Two identical point charges are 3.00 cm apart. Find the charge on each of them if the force of repulsion is 4.00 x 10-7 N

Transcription

1 1 Two identical point charges are 3.00 cm apart. Find the charge on each of them if the force of repulsion is 4.00 x 10-7 N C 7 A small cork with an excess charge of +6 µc (1µ C = 10-6 C) is placed 0.12 m from another cork, which carries a charge of -4.3 µc. (a) What is the magnitude of the electric force between the corks? (b) Is this force attractive or repulsive (c) How man excessive electrons are on the negative cork? (d) How many electrons has the positive cork lost? 2 Two negatively-charged bodies with -5.0 x 10-5 C are 0.20 m from each other. What force acts on each particle? (a) 16 N (b) attractive (c) 2.7 x electrons (d) 3.8 x electrons N A negative charge of -2.0 x 10-4 C and a positive charge of 8.0 x 10-4 C are separated by 0.30 m. What is the force between the two charges? 8 In 1994, element 111 was discovered by an international team of physicists. Its provisional name was unununium (Latin for "one-one-one"). Find the distance between two equal and opposite charges, each having a magnitude equal to the charge of 111 protons, if the magnitude of the electric force between them is 2.0 x N. -15,964 N 1.2 x 10 2 m 4 Two charges, q 1, and q 2, are separated by a distance, d, and exert a force on each other. What new force will exist if q 1, and q 2 are both doubled? 4F 9 In 1990, a French team flew a kite that was 1,034 m long. Imagine two charges, +2.0 nc and -2.8 nc, at opposite ends of the kite. (a) Calculate the magnitude of the electric force between them. (b) If the separation of charges is doubled, what absolute value of equal and opposite charges would exert the same electric force? 5 An electric force of 2.5 x 10-4 N acts between two small equally-charged spheres which are 2.0 cm apart. Calculate the force acting between the spheres if the charge on one of the spheres is doubled and the spheres move to a 5.0-cm separation. (a) 4.7 x N (b) 4.7 x 10-9 C x 10-5 N Coulomb measured the deflection of sphere A when A and B had equal charges and were a distance d apart. He then made the charge on B one third the charge on A. How far apart would the two spheres have to be now for A to have the same deflection it had before? 10 Two small silver spheres, each with a mass of 10.0 g, are separated by 1.00 m. Calculate the fraction of the electrons in one sphere that must be transferred to the other in order to produce an attractive force of N (about 1 ton) between the spheres. (The number of electrons per atom of silver is 47, and the number of atoms per gram is Avogadro s number divided by the molar mass of silver, g/mol.) 2.51 x 10-9 To have the same force with 1/3 the charge, the distance would have to be decreased such that d 2 = (1/3), or 0.58 times as far apart. 11 What must be the distance between point charge q 1 = 26.0 μc and point charge q 2 = -47 µc for the electrostatic force between them to have a magnitude of 5.70 N? 1.39 m

2 12 Identical isolated conducting spheres 1 and 2 have equal charges and are separated by a distance that is large compared with their diameters (figure a). The electrostatic force acting on sphere 2 due to sphere 1 is F. Suppose now that a third identical sphere 3, having an insulating handle and initially neutral, is touched first to sphere 1 (figure b), then to sphere 2 (figure c), and finally removed (figure d). The electrostatic force that now acts on sphere 2 has magnitude F'. What is the ratio F /F. 16 In figure a, particle 1 (of charge q 1 ) and particle 2 (of charge q 2 ) are fixed in place on an x axis, 8.00 cm apart. Particle 3 (of charge q 3 = x C) is to be placed on the line between particles 1 and 2 so that they produce a net electrostatic force F 3net on it. Figure b gives the x component of that force versus the coordinate x at which particle 3 is placed. (a) What is the sign of charge q 1? (b) What is the ratio q 2 /q l? (a) positive; (b) Two tiny, spherical water drops, with identical charges of x C, have a center-to-center separation of 1.00 cm. (a) What is the magnitude of the electrostatic force acting between them? (b) How many excess electrons are on each drop, giving it its charge imbalance? 17 Figure a shows charged particles 1 and 2 that are fixed in place on an x axis. Particle 1 has a charge with a magnitude of q 1 = 8.00e. Particle 3 of charge q 3 = +8.00e is initially on the x axis near particle 2. Then particle 3 is gradually moved in the positive direction of the x axis. As a result, the magnitude of the net electrostatic force F 2,net on particle 2 due to particles 1 and 3 changes. Figure b gives the x component of that net force as a function of the position x of particle 3. The plot has an asymptote of F 2,net = 1.5 X N as x --> infinity. As a multiple of e and including the sign, what is the charge q 2 of particle 2? (a) 8.99 x N; (b) 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? 2.0 N 13e 15 An object with charge +7.5 x 10-7 C is placed at the origin. The position of a second object, charge +1.5 x 10-7 C, is varied from 1.0 cm to 5.0 cm. Draw a graph of the force on the object at the origin. Hint: Pick 5 points determine the force and draw the graph. Refer to Solutions Appendix. 18 Three particles are placed on a straight line. The left particle has a charge of +4.6 x l0-6 C, the middle particle has a charge of -2.3 x l0-6 C, and the right particle has a charge of -2.3 x l0-6 C. The left particle is 12 cm from the middle particle and the right particle is 24 cm from the middle particle. The left particle is now moved directly above the middle particle, still 12 cm away. Find the force on the middle particle. Hard 7.2 to the left or vertical 6.6 N

