c) Does the grain remain on the bee or does it move to the stigma?

Sachin Rao
Sachin Rao
Numerade Educator
03:21

Problem 52

An electron enters a region of uniform electric field with an initial velocity of $40 \mathrm{~km} / \mathrm{s}$ in the same direction as the electric field, which has magnitude $E=50 \mathrm{~N} / \mathrm{C}$. (a) What is the speed of the electron $1.5 \mathrm{~ns}$ after entering this region? (b) How far does the electron travel during the $1.5 \mathrm{~ns}$ interval?

Abhishek Jana
Abhishek Jana
Numerade Educator
07:32

Problem 53

Two large parallel copper plates are $5.0 \mathrm{~cm}$ apart and have a uniform electric field between them as depicted in Fig. $22-60 .$ An electron is released from the negative plate at the same time that a proton is released from the positive plate. Neglect the force of the particles on each other and find their distance from the positive plate when they pass each other. (Does it surprise you that you need not know the electric field to solve this problem?)

TS
Theodore Stenmark
Numerade Educator
04:14

Problem 54

In Fig. 22-61, an electron is shot at an initial speed of $v_{0}=2.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$, at angle $\theta_{0}=$
$40.0^{\circ}$ from an $x$ axis. It moves through a uniform electric field $\vec{E}=(5.00 \mathrm{~N} / \mathrm{C}) \hat{j} .$ A screen for detecting electrons is positioned parallel to the $y$ axis, at distance $x=3.00 \mathrm{~m} .$ In unit-vector notation, what is the velocity of the electron when it hits the screen?

Abhishek Jana
Abhishek Jana
Numerade Educator
03:15

Problem 55

A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, $2.0 \mathrm{~cm}$ away, in a time $1.5 \times 10^{-8} \mathrm{~s}$ (a) What is the speed of the electron as it strikes the second plate? (b) What is the magnitude of the electric field $\vec{E}$ ?

Sachin Rao
Sachin Rao
Numerade Educator
02:50

Problem 56

An electric dipole consists of charges $+2 e$ and $-2 e$ separated by $0.78 \mathrm{~nm}$. It is in an electric field of strength $3.4 \times 10^{6} \mathrm{~N} / \mathrm{C}$. Calculate the magnitude of the torque on the dipole when the dipole moment is (a) parallel to, (b) perpendicular to, and (c) antiparallel to the electric field.

Nishant Kumar
Nishant Kumar
Numerade Educator
03:24

Problem 57

An electric dipole consisting of charges of magnitude $1.50 \mathrm{n} \mathrm{C}$ separated by $6.20 \mu \mathrm{m}$ is in an electric field of strength 1100 $\mathrm{N} / \mathrm{C} .$ What are (a) the magnitude of the electric dipole moment and (b) the difference between the potential energies for dipole orientations parallel and antiparallel to $\vec{E} ?$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:50

Problem 58

A certain electric dipole is placed in a uniform electric field $\vec{E}$ of magnitude $20 \mathrm{~N} / \mathrm{C}$. Figure $22-62$ gives the potential energy $U$ of the dipole versus the angle $\theta$ between $\vec{E}$ and the dipole moment $\vec{p}$. The vertical axis scale is set by $U_{s}=100 \times 10^{-28} \mathrm{~J} .$ What is the magnitude of $\vec{p}$ ?

Abhishek Jana
Abhishek Jana
Numerade Educator
02:55

Problem 59

How much work is required to turn an electric dipole $180^{\circ}$ in a uniform electric field of magnitude $E=46.0 \mathrm{~N} / \mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \times$ $10^{-25} \mathrm{C} \cdot \mathrm{m}$ and the initial angle is $64^{\circ} ?$

Sachin Rao
Sachin Rao
Numerade Educator
01:03

Problem 60

A certain electric dipole is placed in a uniform electric field $\vec{E}$ of magnitude $40 \mathrm{~N} / \mathrm{C}$. Figure $22-63$ gives the magnitude $\tau$ of the torque on the dipole versus the angle $\theta$ between field $\vec{E}$ and the dipole moment $\vec{p}$. The vertical axis scale is set by $\tau_{s}=100 \times 10^{-28} \mathrm{~N} \cdot \mathrm{m} .$ What is the magnitude of $\vec{p}$ ?

