Solved: Figure 22-53a shows a circular disk that is

Figure 22-20 shows three arrangements of electric field lines. In each arrangement, a proton is released from rest at point A and is then accelerated through point B by the electric field. Points A and B have equal separations in the three arrangements. Rank the arrangements according to the linear momentum of the proton at point B, greatest first.

Figure 22-21 shows two square arrays of charged particles. The squares, which are centered on point P, are misaligned. The particles are separated by either d or d/2 along the perimeters of the squares. What are the magnitude and direction of the net electric field at P?

In Fig. 22-22, two particles of charge -q are arranged symmetrically about the y axis; each produces an electric field at point P on that axis. (a) Are the magnitudes of the fields at P equal? (b) Is each electric field directed toward or away from the charge proQUESTIONS 597 Force F has the same direction as If if q is positive and the opposite direction if q is negative. Dipole in an Electric Field When an electric dipole of dipole moment p is placed in an electric field If, the field exerts a torque T on the dipole: (22-34) The dipole has a potential energy U associated with its orientation in the field: U= -p'' (22-38) Th~potential energy is defined to be zero when pis J?erpendicular to E; it is least (U = -pE) when p is aligned with E and greatest (U = pE) when p is directed opposite If. )' p -q -q ducing it? (c) Is the magnitude of the net electric field at P equal to the sum of the magnitUdes E of the two field vectors (is it equal to 2E)? (d) Do the x components of those two field vectors add or cancel? (e) Do their y components add or cancel? (f) Is the direction of the net field at P that of the canL~------~----~~--X I--d d~ Fig. 22-22 Question 3. celing components or the adding components? (g) What is the direction of the net field?

Figure 22-23 shows four situations in which four charged patticles are evenly spaced to the left and right of a central point. The charge values are indicated. Rank the situations according to the magnitude of the net electric field at the central point, greatest first.

Figure 22-24 shows two charged particles fixed in place on an axis. (a) Where on the axis (other " @ than at an infinite distance) is there +q -3q a point at which their net electric Fig. 22-24 Question 5. field is zero: between the charges, to their left, or to their right? (b) Is there a point of zero net electric field anywhere off the axis (other than at an infinite distance)?

In Fig. 22-25, two identical circular nonconducting rings are centered on the same line. For three situations, the uniform charges on rings A and B are, respectively, (1) qo and qo, (2) -qo and -qo, and (3) -qo and qo. Rank the situations according to the magnitude of the net electric field at (a) point PI midway between the rings, (b) point P2 at the center of ring B, and (c) point P3 to the right of ring B, greatest first.

The potential energies associated with four orientations of an electric dipole in an electric field are (1) -SUo, (2) -7Uo, (3) 3Uo, and (4) SUa, where Uois positive. Rank the orientations according to (a) the angle between the electric dipole moment p and the electric field E and (b) the magnitude of the torque on the electric dipole, greatest first.

(a) In the Checkpoint of Section 22-9, if the dipole rotates from orientation 1 to orientation 2, is the work done on the dipole by the field positive, negative, or zero? (b) If, instead, the dipole rotates from orientation 1 to orientation 4, is the work done by the field more than, less than, or the same as in (a)?

Figure 22-26 shows two disks and a flat ring, each with the same uniform charge Q. Rank the objects according to the magnitude of the electric field they create at points P (which are at the same vertical heights), greatest first.

In Fig. 22-27, an electron e travels through a small hole in plate A and then toward plate B. A uniform electric field in the region between the plates then slows the electron without deflecting it. (a) What is the direction of the field? (b) Four other particles simi- e --+t---... larly travel through small holes in ...... -H-- +q2 either plate A or plate B and then +ql--++-- into the region between the plates. ......-H---q3 Three have charges +qb +q2, and -q3' The fourth (labeled n) is a n_-++-__ A B neutron, which is electrically neu- Fig. 22-27 Question 10. tral. Does the speed of each of those four other particles increase, decrease, or remain the same in the region between the plates?

In Fig. 22-28a, a circular plastic rod with uniform charge + Q produces an electric field of magnitude E at the center of curvature (at the origin). In Figs. 22-28b, c, and d, more circular rods, each with identical uniform charges +Q, are added until the circle is complete. A fifth arrangement (which would be labeled e) is like that in d except the rod in the fourth quadrant has charge -Q. Rank the five arrangements according to the magnitude of the electric field at the center of curvature, greatest first.