3 19 A positive charge of 3.0 x 10-6 C is pulled on by two negative charges. One, -2.0 x 10-5 C, is m to the north and the other, -4.0 x 10-6 C, is m to the south. What total force is exerted on the positive charge? 23 Three charges lie along the x-axis. One positive charge, q 1 = 15 µc, is at x= 2.0 m, and another positive charge, q 2 = 6.0 µc, is at the origin. At what point on the x-axis must a negative charge, q 3, be placed so that the resultant force on it is zero? -9 N, North 20 A charge Q 1 = +10 x 10-9 C on the x axis at x = 0, and a second charge Q 2 = +3 x 10-9 C is on the x axis at x = 5 m. A third charge Q 3 = -10 x 10-9 C is placed on the x axis at x = 15 m. Calculate the magnitude and direction of the force on Q 2..8 m from q x 10-9 N toward the right 24 A +2.2 x 10-9 C charge is on the x-axis at x = 1. 5 m, a +5.4 x 10-9 C charge is on the x-axis at x = 2.0 m, and a x 10-9 C charge is at the origin. Find the net force on the charge at the origin. 21 Three charges, each of +80 x 10-6 C, are equally spaced along a straight line, successive charges being 6 m apart. (a) Calculate the force on one of the end charges. (b) Calculate the force on the central charge. 7.3 x 10-8 N along the negative x-axis 22 (a) 2.00 N away from the center (b) 0 newtons on the center charge. Three masses, each of +80 x 10 6 kg, are equally spaced along a straight line, successive masses are 6.00 m apart. (a) Calculate the force on one of the end masses. (b) Calculate the force on the central mass. 25 In 1955, a water bore that was 2,231 m deep was drilled in Montana. Consider two charges, q 2 = 1.60 mc and q 1, separated by a distance equal to the depth of the well. If a third charge, q mc is placed 888 m from q 2 and is between q 2 and q 1, this third charge will be in equilibrium. What is the value of q 1? 366 mc (a) 1.48 x 10 4 N (b) 0 26 The figure shows two protons (symbol p) and one electron (symbol e) on an axis. (a) What is the direction of the electrostatic force on the central proton due to the electron? (b) What is the direction of the electrostatic force on the central proton due to the other proton? (c) What is the direction of the net electrostatic force on the central proton? (a) leftward; (b) leftward; (c) leftward