Abhishek Jana
Abhishek Jana
Numerade Educator
01:42

Problem 61

Find an expression for the oscillation frequency of an electric dipole of dipole moment $\vec{p}$ and rotational inertia $I$ for small amplitudes of oscillation about its equilibrium position in a uniform electric field of magnitude $E$.

Sachin Rao
Sachin Rao
Numerade Educator
02:46

Problem 62

(a) What is the magnitude of an electron's acceleration in a uniform electric field of magnitude $1.40 \times 10^{6} \mathrm{~N} / \mathrm{C} ?$ (b) How long would the electron take, starting from rest, to attain one-tenth the speed of light? (c) How far would it travel in that time?

Abhishek Jana
Abhishek Jana
Numerade Educator
02:42

Problem 63

A spherical water drop $1.20 \mu \mathrm{m}$ in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magnitude $E=462 \mathrm{~N} / \mathrm{C}$ (a) What is the magnitude of the gravitational force on the drop? (b) How many excess electrons does it have?

Sachin Rao
Sachin Rao
Numerade Educator
02:23

Problem 64

Three particles, each with positive charge $Q$, form an equilateral triangle, with each side of length $d$. What is the magnitude of the electric field produced by the particles at the midpoint of any side?

Abhishek Jana
Abhishek Jana
Numerade Educator
02:53

Problem 65

In Fig. $22-64 a$, a particle of charge $+Q$ produces an electric field of magnitude $E_{\text {part }}$ at point $P$, at distance $R$ from the particle. In Fig. $22-64 b$, that same amount of charge is spread uniformly along a circular arc that has radius $R$ and subtends an angle $\underline{\theta .}$ The charge on the arc produces an electric field of magnitude $E_{\text {arc }}$ at its center of curvature $P$. For what value of $\theta$ does $E_{\text {arc }}=$ $0.500 E_{\text {part }} ?$ (Hint: You will probably resort to a graphical solution.)

Sachin Rao
Sachin Rao
Numerade Educator
04:55

Problem 66

A proton and an electron form two corners of an equilateral triangle of side length $2.0 \times 10^{-6} \mathrm{~m}$. What is the magnitude of the net electric field these two particles produce at the third corner?

Abhishek Jana
Abhishek Jana
Numerade Educator
03:51

Problem 67

A charge (uniform linear density $=9.0 \mathrm{nC} / \mathrm{m}$ ) lies on a string that is stretched along an $x$ axis from $x=0$ to $x=3.0 \mathrm{~m}$. Determine the magnitude of the electric field at $x=4.0 \mathrm{~m}$ on the $x$ axis.

Sachin Rao
Sachin Rao
Numerade Educator
04:25

Problem 68

In Fig. 22-65, eight particles form a square in which distance $d=2.0 \mathrm{~cm} .$ The charges are $q_{1}=+3 e$, $q_{2}=+e, q_{3}=-5 e, q_{4}=-2 e, q_{5}=+3 e$ $q_{6}=+e, q_{7}=-5 e$, and $q_{8}=+e$. In unitvector notation, what is the net electric field at the square's center?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:44

Problem 69

Two particles, each with a charge of magnitude $12 \mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the third vertex if (a) both charges are positive and (b) one charge is positive and the other is negative?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
04:44

Problem 70

The following table gives the charge seen by Millikan at different times on a single drop in his experiment. From the data, calculate the elementary charge $e$.
$$
\begin{array}{lll}
\hline 6.563 \times 10^{-19} \mathrm{C} & 13.13 \times 10^{-19} \mathrm{C} & 19.71 \times 10^{-19} \mathrm{C} \\
8.204 \times 10^{-19} \mathrm{C} & 16.48 \times 10^{-19} \mathrm{C} & 22.89 \times 10^{-19} \mathrm{C} \\
11.50 \times 10^{-19} \mathrm{C} & 18.08 \times 10^{-19} \mathrm{C} & 26.13 \times 10^{-19} \mathrm{C} \\
\hline
\end{array}
$$

Abhishek Jana
Abhishek Jana
Numerade Educator
03:37

Problem 71

A charge of $20 \mathrm{nC}$ is uniformly distributed along a straight rod of length $4.0 \mathrm{~m}$ that is bent into a circular arc with a radius of $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the center of curvature of the arc?