Figure 22-34 shows an uneven arrangement of electrons (e) and protons (p) on a circular arc of radius r = 2.00 cm, with angles 8j = 30.0, 82 = 50.0, 83 = 30.0, and 84 = 20.0. 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?

Figure 22-35 shows a proton (p) on the central axis through a disk with a uniform charge density due to excess electrons. Three of those electrons are shown: electron ec at the disk center and electrons es at opposite sides of the disk, at ra)' Fig. 22-34 Problem 12. z " ,. r " , r---R -40. I .-R----1 dius R from the center. The proton Fig. 22-35 Problem 13. is initially at distance z = R = 2.00 cm from the disk. At that location, what are the magnitudes of (a) the electric field Ec due to electron ec and (b) the net electric field Esnet due to electrons es? The proton is then moved to z = RIlO.O. What then are the magnitudes of (c) ~ and (d) Es 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 Ec increases. Why does the magnitude of IL,net decrease, as we see from (b) and (d)?

In Fig. 22-36, particle 1 of charge qj = -5.00q and particle 2 of charge qz = +2.00q 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 Fig. 22-36 particles zero? (b) Sketch the net electric field lines

In Fig. 22-37, the three particles are fixed in place and have charges ql = q2 = +e and q3 = +2e. Distance a = 6.00 /Lm, What are the (a) magnitude and (b) direction of the net electric field at point P due to the particles?

Figure 22-38 shows a plastic rllig of radius R =50.0 cm. Two small charged beads are on the ring: Bead 1 of charge +2.00/LC is fixed in place at the left side; bead 2 of charge +6.00/LC can be moved Fig. 22-37 Problem 15. 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 8 should bead 2 be positioned such that E = 2.00 X 105 N/C?

1\\'0 charged beads are on the plastic ring in Fig. 22-39a. Bead 2, which is not shown, is fixed in place on the ring, which has radius R = 60.0 cm. Bead 1 is initially on the x axis at angle 8 = 00 It is then moved to the opposite side, at angle 8 = 1800 , through the first and second quadrants of the xy coordinate system. Figure 22- 39b gives the x component of the net electric field produced at the origin by the two beads as a function of 8, and Fig. 22-39c gives the y component. The vertical axis scales are set by En = 5.0 X 104 N/C and Eys = -9.0 X 104 N/C. (a) At what angle 8 is bead 2 located? What are the charges of (b) bead 1 and (c) bead 2?

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 Enext in the expression __ 1_~ ? E - 2 3 + Enext

Figure 22-40 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?

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 = 5.00d from the dipole center (d is the separation distance between the particles of the dipole). Let Eappr be the magnitude of the field at point P as approximated by Eqs. 22-8 and 22-9. Let E act be the actual magnitude. What is the ratio Eappr/Eact?

lectric quadrupole. Figure 22-41 shows an 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 in which Q (= 2qd Z) is known as the quadrupole moment of the charge distribution.

Density, density, density. (a) A charge -300e is uniformly distributed along a circular arc of radius 4.00 cm, which subtends an angle of 400 What is the linear charge density along the arc? (b) A charge - 300e is uniformly distributed over one face of a circular disk of radius 2.00 cm. What is the surface charge density over that face? (c) A charge - 300e is uniformly distributed over the surface of a sphere of radius 2.00 cm. What is the surface charge density over that surface? (d) A charge -300e is uniformly spread through the volume of a sphere of radius 2.00 cm. What is the volume charge density in that sphere?

Figure 22-42 shows two parallel nonconducting rings with their central axes along a common line. Ring 1 has uniform charge qj and radius R; ring 2 has uniform charge qz and the same radius R. The rings are separated by distance d = 3.00R. The net electric field at point P on the common line, at distance R from ring 1, is zero. What is the ratio q/qz?

A thin nonconducting rod with a uniform distribution of positive charge Q is bent into a circle of radius R (Fig. 22-43). The central perpendicular axis through the ring is a z axis, with the origin at the center of the ring. What is the magniFig. 22-42 Problem 23. z tude of the electric field due to the Fig. 22-43 Problem 24. rod at (a) z = 0 and (b) z = oo? (c) In terms of R, at what positive value of z is that magnitude maximum? (d) If R = 2.00 cm and Q = 4.00 j.LC, what is the maximum magnitude?

Figure 22-44 shows three circular arcs centered on the origin of a coordinate system. On each arc, the uniformly distributed charge is given in terms of Q = 2.00 j.Lc. The radii are given in terms of R = 10.0 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?