4 27 The figure shows four situations in which charged particles are fixed in place on an axis. In which situations is there a point to the left of the particles where an electron will be in equilibrium? 30 Particles of charge +75, + 48, and 85 μc are placed in a line as shown in the figure. The center one is 0.35 m from each of the others. Calculate the net force on each charge due to the other two. a and b F +75 = -1.5 x 10 2 N F +48 = +5.6 x 10 2 N F -85 = -4.2 x 10 2 N 28 In the figure, four particles are fixed along an x axis, separated by distances d = 2.00 cm. The charges are q 1 = +2e, q 2 = -e, q 3 = +e, and q 4 = +4e, with e = 1.60 x C. (a) In unit-vector notation, what is the net electrostatic force on particle 1 due to the other particles? (b) In unit-vector notation, what is the net electrostatic force on particle 2 due to the other particles? 31 Two charges q 1 and q 2, lie on the x-axis. The first charge is at the origin and the second is at x = 1.0 m determine the equilibrium position for a third charge q 3, with respect to q 1 and q 2 for each of the following cases (a) q 1 = + 10 µc, q 2 =+7.5 µc (b) q 1 = µc, q 2 =+5.2 µc (c) q 1 = -3.7 nc, q 2 =-5.2 nc (a) (3.52 x N)ˆi ; (b) 0 (a) 0.55 m form q 1 (b) 0.46 from q 1 (c) 0.46 m from q 1 29 Figure a shows two particles fixed in place: a particle of charge q 1 = +8q at the origin and a particle of charge q 2 = -2q at x L. (a) At what point (other than infinitely far away) can a proton be placed so that it is in equilibrium (the net force on it is zero)? (b) Is that equilibrium stable or unstable? 32 A charge q 1 of x 10-9 C and a charge q 2 of x 10-9 C are separated by a distance of 60.0 cm. Where could a third charge be placed so that the net electric force on it is zero? 35.2 cm from q 1 (24.8 cm from q 2 ) 33 Three point charges lie along the y-axis. A charge of q 1 = -9.0 µc is at y = 6.0 m, and a charge of q 2 = -8.0 µc is at y= -4.0 m. The net electric force on the third charge is zero. Where is this charge located? (a) x = 2L (b) Unstable y = 0.8 m from the zero point or 5.1 from one end 34 Hans Langseth's beard measured 5.33 m in Consider two charges, q 1 = 2.5 nc and q 2 = 8.0 nc, separated by the length of Langseth's beard. How far from q 1 should a third charge of 1.0 nc be placed so that no force is exerted on the third charge? 1.9 m

5 35 In more than 30 years, Albert Klein, of California, drove 2.5 x 10 6 km in one automobile. Consider two charges, q 1 = 2.0 C and q 2 = 6.0 C, separated by Klein's total driving distance. A third charge, q 3 = 4.0 C, is placed on the line connecting q 1 and q 2. How far from q 1 should q 3 be placed for q 3 to be in equilibrium? 38 Two small beads having positive charges 3q and q are fixed at the opposite ends of a horizontal, insulating rod, extending from the origin to the point x = d. As shown in the figure, a third small charged bead is free to slide on the rod. (a) At what position is the third bead in equilibrium? (b) Can it be in stable equilibrium? 9.3 x 10 8 m 36 A charge of x 10-9 C is placed at the origin, and another charge of +4 x 10-9 C is placed at x = 1.5 m. find the point between these two charges where a charge of x 10-9 C should be paced so that the net electric force on it is zero. (a) =.634d (b) stable if the third bead has a positive charge x = 0.64 m 39 Two known charges, 12.0 µc and 45.0 µc, and an unknown charge are located on the x axis. The charge 12.0 µc is at the origin, and the charge 45.0 µc is at x = 15.0 cm. The unknown charge is to be placed so that each charge is in equilibrium under the action of the electric forces exerted by the other two charges. (a) Is this situation possible? (b) Is it possible in more than one way? (c) Find the required location, magnitude, and sign of the unknown charge. 37 The figure shows two charged particles on an axis. The charges are free to move. However, a third charged particle can be placed at a certain point such that all three particles are then in equilibrium. (a) Is that point to the left of the first two particles, to their right, or between them? (b) Should the third particle be positively or negatively charged? (c) Is the equilibrium stable or unstable? 40 (a) (b) (c) -16 cm 51.3 µc In the fiure, three charged particles lie on an x axis. Particles 1 and 2 are fixed in place. Particle 3 is free to move, but the net electrostatic force on it from particles 1 and 2 happens to be zero. If L 23 = L 12, what is the ratio q l /q 2? (a) between; (b) positively charged; (c) unstable Point charges of +6.0 μc and -4.0 μc are placed on an x axis, at x = 8.0 m and x = 16 m, respectively. What charge must be placed at x = 24 m so that any charge placed at the origin would experience no electrostatic force? -45 µc

