Two parallel circular rings of radius R have their

What is the magnitude of the electric force of attraction between an iron nucleus (q = +26e) and its innermost electron if the distance between them is 1.5 X 10-12 m?

How many electrons make up a charge of 38.0 fiCl

What is the magnitude of the force a +25 /jlC charge exerts on a +2.5 mC charge 28 cm away?

What is the repulsive electrical force between two protons 4.0 X 1 0 15 m apart from each other in an atomic nucleus?

When an object such as a plastic comb is charged by rubbing it with a cloth, the net charge is typically a few microcoulombs. If that charge is 3.0 /jlC, by what percentage does the mass of a 35-g comb change during charging?

Two charged dust particles exert a force of 3.2 X 10 2 N on each other. What will be the force if they are moved so they are only one-eighth as far apart?

Two charged spheres are 8.45 cm apart. They are moved, and the force on each of them is found to have been tripled. How far apart are they now?

A person scuffing her feet on a wool rug on a dry day accumulates a net charge of 46 /jlC. H o w many excess electrons does she get, and by how much does her mass increase?

What is the total charge of all the electrons in a 15-kg bar of gold? What is the net charge of the bar? (Gold has 79 electrons per atom and an atomic mass of 197 u.)

Compare the electric force holding the electron in orbit (r = 0.53 X 10-10m) around the proton nucleus of the hydrogen atom, with the gravitational force between the same e le c tro n and n ro to n . W h a t is th e ra tio o f these

Two positive point charges are a fixed distance apart. The sum of their charges is QT. What charge must each have in order to (a) maximize the electric force between them, and (b) minimize it?

Particles of charge +75, +48, and 85 jllC are placed in a line (Fig. 21-52). The center one is 0.35 m from each of the others. Calculate the net force on each charge due to the other two.FIGURE 21-52 Problem 12.

Three charged particles are placed at the corners of an equilateral triangle of side 1.20 m (Fig. 21-53). The charges are +7.0 (jlC, S.OfxC, and -6.0 pC. Calculate the magnitude and direction of the net force on each due to the other two. FIGURE 21-53 Problem 13.

Two small nonconducting spheres have a total charge of 90.0 /JLC. (a) When placed 1.16 m apart, the force each exerts on the other is 12.0 N and is repulsive. What is the charge on each? (b) What if the force were attractive?

A charge of 4.15 mC is placed at each corner of a square 0.100 m on a side. Determine the magnitude and direction

Two negative and two positive point charges (magnitude Q = 4.15 mC) are placed on opposite corners of a square as shown in Fig. 21-54. Determine the magnitude and direction of the force on each charge. FIGURE 21-54 Problem 16.

A charge Q is transferred from an initially uncharged plastic ball to an identical ball 12 cm away. The force of attraction is then 17 mN. How many electrons were transferred from one ball to the other?

Two charges, Q0 and ~4Q0, are a distance apart. These two charges are free to move but do not because there is a third charge nearby. What must be the magnitude of the third charge and its placement in order for the first two to be in equilibrium?

Two positive charges +Q are affixed rigidly to the x axis, one at x = +d and the other at x = d. A third charge +q of mass ra, which is constrained to move only along the x axis, is displaced from the origin by a small distance s d and then released from rest, (a) Show that (to a good approximation) +q will execute simple harmonic motion and determine an expression for its oscillation period T. (b) If these three charges are each singly ionized sodium atoms (q = Q = +e) at the equilibrium spacing d = 3 X 10-10m typical of the atomic spacing in a solid, find T in picoseconds.

Two small charged spheres hang from cords of equal length as shown in Fig. 21-55 and make small angles $i and 02 with the vertical, (a) If Qi = Q, Qi = 20, and m1 = m2 = ra, determine the ratio 61/ 62- (b) If <2i = Qi Qi = 2<2, mi = ra, and ra2 = 2ra, determine the ratio 0i/02. (c) Estimate the distance between the spheres for each case. FIGURE 21-55 Problem 20.