In Fig. 22-45, a thin glass rod forms a semicircle of radius r = 5.00 cm. Charge is uniformly distributed along the rod, with +q = 4.50 y 3R 2R R pC in the upper half and -q = -4.50 Fig. 22-44 Problem 25. 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 E at P, the center of the semicircle?

In Fig. 22-46, two curved plastic rods, one of charge +q and the other of charge -q, form a circle of radius R = 8.50 cm in an xy plane. The x axis passes through both of the connecting points, and the charge is distributed uniformly on both rods. If q = 15.0 pC, what are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the electric field E produced at P, the center of the circle?

Charge is uniformly distributed around a ring of radius R = 2.40 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 Emaximum?

Figure 22-47 a shows a nonconducting rod with a uniformly distributed charge +Q. The rod forms a half-circle with radius Rand produces an electric field of magnitude Eare at its center of curvature P. If the arc is collapsed to a point at distance R from P (Fig. 22-47b), by what factor is the magnitude of the electric field at P multiplied?

Figure 22-48 shows two concentric rings, of radii Rand R' = 3.00R, that lie on the same plane. Point P lies on the central z axis, at distance D = 2.00R 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 Pis zero?

In Fig. 22-49, a nonconducting rod of length L = 8.15 cm has a charge -q = -4.23 fC uniformly distributed along its length. (a) What is the linear charge density of the PROBLEMS 601 Fig. 22-49 Problem 31. 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 cm from the rod? What is the electric field magnitude produced at distance a = 50 m by (d) the rod and (e) a particle of charge -q = -4.23 fC that replaces the rod?

In Fig. 22-50, positive charge q = 7.81 pC is spread uniformly along a thin nonconducting rod of length L = 14.5 cm. What are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the electric R field produced at point P, at dis- G+DD+:::;:.3+~DDDD+J-x tance R = 6.00 cm from the rod ~I'---L----<' I along its perpendicular bisector?

In Fig. 22-51, a "semi-infinite" nonconducting rod (that is, infinite in one direction only) has uniform linear charge densi.!}' A. Show that the electric field Ep at point P makes an angle of 45 with the rod and that this result is independent of the distance R. (Hint: Separately find the component of Ep parallel to the rod and the component perpendicular to the rod.)

A disk of radius 2.5 cm has a surface charge density of 5.3 /-LC/m2 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 cm from the disk?

At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius 0.600 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?

A circular plastic disk with radius R = 2.00 cm has a uniformly distributed charge Q = +(2.00 X 106)e on one face. A circular ring of width 30 /-Lm is centered on that face, with the center of that width at radius r = 0.50 cm. In coulombs, what charge is contained within the width of the ring?

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 2.00R from the disk (Fig. 22-52a). Cost analysis suggests that you switch to a ring of the same outer radius R but with inner radius R/2.00 (Fig. 22-52b). Assume z p (a) z p (b) 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?

Figure 22-53a 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 22-53b gives the magnitude of the electric field along that axis in terms of the maximum magnitUde Em at the disk surface. The z axis scale is set by Zs = 8.0 cm.What is the radius of the disk?

In Millikan's experiment, an oil drop of radius 1.64 [Lm and density 0.851 g/cm3 is suspended in chamber C (Fig. 22-14) when a downward electric field of 1.92 X 105 N/C is applied. Find the charge on the drop, in terms of e.

An electron with a speed of 5.00 X 108 cm/s enters an electric field of magnitude 1.00 X 103 N/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 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?

A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge -2.0 X 10-9 C is acted on by a downward electrostatic force of 3.0 X 10-6 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 ~l on the proton placed in this field? (d) What is the magnitude of the gravitational force F; on the proton? (e) What is the ratio Fell Fg in this case?

Humid air breaks down (its molecules become ionized) in an electric field of 3.0 X 106 N/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?

An electron is released from rest in a uniform electric field of magnitude 2.00 X 104 N/C. Calculate the acceleration of the electron. (Ignore gravitation.)

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

An electron on the axis of an electric dipole is 25 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 X 10-29 C m? Assume that 25 nm is much larger than the dipole charge separation.

An electron is accelerated eastward at 1.80 X 109 m/s2 by an electric field. Determine the field (a) magnitUde and (b) direction.

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 X 104 N/C? (b) What speed would the proton attain if the field accelerated the proton through a distance of 1.00 cm?

In Fig. 22-54, 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 [LClm2. What is the magnitude of the electron's initial acceleration if it is released at a distance (a) R, (b) R1100, and (c) RI1000 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?