6 42 Two charges, Q 0 and 3Q 0, are a distance l apart. These two charges are free to move but do not because there is a third charge nearby. What must be the charge and placement of the third charge for the first two to be in equilibrium? 46 In which direction will the electric force from the two equal positive charges pull the negative charge shown in the figure below? 0.40 Q l from - Q 0 towards -3Q 0 43 A triangle ABC, marked out on a flat surface, has sides of the following lengths: AB = 4 m, BC = 5 m, and AC = 3 m. At the corners are the following charges: -40 x 10-6 C at A, -160 x 10-6 C at B, and +90 x 10-6 C at (C) What are the magnitude and direction of the net force on the charge at A? to the left N, 45 degrees from AC, 135 degrees from AB Equal masses of +8 x 10 6 kg are placed at the three corners of an equilateral triangle 200 m on a side. Calculate the magnitude and direction of the force on one of the masses. 47 American athlete Jesse Castenada walked km in 24 h in 1976, setting a new record. Consider an equilateral triangle with a perimeter equal to the distance Castenada walked. Suppose the charges are placed at the following vertices of the triangle: q 1 = 8.8 nc at the bottom left vertex, q 2 = -2.4 nc at the bottom right vertex, and q 3 = 4.0 nc at the top vertex. Find the magnitude and direction of the resultant electric force acting on q N away from the triangle, perpendicular to the opposite base. 4.7 x below the positive x axis 45 Consider three point charges at the corners of a triangle, as shown in the figure, where q 1 = 6.00 x 10-9 C, q 2 = x 10-9 C, and q 3 = 5.00 x 10-9 C. Find the magnitude and direction of the resultant force on q The figure here shows three arrangements of an electron e and two protons p. (a) Rank the arrangements according to the magnitude of the net electrostatic force on the electron due to the protons, largest first. (b) In situation c, is the angle between the net force on the electron and the line labeled d less than or more than 45 0? (a) a, c, b; (b) less than 7.16 x 10-9 N θ = or x 10-9 i x 10-9 j

7 49 The figure shows four situations in which five charged particles are evenly spaced along an axis. The charge values are indicated except for the central particle, which has the same charge in all four situations. Rank the situations according to the magnitude of the net electrostatic force on the central particle, greatest first. 52 In figure a, particles 1 and 2 have charge 20.0 μc each and are held at separation distance d = 1.50 m. (a) What is the magnitude of the electrostatic force on particle 1 due to particle 2? In figure b, particle 3 of charge 20.0 μc is positioned so as to complete an equilateral triangle. (b) What is the magnitude of the net electrostatic force on particle 1 due to particles 2 and 3? 3, 1, 2, 4 (zero) (a) 1.60 N; (b) 2.77 N 50 The figure shows four arrangements of charged particles. Rank the arrangements according to the magnitude of the net electrostatic force on the particle with charge +Q, greatest first. 53 In the figure, particle 1 of charge +1.0 μc and particle 2 of charge -3.0 μc are held at separation L = 10.0 cm on an x axis. Assume particle 3 of unknown charge q 3 is to be located such that the net electrostatic force on it from particles 1 and 2 is zero. (a) What must be the x coordinates of particle 3? (b) What must be the y coordinates of particle 3? (a) 14 cm; (b) 0 a and d tie, then b and c tie 51 Three point charges are located at the corners of an equilateral triangle as shown in the figure. Calculate the resultant electric force on the 7.00 µc charge N i N j N at an angle of 330 0

8 54 In the figure, particles 1 and 2 of charge q 1 = q 2 = x C are on a y axis at distance d = 17.0 cm from the origin. Particle 3 of charge q 3 = x C is moved gradually along the x axis from x = 0 to x = +5.0 m. (a) At what values of x will the magnitude of the electrostatic force on the third particle from the other two particles be minimum and (b) At what values of x will the magnitude of the electrostatic force on the third particle from the other two particles be maximum? (c) What is the minimum magnitudes? (d) What is the maximum magnitudes? 56 In the figure, three identical conducting spheres form an equilateral triangle of side length d = 20.0 cm. The sphere radii are much smaller than d, and the sphere charges are q A = nc, q B = nc, and q C = nc. (a) What is the magnitude of the electrostatic force between spheres A and C? The following steps are then taken: A and B are connected by a thin wire and then disconnected; B is grounded by the wire, and the wire is then removed; B and C are connected by the wire and then disconnected. (b) What now is the magnitude of the electrostatic force between spheres A and C? (c) What now is the magnitude of the electrostatic force between spheres B and C? (a) 0; (b) 12 cm; (c) 0; (d) 4.9 x N (a) 3.60 µn; (b) 2.70 µn; (c) 3.60 µn 55 In the figure, particle 1 of charge μc and particle 2 of charge μc are held at separation L = 20.0 cm on an x axis. (a) In unit-vector notation, what is the net electrostatic force on particle 3, of charge q3 = 20.0μC, if particle 3 is placed at x = 40.0 cm? (b) In unit-vector notation, what is the net electrostatic force on particle 3, of charge q3 = 20.0μC, if particle 3 is placed at x = 80.0 cm? (c) What should be the x coordinates of particle 3 if the net electrostatic force on it due to particles 1 and 2 is zero? (d) What should be the y coordinates of particle 3 if the net electrostatic force on it due to particles 1 and 2 is zero? 57 The initial charges on the three identical metal spheres in the figure are the following: sphere A, Q; sphere B, -Q/4; and sphere C, Q/2, where Q = 2.00 x C. Spheres A and B are fixed in place, with a center-to-center separation of d = 1.20 m, which is much larger than the spheres. Sphere C is touched first to sphere A and then to sphere B and is then removed. What then is the magnitude of the electrostatic force between spheres A and B? 4.68 x N (a) (89.9N)ˆi; (b) (-2.50 N)ˆi; (c) 68.3 cm; (d) 0