What are the magnitude and direction of the electric force on an electron in a uniform electric field of strength 1920 N/C that points due east?

A proton is released in a uniform electric field, and it experiences an electric force of 2.18 X 10-14 N toward the south. What are the magnitude and direction of the electric field?

Determine the magnitude and direction of the electric field 16.4 cm directly above an isolated 33.0 X 10-6 C charge.

A downward electric force of 8.4 N is exerted on a -8.8 jjlC charge. What are the magnitude and direction of the electric field at the position of this charge?

The electric force on a +4.20-/aC charge is F = (7.22 X 10-4N)j. What is the electric field at the position of the charge?

What is the electric field at a point when the force

Determine the magnitude of the acceleration experienced by an electron in an electric field of 576 N/C. How does the direction of the acceleration depend on the direction of the field at that point?

Determine the magnitude and direction of the electric field at a point midway between a -8.0jitC and a +5.8 /jlC charge 8.0 cm apart. Assume no other charges are nearby.

Draw, approximately, the electric field lines about two point charges, +Q and 3Q, which are a distance apart.

What is the electric field strength at a point in space where a proton experiences an acceleration of 1.8 million gs?

A long uniformly charged thread (linear charge density A = 2.5 C/m) lies along the x axis in Fig. 21-56. A small charged sphere (Q = -2.0 C) is at the point x = 0 cm, y = -5.0 cm. What is the electric field at the {joint x = 7.0 cm, y = 7.0 cm? Fthread and Eq represent fields due to the long thread and the charge Q, respectively.FIGURE 21-56 Problem 31.

The electric field midway between two equal but opposite point charges is 586 N/C, and the distance between the charges is 16.0 cm. What is the magnitude of the charge on each?

Calculate the electric field at one corner of a square 1.22 m on a side if the other three corners are occupied by 2.25 X 10-6 C charges.

Calculate the electric field at the center of a square 52.5 cm on a side if one corner is occupied by a -38.6 /jlC charge and the other three are occupied by 27.0 /xC charges.

Determine the direction and magnitude of the electric field at the point P in Fig. 21-57. The charges are separated by a distance 2a, and point P is a distance x from the midpoint between the two charges. Express your answer in terms of Q, x, a, and k.FIGURE 21-57 Problem 35.

Two point charges, Q\ = 25 /jlC and Q2 = +45 fxC, are separated by a distance of 12 cm. The electric field at the point P (see Fig. 21-58) is zero. How far from Qi is P? FIGURE 21-58 _ Problem 36. P Qi 12 cm Q i -25 ix C +45 jxC

A very thin line of charge lies along the x axis from x = 00 to x = + 00. Another similar line of charge lies along the y axis from y 00 to y = + 00. Both lines have a uniform charge per length A. Determine the resulting electric field magnitude and direction frelative to the r axis'!

"(a) Determine the electric field E at the origin 0 in Fig. 21-59 due to the two charges at A and B. (b) Repeat, but let the charge at B be reversed v in sign. FIGURE 21-59 Problem 38."

Draw, approximately, the electric field lines emanating from a uniformly charged straight wire whose length is not great. The spacing between lines near the wire should be much less than . [Hint. Also consider points very far from the wire.

Two parallel circular rings of radius R have their centers on the x axis separated by a distance as shown in Fig. 21-60. If each ring carries a uniformly v distributed charge Q, find the electric field, E(jc), at points along the x axis.FIGURE 21-60 Problem 40.

You are given two unknown point charges, Q1 and Q2. At a point on the line joining them, one-third of the way from Qi to Q2, the electric field is zero (Fig. 21-61). What is the ratio Q\IQ{1 FIGURE 21-61 Problem 41.

Use Coulombs law to determine the magnitude and direction of the electric field at points A and B in Fig. 21-62 due to the two positive charges (Q = 5.7 /iC) shown. Are your results consistent with Fig. 21-34b? B A f 1 '} 5 1 Qi -------- 5.0 cm FIGURE 21-62 Problem 42.