Sachin Rao
Sachin Rao
Numerade Educator
03:00

Problem 72

An electron is constrained to the central axis of the ring of charge of radius $R$ in Fig. $22-11$, with $z<R .$ Show that the electrostatic force on the electron can cause it to oscillate through the ring center with an angular frequency
$$
\omega=\sqrt{\frac{e q}{4 \pi \varepsilon_{0} m R^{3}}}
$$
where $q$ is the ring's charge and $m$ is the electron's mass.

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
06:21

Problem 73

The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \mathrm{i}+3.0 \mathrm{j}) \mathrm{N} / \mathrm{C}$ at the point $(3.0,3.0) \mathrm{cm}$ and $100 \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}$ at the point $(2.0,0) \mathrm{cm} .$ What are the (a) $x$ and (b) $y$ coordinates of the particle? (c) What is the charge of the particle?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
04:00

Problem 74

(a) What total (excess) charge $q$ must the disk in Fig. $22-15$ have for the electric field on the surface of the disk at its center to have magnitude $3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}$, the $E$ value at which air breaks down electrically, producing sparks? Take the disk radius as $2.5 \mathrm{~cm}$.
(b) Suppose each surface atom has an effective cross-sectional area of $0.015 \mathrm{~nm}^{2}$. How many atoms are needed to make up the disk surface? (c) The charge calculated in (a) results from some of the surface atoms having one excess electron. What fraction of these atoms must be so charged?

Abhishek Jana
Abhishek Jana
Numerade Educator
01:59

Problem 75

In Fig. 22-66, particle 1 (of charge $+1.00 \mu \mathrm{C}$ ), particle 2 (of charge $+1.00 \mu \mathrm{C}$ ), and particle 3 (of charge $Q$ ) form an equilateral triangle of edge length $a .$ For what value of $Q$ (both sign and magnitude) does the net electric field produced by the particles at the center of the triangle vanish?

Sachin Rao
Sachin Rao
Numerade Educator
01:08

Problem 76

In Fig. $22-67$, an electric dipole swings from an initial orientation $i\left(\theta_{i}=20.0^{\circ}\right)$ to a final orientation $f\left(\theta_{f}=20.0^{\circ}\right)$ in a uniform external electric field $\vec{E} .$ The electric dipole moment is $1.60 \times 10^{-27} \mathrm{C} \cdot \mathrm{m} ;$ the field magnitude is $3.00 \times 10^{6} \mathrm{~N} / \mathrm{C}$. What is the change in the dipole's potential energy?

Abhishek Jana
Abhishek Jana
Numerade Educator
06:09

Problem 77

A particle of charge $-q_{1}$ is at the origin of an $x$ axis (a) At what location on the axis should a particle of charge $-4 q_{1}$ be placed so that the net electric field is zero at $x=2.0 \mathrm{~mm}$ on the axis? (b) If, instead, a particle of charge $+4 q_{1}$ is placed at that location, what is the direction (relative to the positive direction of the $x$ axis) of the net electric field at $x=2.0 \mathrm{~mm} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
06:37

Problem 78

Two particles, each of positive charge $q$, are fixed in place on a $y$ axis, one at $y=d$ and the other at $y=-d$. (a) Write an expression that gives the magnitude $E$ of the net electric field at points on the $x$ axis given by $x=\alpha d$. (b) Graph $E$ versus $\alpha$ for the range $0<$ $\alpha<4$. From the graph, determine the values of $\alpha$ that give (c) the maximum value of $E$ and $(\mathrm{d})$ half the maximum value of $E .$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
05:35

Problem 79

A clock face has negative point charges $-q,-2 q,-3 q, \ldots$, $-12 q$ fixed at the positions of the corresponding numerals. The clock hands do not perturb the net field due to the point charges. At what time does the hour hand point in the same direction as the electric field vector at the center of the dial? (Hint: Use symmetry.)