9 58 The figure shows four situations in which particles of charge +q or -q are fixed in place. In each situation, the particles on the x axis are equidistant from the y axis. First, consider the middle particle in situation 1; the middle particle experiences an electrostatic force from each of the other two particles. (a) Are the magnitudes F of those forces the same or different? (b) Is the magnitude of the net force on the middle particle equal to, greater than, or less than 2F? (c) Do the x components of the two forces add or cancel? (d) Do their y components add or cancel? (e) Is the direction of the net force on the middle particle that of the canceling components or the adding components? (f) What is the direction of that net force? Now consider the remaining situations: (g) What is the direction of the net force on the middle particle in situation 2? (h) What is the direction of the net force on the middle particle in situation 3? (i) What is the direction of the net force on the middle particle in situation 4? (In each situation, consider the symmetry of the charge distribution and determine the canceling components and the adding components.) 60 The cinema screen installed at the Science Park, in Taejon, Korea, is 24.7 m high and 33.3 m wide. Consider the arrangement of charges shown below. If q 1 = 2.00 nc, q 2 = nc, and q 3 = 4.00 nc, find the magnitude and direction of the resultant electric force on q x N In 1913, a special postage stamp was issued in China. It was 248 mm long and 70.0 mm wide. Suppose equal charges of 1.0 nc are placed in the corners of this stamp. Find the magnitude and direction of the resultant electric force acting on the upper right corner (assume the widest part of the stamp is aligned with the x-axis) 1.8 x 10-6 N 81 0 above the positive x-axis 59 (a) same; (b) less than; (c) cancel; (d) add; (e) adding components; (f) positive y direction; (g) negative y direction; (h) positive x direction; (i) negative x direction Three charged particles are placed at the corners of an equilateral triangle of side 1.20 m as shown in the figure. The charges are +4.0 μc, 8.0 μc, and 6.0 μc. Calculate the magnitude and direction of the net force on each due to the other two. 62 In 1993, a chocolate chip cookie was baked in Arcadia, California. It contained about three million chips and was 10.7 m long and 8.7 m wide. Suppose four charges are placed in the corners of that cookie as follows: q 1 = nc at the lower left corner, q 2 = 5.6 nc at the upper left corner, q 3 = 2.8 nc at the upper right corner, and q 4 = 8.4 nc at the lower right corner. (a) Draw a picture of the rectangular cooke (b) Find the magnitude and direction of the resultant electric force acting on q 1. (a) Picture (b) 1.28 x 10-8 N 45 0 above the positive x-axis

10 63 In 1988, a giant firework was exploded at the Lake Toya festival, in Japan. The shell had a mass of about 700 kg and produced a fireball 1.2 km in diameter. Consider a circle with this diameter. Suppose four charges are placed on the circle's perimeter so that the lines between them form a square with sides parallel to the x- or y-axes. The charges have the following strengths and locations: q 1 = 16.0 mc at the upper left "corner," q 2 = 2.4 mc at the upper right corner, q 3 = -3.2 mc at the lower right corner, and q 4 = -4.0 mc at the lower left corner. Find the magnitude and direction of the resultant electric force acting on q 1. (Hint: Find the distances between the charges first.) 67 In the figure, a central particle of charge -2q is surrounded by a square array of charged particles, separated by either distance d or d/2 along the perimeter of the square. What are the magnitude and direction of the net electrostatic force on the central particle due to the other particles? (Hint: Consideration of symmetry can greatly reduce the amount of work required here.) 1.0 N 77 0 below the negative x axis 64 A charge of 6.00 mc is placed at each corner of a square m on a side. Determine the magnitude and direction of the force on each charge. 6kq 2 /d 2, leftward 65 A charge of 6.00 mc is placed at each corner of a square m on a side. Two of the positive charges, on opposite corners, are replaced by negative charges of the same magnitude as shown in the figure. Determine the magnitude and direction of the force on each charge. 68 Consider a regular polygon with 29 sides. The distance from the center to each vertex is a. Identical charges q are placed at 28 vertices of the polygon. A single charge Q is placed at the center of the polygon. What is the magnitude and direction of the force experienced by the charge Q? 0 69 The figure shows an arrangement of four charged particles, with angle θ = = and distance d = 2.00 cm. Particle 2 has charge q 2 = x C; particles 3 and 4 have charges q 3 = q 4 = x C. (a) What is distance D between the origin and particle 2 if the net electrostatic force on particle 1 due to the other particles is zero? (b) If particles 3 and 4 were moved closer to the x axis but maintained their symmetry about that axis, would the required value of D be greater than, less than, or the same as in part (a)? 2.96 x 10 7 N For each charge, the net force will be the magnitude of 2.96 x 10 7 N and each net force will lie along the line from the charge inwards towards the center of the square. 66 When more than one charged object is present in an area, how can the total electric force on one of the charged objects be found? (a) 1.92 cm; (b) less than Each force exerted on an object is found, then the forces are added together vectorially.