(a) Two equal charges Q are positioned at points (x = , y = 0) and (jt = -, y = 0). Determine the electric field as a function of y for points along the y axis. (b) Show that the field is a maximum at y = f/V2.

At what position, x = xM, is the magnitude of the electric field along the axis of the ring of Example 21-9 a maximum?

Estimate the electric field at a point 2.40 cm perpendicular to the midpoint of a uniformly charged 2.00-m-long thin wire carrying a total charge of 4.75 /xC.

The uniformly charged straight wire in Fig. 21-29 has the length , where point 0 is at the midpoint. Show that the field at point P, a perpendicular distance x from 0, is given by

Use your result from Problem 46 to find the electric field (magnitude and direction) a distance z above the center of a square loop of wire, each of whose sides has length and uniform charge per length A (Fig. 21-63).FIGURE 21-63 Problem 47.

Determine the direction and magnitude of the electric field at the point P shown in Fig. 21-64. The two charges are separated by a distance of 2a. Point P is on the perpendicular bisector of the line joining the charges, a distance x from the midpoint between them. Express your answers in terms of Q, x, a, and k. FIGURE 21-64 Problem 48.

A thin rod bent into the shape of an arc of a circle of radius R carries a uniform charge per unit length A. The arc subtends a total angle 2 0O, symmetric about the x axis, as shown in Fig. 21-65. Determine the electric field E at the origin 0. FIGURE 21-65 Problem 49.

A thin glass rod is a semicircle of radius R, Fig. 21-66. A charge is nonuniformly distributed along the rod with a linear charge density given by A = A0 sin 6, where A0 is a positive constant. Point P is at the center of the semicircle. (a) Find the electric field E (magnitude and direction) at point P. [Hint: Remember sin(6) = sin 6, so the two halves of the rod are oppositely charged.] (b) Determine the acceleration (magnitude and direction) of an electron placed at point P, assuming R = 1.0 cm and A0 = 1.0/xC/m. H FIGURE 21-66 Problem 50.

"Suppose a uniformly charged wire starts at point 0 and rises vertically along the positive y axis to a length . (a) Determine the components of the electric field Ex and Ey at point (x, 0). That is, calculate E near one end of a long wire, in the plane perpendicular to the wire. (b) If the wire extends from y = 0 to y = oo, so that = oo, show"

Suppose in Example 21-11 that x = 0.250 m, Q = 3.15 nC, and that the uniformly charged wire is only 6.50 m long and extends along the y axis from y = -4.00 m to y = +2.50 m. (a) Calculate Ex and Ey at point P. (b) Determine what the error would be if you simply used the result of Example 21-11, E = A/277e0*. Express this error as (Ex - E)/E and Ey/E. 53. (Ill) A thin rod of length carries

A thin rod of length carries a total charge Q distributed uniformly along its length. See Fig. 21-67. Determine the electric field along the axis of the rod starting v at one endthat is, find E(x) u----p---- ^ for x > 0 in Fig. 21-67. " C Q b FIGURE 21-67 Problem 53.

Uniform plane of charge. Charge is distributed uniformly over a large square plane of side , as shown in Fig. 21-68. The charge per unit area (C/m2) is cr. Determine the electric field at a point P a distance z above the center of the plane, in the limit > oo. [Hint. Divide the plane into long narrow strips of width dy, and use the result of Example 21-11; then sum the fields due to each strip to get the total field at P.]FIGURE 21-68 Problem 54.

Suppose the charge Q on the ring of Fig. 21-28 was all distributed uniformly on only the upper half of the ring, and no charge was on the lower half. Determine the electric field E at P. (Take y vertically upward.)

An electron with speed v0 = 27.5 X 106m/s is traveling parallel to a uniform electric field of magnitude E = 11.4 X 103N/C. (a) How far will the electron travel before it stops? (b) How much time will elapse before it returns to its starting point?

An electron has an initial velocity v0 = (8.0 X 104m/s)j. It enters a region where E = (2.0i + 8.0j) X 104N/C. (a) Determine the vector acceleration of the electron as a function of time, (b) At what angle 6 is it moving (relative to its initial direction) at t = 1.0 ns?