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
00:56

Problem 80

Calculate the electric dipole moment of an electron and a proton $4.30 \mathrm{~nm}$ apart.

Abhishek Jana
Abhishek Jana
Numerade Educator
01:50

Problem 81

An electric field $\vec{E}$ with an average magnitude of about $150 \mathrm{~N} / \mathrm{C}$ points downward in the atmosphere near Earth's surface. We wish to "float" a sulfur sphere weighing $4.4 \mathrm{~N}$ in this field by charging the sphere. (a) What charge (both sign and magnitude) must be used? (b) Why is the experiment impractical?

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
03:17

Problem 82

A circular rod has a radius of curvature $R=9.00 \mathrm{~cm}$ and $a$ uniformly distributed positive charge $Q=6.25 \mathrm{pC}$ and subtends an angle $\theta=2.40 \mathrm{rad}$. What is the magnitude of the electric field that $Q$ produces at the center of curvature?

Abhishek Jana
Abhishek Jana
Numerade Educator
05:13

Problem 83

An electric dipole with dipole moment
$$
\vec{p}=(3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}})\left(1.24 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}\right)
$$
is in an electric field $\vec{E}=(4000 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{i}}$. (a) What is the potential energy of the electric dipole? (b) What is the torque acting on it?
(c) If an external agent turns the dipole until its electric dipole moment is
$$
\vec{p}=(-4.00 \hat{\mathrm{i}}+3.00 \hat{\mathrm{j}})\left(1.24 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}\right)
$$
how much work is done by the agent?

Sachin Rao
Sachin Rao
Numerade Educator
07:40

Problem 84

In Fig. $22-68$, a uniform, upward electric field $\vec{E}$ of magnitude $2.00 \times$ $10^{3} \mathrm{~N} / \mathrm{C}$ has been set up between two horizontal plates by charging the lower plate positively and the upper plate negatively. The plates have length $L=10.0 \mathrm{~cm}$ and separation $d$ $=2.00 \mathrm{~cm}$. An electron is then shot between the plates from the left edge of the lower plate. The initial velocity $\vec{v}_{0}$ of the electron makes an angle $\theta=45.0^{\circ}$ with the lower plate and has a magnitude of $6.00 \times 10^{6} \mathrm{~m} / \mathrm{s} .$ (a) Will the electron strike one of the plates? (b) If so, which plate and how far horizontally from the left edge will the electron strike?

Abhishek Jana
Abhishek Jana
Numerade Educator
04:05

Problem 85

For the data of Problem 70 , assume that the charge $q$ on the drop is given by $q=n e$, where $n$ is an integer and $e$ is the elementary charge. (a) Find $n$ for each given value of $q .$ (b) Do a linear regression fit of the values of $q$ versus the values of $n$ and then use that fit to find $e$

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
02:13

Problem 86

In Fig. 22-66, particle 1 (of charge $+2.00 \mathrm{pC}$ ), particle 2 (of charge $-2.00 \mathrm{pC}$ ), and particle 3 (of charge $+5.00 \mathrm{pC}$ ) form an equilateral triangle of edge length $a=9.50 \mathrm{~cm} .$ (a) Relative to the positive direction of the $x$ axis, determine the direction of the force $\vec{F}_{3}$ on particle 3 due to the other particles by sketching electric field lines of the other particles. (b) Calculate the magnitude of $\vec{F}_{3}$.

Abhishek Jana
Abhishek Jana
Numerade Educator
07:19

Problem 87

In Fig. $22-69$, particle 1 of charge $q_{1}=1.00 \mathrm{pC}$ and particle 2 of charge $q_{2}=-2.00 \mathrm{pC}$ are fixed at a distance $d=5.00 \mathrm{~cm}$ apart. In unit-vector notation, what is the net electric field at points (a) $A$,
(b) $B$, and (c) $\underline{C \text { ? }}$
(d) Sketch the electric field lines.

Sachin Rao
Sachin Rao
Numerade Educator