11 70 In the figure, six charged particles surround particle 7 at radial distances of either d = 1.0 cm or 2d, as drawn. The charges are q 1 = +2e, q 2 = +4e, q 3 = +e, q 4 = +4e, q 5 = +2e, q 6 = +8e, q 7 = +6e, with e = 1.60 x C. What is the magnitude of the net electrostatic force on particle 7? 73 In the figure, a central particle of charge -q is surrounded by two circular rings of charged particles. What are the magnitude and direction of the net electrostatic force on the central particle due to the other particles? (Hint: Consideration of symmetry can greatly reduce the amount of work required here.) 0 71 In the figure, particles 2 and 4, of charge -e, are fixed in place on a y axis, at y 2 = cm and y 4 = 5.00 cm. Particles 1 and 3, of charge -e, can be moved along the x axis. Particle 5, of charge +e, is fixed at the origin. Initially particle 1 is at x 1 = cm and particle 3 at x 3 = 10.0 cm. (a) To what x value must particle 1 be moved to rotate the direction of the net electric force F net on particle 5 by 30 0 counterclockwise? (b) With particle 1 fixed at its new position, to what x value must you move particle 3 to rotate F net back to its original direction? 74 2kq 2 /r 2, up the page Four identical particles, each having charge +q, are fixed at the corners of a square of side L. A fifth point charge Q lies a distance z along the line perpendicular to the plane of the square and passing through the center of the square as shown in the figure. (a) Show that the force exerted on Q by the other four charges is reduces to F = (constant) z. (b) Why does this imply that the motion of Q is simple harmonic, and what is the period of this motion if the mass of Q is m? 72 The figure shows four identical conducting spheres that are actually well separated from one another. Sphere W (with an initial charge of zero) is touched to sphere A and then they are separated. Next, sphere W is touched to sphere B (with an initial charge of -32e) and then they are separated. Finally, sphere W is touched to sphere C (with an initial charge of +48e), and then they are separated. The final charge on sphere W is +18e. What was the initial charge on sphere A? 16e

12 75 Eight point charges, each of magnitude q, are located on the corners of a cube of edge s, as shown in the figure. (a) Determine the x, y, and z components of the resultant force exerted by the other charges on the charge located at point A. (b) What are the magnitude and direction of this resultant force? 77 In the figure, particle 1 of charge +4e is above a floor by distance d 1 = 2.00 mm and particle 2 of charge +6e is on the floor, at distance d 2 = 6.00 mm horizontally from partcle 1. What is the x component of the electrostatic force on particle 2 due to particle 1? F x = 1.31 x N 76 In crystals of the salt cesium chloride, cesium ions Cs + form the eight corners of a cube and a chlorine ion Cl - is at the cube's center (see figure). The edge length of the cube is 0.40 nm. The Cs + ions are each deficient by one electron (and thus each has a charge of +e), and the Cl - ion has one excess electron (and thus has a charge of -e). (a) What is the magnitude of the net electrostatic force exerted on the Cl - ion by the eight Cs + ions at the corners of the cube? (b) If one of the Cs + ions is missing, the crystal is said to have a defect; what is the magnitude of the net electrostatic force exerted on the Cl - ion by the seven remaining Cs + ions? 78 In the figure, all four particles are fixed in the xy plane, and q 1 = x C, q 2 = x C, q 3 = x C, q 4 = x Q, θ, = , d 1 = 3.00 cm, and d 2 = d 3 = 2.00 cm. (a) What is the magnitude of the net electrostatic force on particle 4 due to the other three particles? (b) What is the direction of the net electrostatic force on particle 4 due to the other three particles? (a) 6.16 x N; (b) An alpha particle (charge = +2.0e) is sent at high speed toward a gold nucleus (charge = +79e). What is the electric force acting on the alpha particle when it is 2.0 x m from the gold nucleus? (a) 0; (b) 1.9 x 10-9 N 91 N (repulsive) 80 The CN Tower, in Toronto, Canada, is 553 m tall. Suppose two balls, each with a mass of 5.00 kg and a charge of 40.0 mc, are placed at the top and bottom of the tower, respectively. The ball at the top is then dropped. At what height is the acceleration on the ball zero? 542