An electron moving to the right at 7.5 X 105m/s enters a uniform electric field parallel to its direction of motion. If the electron is to be brought to rest in the space of 4.0 cm, (a) what direction is required for the electric field, and (b) what is the strength of the field?

At what angle will the electrons in Example 21-16 leave the uniform electric field at the end of the parallel plates (point P in Fig. 21-41)? Assume the plates are 4.9 cm lone and E = 5.0 X 103N/C. Tenore fringing of

An electron is traveling through a uniform electric field. The field is constant and given by E = (2.00 X 10-11 N/C)i - (1.20 X 10-11 N/C)j. At t = 0, the electron is at the origin and traveling in the x direction with a speed of 1.90 m/s. What is its position 2.00 s later?

A positive charge q is placed at the center of a circular ring of radius R. The ring carries a uniformly distributed negative charge of total magnitude Q. (a) If the charge q is displaced from the center a small distance x as shown in Fig. 21-69, show that it will undergo simple harmonic motion when released. (b) If its mass is m, what is its period? FIGURE 21-69 Problem 61.

A dipole consists of charges +e and e separated by 0.68 nm. It is in an electric field E = 2.2 X 104N/C. (a) What is the value of the dipole moment? (b) What is the torque on the dipole when it is perpendicular to the field? (c) What is the torque on the dipole when it is at an angle of 45 to the field? (d) What is the work required to rotate the dipole from being oriented parallel to the field to being antiparallel to the field?

The HC1 molecule has a dipole moment of about 3.4 X 10_30C-m. The two atoms are separated by about 1.0 X 10-10m. (a) What is the net charge on each atom? (b) Is this equal to an integral multiple of el If not, explain. (c) What maximum torque would this dipole experience in a 2.5 X 104N/C electric field? (d) How much energy would be needed to rotate one molecule 45 from its equilibrium position of lowest potential energy?

Suppose both charges in Fig. 21-45 (for a dipole) were positive, (a) Show that the field on the perpendicular bisector, for r , is given by (l/477e0)(2<2/r2). (b) Explain why the field decreases as 1/r2 here whereas for a dipole it decreases as 1/r3.

An electric dipole, of dipole moment p and moment of inertia I, is placed in a uniform electric field E. (a) If displaced by an angle 6 as shown in Fig. 21-44 and released, under what conditions will it oscillate in simple harmonic motion? (b) What will be its frequency?

Suppose a dipole p is placed in a nonuniform electric field E = Ei that points along the x axis. If E depends only on x, show that the net force on the dipole is where dE/dx is the gradient of the field in the x direction.

(a) Show that at points along the axis of a dipole (along the same line that contains +<2 and Q), the electric field has magnitude E = 1 2p 47re0 r3 for r (Fig. 21-45), where r is the distance from a n o in t to th e ce n te r o f th e d in o le . (h \ Tn w h a t d ire c tio n does

How close must two electrons be if the electric force between them is equal to the weight of either at the Earths surface?

Given that the human body is mostly made of water, estimate the total amount of positive charge in a 65-kg person.

A 3.0-g copper penny has a positive charge of 38 fiC. What fraction of its electrons has it lost?

Measurements indicate that there is an electric field surrounding the Earth. Its magnitude is about 150 N/C at the Earths surface and points inward toward the Earths center. What is the magnitude of the electric charge on the Earth? Is it positive or negative? [Hint: The electric field outside a uniformly charged sphere is the same as if all the charge were concentrated at its center.]

(a) The electric field near the Earths surface has magnitude of about 150 N/C. What is the acceleration experienced by an electron near the surface of the Earth? (b) What about a proton? (c) Calculate the ratio of each acceleration to g = 9.8 m/s2.

A water droplet of radius 0.018 mm remains stationary in the air. If the downward-directed electric field of the Earth is 150 N/C, how many excess electron charges must the water droplet have?