A 10.0 g block with a charge of +8.00 X 10-5 C is placed in an electric field E = (3000i - 600]) N/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 s?

At some instant the velocity components of an electron moving between two charged parallel plates are v, = 1.5 X 105 mls and Vy = 3.0 X 103 m/s. Suppose the electric field between the plates is given by E = (120 N/C)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 cm?

Assume that a honeybee is a sphere of diameter 1.000 cm with a charge of +45.0 pC uniformly spread over its surface. Assume also that a spherical pollen grain of diameter 40.0 [Lm is electrically held on the surface of the sphere because the bee's charge induces a charge of -1.00 pC on the near side of the sphere and a charge of + 1.00 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 mm from the tip of a flower's stigma and that the tip is a particle of charge -45.0 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?

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

Two large parallel copper plates are 5.0 cm apart and have a uniform electric field between them as depicted in Fig. 22-55. 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 Positive plate ~ E Negative plate Fig. 22-55 Problem 53. pass each other. (Does it surprise you that you need not know the electric field to solve this problem?)

In Fig. 22-56, an electron is shot at an initial speed of 1'0 = 2.00 X 106 mis, at angle 80 = 40.0 from an x axis. It moves through a uniform electric field )' Detecting screen E = (5.00 N/C)]. A screen for de- Fig.22-56 Problem 54. tecting electrons is positioned parallel to the y axis, at distance x = 3.00 m. In unit-vector notation, what is the velocity of the electron when it hits the screen?

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 cm away, in a time 1.5 X 10-8 s. (a) What is the speed of the electron as it strikes the second plate? (b) What is the magnitude of the electric field E?

An electric dipole consists of charges +2e and -2e separated by 0.78 nm. It is in an electric field of strength 3.4 X 106 N/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.

An electric dipole consisting of charges of magnitude 1.50 nC separated by 6.20 ,urn is in an electric field of strength 1100 N/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 If?

A certain electric dipole is placed in a uniform electric field E of magnitude 20 N/C. Figure 22-57 gives the potential energy U of the dipole versus the angle 8 between E and the dipole moment p. The vertical axis scale is set by Us = 100 X 10-28 1. What is the magnitude of p?

How much work is required to turn an electric dipole 180 in a uniform electric field of magnitude E = 46.0 N/C if p = 3.02 X 10-25 C . m and the initial angle is 64?

A certain electric dipole is placed in a uniform electric field E of magnitude 40 N/C. Figure 22-58 gives ilCN l- 0 I I e Fig. 22-58 Problem 60. the magnitude 7 of the torque on the dipole versus the angle 8 between field E and the dipole moment p. The vertical axis scale is set by 7s = 100 X 10-28 N m. What is the magnitude of p?

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

(a) What is the magnitude of an electron's acceleration in a unifmm electJic field of magnitude 1.40 X 106 N/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?

A spherical water drop 1.20 ,urn in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magniPROBLEMS 603 tude E = 462 N/C. (a) What is the magnitude of the gravitational force on the drop? (b) How many excess electrons does it have?

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?

In Fig. 22-59a, a particle of charge +Q produces an electric field of magnitude Epart at point P, at distance R from the particle. In Fig. 22-59b, that same amount of charge is spread uniformly along a circular arc that has radius Rand sub tends an angle 8. The charge on the arc produces an electric field of magnitude Earc at its center of curvature P. For what value of 8 does Earc = 0.500Epart? (Hint: You will probably resort to a graphical solution.)

A proton and an electron form two corners of an equilateral triangle of side length 2.0 X 10-6 m. What is the magnitude of the net electric field these two particles produce at the third corner?

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

In Fig. 22-60, eight particles form q1 q2 a square in which distance d = 2.0 cm. d d The charges are ql = +3e, q2 = +e, d q3 = -5e, q4 = -2e, q5 = +3e, q6 = +e, q7 = -5e, and q8 = +e. In unit-vector notation, what is the net electric field at the square's center?

Two particles, each with a charge of magnitude 12 nC, are at two of the vertices of an equilateral triangle with edge length 2.0 ill. What is the magnitude of the electric field at the third vertex if (a) both charges are positive q8 d q7 )' Lx d d Fig. 22-60 Problem 68. and (b) one charge is positive and the other is negative?

In one of his experiments, Millikan observed that the following measured charges, among others, appeared at different times on a single drop: 6.563 X 10-19 C 8.204 X 10- 19 C 11.50 X 10-19 C 13.13 X 10-19 C 16.48 X 10-19 C 18.08 X 10- 19 C 19.71 X 10-19 C 22.89 X 10- 19 C 26.13 X 10- 19 C What value for the elementary charge e can be deduced from these data?