13 81 Two identical particles, each having charge +q, are fixed in space and separated by a distance d. A third point charge Q is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance x from the midpoint between the two fixed charges shown in the figure. (a) Show that if x is small compared with d, the motion of Q will be simple harmonic along the perpendicular bisector. Determine the period of that motion. (b) How fast will the charge Q be moving when it is at the midpoint between the two fixed charges, if initially it is released at a distance a << d from the midpoint? 85 In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is m. (a) Find the electric force between the two. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron? (a) 8.22 x 10-8 N (b) 2.19 x 10 6 m/s 86 A particle of charge Q is fixed at the origin of an xy coordinate system. At t = 0 a particle (m = g, q = 4.00 μc) is located on the x axis at x = 20.0 cm, moving with a speed of 50.0 m/s in the positive y direction. For what value of Q will the moving particle execute circular motion? (Neglect the gravitational force on the particle.) µc 82 Two equally charged particles are held 3.2 x 10-3 m apart and then released from rest. The initial acceleration of the first particle is observed to be 7.0 m/s 2 and that of the second to be 9.0 m/s 2. The mass of the first particle is 6.3 x 10-7 kg. (a) What is the mass of the second particle? (b) What is the magnitude of the charge of each particle? 87 Compare the electric force holding the electron in orbit (r = 0.53 x m) around the proton nucleus of the hydrogen atom, with the gravitational force between the same electron and proton. What is the ratio of these two forces? 2.3 x (a) 4.9 x 10-7 kg; (b) 7.1 x C Two point charges of 30 nc and -40 nc are held fixed on an x axis, at the, origin and at x = 72 cm, respectively. A particle with a charge of 42 μc is released from rest at x = 28 cm. If the initial acceleration of the particle has a magnitude of 100 km/s 2, what is the particle's mass? 88 The two pith balls in the figure each have a mass of 1.0 g and equal charges. One pith ball is suspended by an insulating thread. The other is brought to 3.0 cm from the suspended ball. The suspended ball is now hanging with the thread forming an angle of 30.0 with the vertical. The ball is in equilibrium with F E mg, and T adding vectorial to yield zero. (a) Calculate mg. (b) Calculate F E (c) Calculate the charge on the balls. 2.2 x 10-6 kg 84 Consider two electric dipoles in empty space. Each dipole has zero net charge. Does an electric force exist between the dipoles-that is, can two objects with zero net charge exert electric forces on each other? If so, is the force one of attraction or of repulsion? (a) 9.8 x 10-3 N (b) 5.7 x 10-3 N (c) 2.4 x 10-8 C

14 89 Two Ping-Pong balls painted with aluminum paint are suspended from the same point by threads 50 cm long. The mass of each ball is 20 g. When equal charges are given to the two balls, they come to rest in an equilibrium position in which their centers are 60 cm apart. Calculate the charge on each ball. 93 In the Figure, two tiny conducting balls of identical mass m and identical charge q hang from nonconducting threads of length L. Assume that θ is so small that tan θ can be replaced by its approximate equal, sin θ. (a) Show that gives the equilibrium separation x of the balls you can deduce the given equation. (b) If L = 120 cm, m = 10 g, and x = 5.0 cm, what is q? 2.42 x 10-6 C 90 Two small spheres each having a mass of 0.10 g are suspended from the same point on silk threads 20 cm long. When given equal charges, they repel each other, coming to rest 24 cm apart. Find the charge on each sphere? 6.86 x 10-8 C (b) 2.4 x 10-8 C 91 Two small metallic spheres, each with a mass of 0.20 g, are suspended as pendulums by light strings from a common point. They are given the same electric charge, and the two come to equilibrium when each string is at an angle of with the vertical. If the string is 30.0 cm long, what is the magnitude of the charge on each sphere? 94 Explain the following question. (a) Explain what happens to the balls of the figure if one of them is discharged (loses its charge q to, say, the ground). (b) Find the new equilibrium separation x, using the given values of L and m and the computed value of q x 10-9 C 92 Two small spheres of mass m are suspended from strings of length l that are connected at a common point. One sphere has charge Q; the other has charge 2Q. The strings make angles θ 1 and θ 2 with the vertical. (a) How are θ 1 and θ 2 related? (b) Assumeθ 1 and θ 2 are small. Show that the distance r between the spheres is given by (a) (b) 3.1 cm