Estimate the net force between the CO group and the HN group shown in Fig. 21-70. The C and O have charges 0.40e, and the H and N have charges 0.20e, where e = 1.6 X 10-19 C. [Hint: Do not include the internal forces between C and O, or between H and N.]FIGURE 21-70 Problem 74.

Suppose that electrical attraction, rather than gravity, were responsible for holding the Moon in orbit around the Earth. If equal and opposite charges Q were placed on the Earth and the Moon, what should be the value of Q to maintain the present orbit? Use data given on the inside front cover of this book. Treat the Earth and Moon as point particles.

In a simple model of the hydrogen atom, the electron revolves in a circular orbit around the proton with a speed of 2.2 X 106m/s. Determine the radius of the electrons orbit. [Hint. See Chapter 5 on circular motion.]

A positive point charge Qi = 2.5 X 10-5 C is fixed at the origin of coordinates, and a negative point charge Q2 = 5.0 X 10_6C is fixed to the x axis at x = +2.0 m. Find the location of the place(s) along the x axis where the electric field due to these two charges is zero.

When clothes are removed from a dryer, a 40-g sock is stuck to a sweater, even with the sock clinging to the sweater's underside. Estimate the minimum attractive force between the sock and the sweater. Then estimate the minimum charge on the sock and the sweater. Assume the charging came entirely from the sock rubbing against the sweater so that they have equal and opposite charges, and approximate the sweater as a flat sheet o f uniform rharop

A small lead sphere is encased in insulating plastic and suspended vertically from an ideal spring (spring constant k = 126 N/m) as in Fig. 21-71. The total mass of the coated sphere is 0.650 kg, and its center lies 15.0 cm above a tabletop when in equilibrium. The sphere is pulled down 5.00 cm below equilibrium, an electric charge Q = -3.00 X 10-6 C is deposited on it, and then it is released. Using what you know about harmonic oscillation, write an expression for the electric field strength as a function of time that would be measured at the point on the tabletop (P) directly below the sphere. FIGURE 21-71 Problem 79.

A large electroscope is made with leaves that are 78-cmlong wires with tiny 24-g spheres at the ends. When charged, nearly all the charge resides on the spheres. If the wires each make a 26 angle with the vertical (Fig. 21-72), what total charge Q must have been applied to the electroscope? Ignore the mass of the wires. FIGURE 21-72 Problem 80.

Dry air will break down and generate a spark if the electric field exceeds about 3 X 106N/C. How much charge could be packed onto a green pea (diameter 0.75 cm) before the pea spontaneously discharges? [Hint: Eqs. 21-4 work outside a sphere if r is measured from its center.]

Two point charges, Qi = -6.7jitC and Q2 = 1.8 (jlC, are located between two oppositely charged parallel plates, as shown in Fig. 21-73. The two charges are separated by a distance of x = 0.34 m. Assume that the electric field produced by the charged plates is uniform and equal to E = 73,000 N/C. Calculate the net electrostatic force on Q1 and give its direction.FIGURE 21-73 Problem 82.

Packing material made of pieces of foamed polystyrene can easily become charged and stick to each other. Given that the density of this material is about 35 kg/m3, estimate how much charge might be on a 2.0-cm-diameter foamed polystyrene sphere, assuming the electric force between two spheres stuck together is equal to the weight of one sphere.

One type of electric quadrupole consists of two dipoles placed end to end with their negative charges (say) overlapping; that is, in the center is 2Q flanked (on a line) by a +Q to either side (Fig. 21-74). Determine the electric field E at points along the perpendicular bisector and show that E decreases as 1/r4. Measure r from the - 2 Q charge and assume r .

Suppose electrons enter a uniform electric field midway between two plates at an angle 0O to the horizontal, as shown in Fig. 21-75. The path is symmetrical, so they leave at the same angle 0O and just barely miss the top plate. What is 0 O? Ignore fringing of the field.FIGURE 21-75 Problem 85.