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

An electron is constrained to the central axis of the ring of charge ofradius R in Fig. 22-10, with z q R. Show that the electrostatic force on the electron can cause it to oscillate through the ring center with an angular frequency w = ) 47rs::R3 ' where q is the ring's charge and m is the electron's mass.

The electric field in an xy plane produced by a positively charged particle is 7.2( 4.oi + 3.0)) N/C at the point (3.0,3.0) cm and 100i N/C at the point (2.0,0) cm. What are the (a) x and (b) y coordinates of the particle? (c) What is the charge of the particle?

(a) What total (excess) charge q must the disk in Fig. 22-13 have for the electric field on the surface of the disk at its center to have magnitude 3.0 X 106 N/C, the E value at which air breaks down electrically, producing sparks? Take the disk radius as 2.5 cm, and use the listing for air in Table 22-1. (b) Suppose each surface atom has an effective cross-sectional area of 0.015 nm2 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?

In Fig. 22-61, particle 1 (of charge + 1.00 ftC), particle 2 (of charge + 1.00 ftC), 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 Fig. 22-61 Problems produced by the particles at the center 75 and 86. of the triangle vanish?

In Fig. 22-62, an electric dipole swings from an initial orientation i (Oi = 20.00 ) to a fi- ~ nal orientation f (Of = 20.0 0 ) in a uniform ex- Pt ternal electric field E. The electric dipole moment is 1.60 X 10-27 C . m; the field magnitude is 3.00 X 106 N/C. What is the change in the dipole's potential energy?

A particle of charge -qj is at the origin of an x axis. (a) At what location on the axis should a particle of charge -4qj be placed so that the net electric field is zero at x = 2.0 mm Fig. 22-62 Problem 76. on the axis? (b) If, instead, a particle of charge +4qj 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 mm?

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 = ad. (b) Graph E versus a for the range 0 < a < 4. From the graph, determine the values of a that give (c) the maximum value of E and (d) half the maximum value of E.

A clock face has negative point charges -q, -2q, -3q, ... , -12q 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.)

Calculate the electric dipole moment of an electron and a proton 4.30 nm apart.

An electric field E with an average magnitude of about 150 N/C points downward in the atmosphere near Earth's surface. We wish to "float" a sulfur sphere weighing 4,4 N in this field by charging the sphere. (a) What charge (both sign and magnitude) must be used? (b) Why is the experiment impractical

A circular rod has a radius of curvature R = 9.00 cm and a uniformly distributed positive charge Q = 6.25 pC and sub tends an angle 0 = 2,40 rad. What is the magnitude of the electric field that Q produces at the center of curvature?

An electric dipole with dipole moment p = (3.00i + 4.00))(1.24 X 10-30 C' m) is in an electric field E = (4000 N/C)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 p = (-4.00i + 3.00))(1.24 X 10-30 C m), how much work is done by the agent?

In Fig. 22-63, a uniform, upward electric field E of magnitude 2.00 X 103 N/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 cm and separation d = 2.00 cm. An electron is then shot between the plates from the left edge of the lower plate. The initial velocity Vo of the electron makes an angle 0 = 45.0 0 with the lower plate and has a magnitude of 6.00 X 106 m/s. (a) Will the electron strike one of the plates? (b) If so, which plate ~ d and how far horizontally from the 5<: E I 1'0 e left edge will the electron strike?

For the data of Problem 70, as- k
.---- L ---~.I sume that the charge q on the drop Fig. 22-63 Problem 84. is given by q = ne, 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.

In Fig. 22-61, particle 1 (of charge +2.00 pC), particle 2 (of charge -2.00 pC), and particle 3 (of charge +5.00 pC) form an equilateral triangle of edge length a = 9.50 cm. (a) Relative to the positive direction of the x axis, determine the direction of the force l{ on particle 3 due to the other particles by sketching electric field lines of the other particles. (b) Calculate the magnitude of l{.

In Fig. 22-64, particle 1 of charge ql = 1.00 pC and particle 2 of charge q2 = -2.00 pC are fixed at a distance d = 5.00 cm apart. In unit-vector notation, what is the net electric field at points (a) A, (b) B, and (c) C? (d) Sketch the electric field lines.

In Fig. 22-8, let both charges be positive. Assuming z ~ d, show that E at point P in that figure is then given by E=_l_ !!L 47rso Z2