An electron moves in a circle of radius r around a very long uniformly charged wire in a vacuum chamber, as shown in Fig. 21-76. The charge density on the wire is A = 0.14 piC/ni. (a) What is the electric field at the electron (magnitude and direction in terms of r and A)? (b) What is the speed of the electron?FIGURE 21-76 Problem 86.

Three very large square planes of charge are arranged as shown (on edge) in Fig. 21-77. From left to right, the planes have charge densities per unit area of -0.50 /LtC/m2, +0.25 ptC/m2, and -0.35 /zC/m2. Find the total electric field (direction and magnitude) at the points A, B, C, and D. Assume the plates are much larger than the distance AD.FIGURE 21-77 Problem 87.

A point charge (m = 1.0 g) at the end of an insulating cord of length 55 cm is observed to be in equilibrium in a uniform horizontal electric field of 15,000 N/C, when the pendulums position is as shown in Fig. 21-78, with the charge 12 cm above the lowest (vertical) position. If the field points to the right in Fig. 21-78, determine the magnitude and sign of the point charge.FIGURE 21-78 Problem 88.

Four equal positive point charges, each of charge 8.0 jjlC, are at the corners of a square of side 9.2 cm. What charge should be placed at the center of the square so that all charges are at equilibrium? Is this a stable or unstable equilibrium (Section 12-3) in the plane?

Two small, identical conducting spheres A and B are a distance R apart; each carries the same charge Q. (a) What is the force sphere B exerts on sphere A? (b) An identical sphere with zero charge, sphere C, makes contact with sphere B and is then moved very far away. What is the net force now acting on sphere A? (c) Sphere C is brought back and now makes contact with sphere A and is then moved far away. What is the force on sphere A in this third case?

A point charge of mass 0.210 kg, and net charge +0.340 /iC, hangs at rest at the end of an insulating cord above a large sheet of charge. The horizontal sheet of fixed uniform charge creates a uniform vertical electric field in the vicinity of the point charge. The tension in the cord is measured to be 5.18 N. (a) Calculate the magnitude and direction of the electric field due to the sheet of charge (Fig. 21-79). (b) What is the surface charge density a (C/m2) on the sheet?FIGURE 21-79 Problem 91.

A one-dimensional row of positive ions, each with charge +Q and separated from its neighbors by a distance d, occupies the right-hand half of the x axis. That is, there is a +Q charge at x = 0, x = +d, x = +2d, x = +3d, and so on out to oo. (a) If an electron is placed at the position x = d, determine F, the magnitude of force that this row of charges exerts on the electron, (b) If the electron is instead placed at x = 3d, what is the value of F? [Hint: The infinite sum where n is a positive integer.]

A thin ring-shaped object of radius a contains a total charge Q uniformly distributed over its length. The electric field at a point on its axis a distance x from its center is given in Example 21-9 as E = 1 Qx 47re0 (x2 + a2)2 (a) Take the derivative to find where on the x axis (x > 0) Ex is a maximum. Assume Q = 6.00 fxC and a = 10.0 cm. (b) Calculate the electric field for x = 0 to x = +12.0 cm in steps of 0.1 cm, and make a graph of the electric field. Does the maximum of the graph coincide with the maximum of the electric field you obtained analytically? Also, calculate and graph the electric field (c) due to the ring, and (d) due to a point charge Q = 6.00 /xC at the center of the ring. Make a single graph, from x = 0 (orx = 1.0 cm) out to x = 50.0 cm in 1.0 cm steps, with two curves of the electric fields, and show that both fields converge at large distances from the center. (e) At what distance does the electric field of the ring differ from that of the point charge by 10%?

An 8.00 /iC charge is on the x axis of a coordinate system at x = +5.00 cm. A -2.00/xC charge is at x = -5.00 cm. (a) Plot the x component of the electric field for points on the x axis from x = -30.0 cm to x = +30.0 cm. The sign of Ex is positive when E points to the right and negative when it points to the left, (b) Make a plot of Ex and Ey for points on the y axis from y = -30.0 to +30.0 cm.