15 95 A large electroscope is made with leaves that are 78 cm long 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 30 angle with the vertical as shown in the figure, what total charge Q must have been applied to the electroscope? Ignore the mass of the wires. 99 In 1974, an emerald with a mass of kg was found in Brazil. Suppose you want to hang this emerald on a string that is 80.0 cm long and has a breaking strength of N. To hang the jewel safely, you remove a certain charge from the emerald and place it at the pivot point of the string. What is the minimum possible value of this charge? 1.0 x 10-5 C 100 Suppose someone proposes the idea that people are bound to the Earth by electric forces rather than by gravity. How could you prove this idea is wrong? x 10-6 C A small plastic sphere coated with a thin metalized surface has mass 0.05 g and carries a charge of +8 x 10-9 C. It is suspended by a light insulating thread at a point 3 cm below the center of a small fixed conducting sphere carrying -5 x 10-9 C. The thread is cut. (a) What is the electrostatic acceleration upward? (b) What is the net or observed acceleration? 101 An electron is in a vacuum near Earth's surface and located at y = 0 on a vertical y axis. At what value of y should a second electron be placed such that its electrostatic force on the first electron balances the gravitational force on the first electron? -5.1 m 102 How close must two electrons be if the electric force between them is equal to the weight of either at the Earth s surface? (a) 8.0 m/s 2 (b) 1.8 m/s 2 downward 5.08 m 97 An electron is released above the Earth's surface. A second electron directly below it exerts just enough of an electric force on the first electron to cancel the gravitational force on it. Find the distance between the two electrons m 103 Coulombs Law and the Universal gravitational Law are very similar. (a) Calculate the repulsive Coulomb force between two protons 4 x m apart inside a nucleus. (b) Calculate the gravitational force of attraction between the two protons. (c) Why doesn't a force of such magnitude, acting on a proton of mass about 1.67 x g, cause the nucleus to fly apart? 98 The most accurate balance can measure objects with masses as small as 1.0 x kg, which is less than the mass of the ink in the period at the end of this sentence. Suppose a mass this small is suspended by electric repulsion over a charge of -4.0 nc. How many extra electrons must be placed on the mass so that it will float 2.0 cm directly over the -4.0 nc charge? 6.8 x 10 3 electrons 104 (a) 14.4 N (b) 1.17 x N (c) Four objects of equal mass and equal charge are placed at the corners of a square with 4 meter sides. A charge of 12 x 10-5 C are placed on each object. What is the mass of each object if the Gravitational Force on each object is equal to the electrostatic force x 10 6

16 105 The electron and proton of a hydrogen atom are separated on average by a distance of about 5.3 x m. Find the magnitudes of the electric force and the gravitational force that each particle exerts on the other. 110 Little Pumpkin, a miniature horse owned by J. C. Williams, Jr., of South Carolina, had a mass of about 9.00 kg. Consider Little Pumpkin on a twin-pan balance. If the mass on the other pan is 8.00 kg, what equal and opposite charges must be placed as shown in the diagram below to maintain equilibrium? Assume r = 1 m. F electric = -8.2 x 10-8 N F g = 3.6 x N 106 The moon (m = 7.36 x kg) is bound to Earth (m = 5.98 x kg) by gravity. The moon is 3.82 x 10 8 m from the earth. If, instead, the force of attraction were the result of each having a charge of the same magnitude but opposite in sign, find the quantity of charge that would have to be placed on each to produce the required force x C 3.30 x 10-5 C 107 the following questions. (a) Two protons in a molecule are separated by m. Find the electric force exerted by one proton on the other. (b) How does the magnitude of this force compare to the magnitude of the gravitational force between the two protons? (c) What If? What must be the charge-to-mass ratio of a particle if the magnitude of the gravitational force between two of these particles equals the magnitude of electric force between them? 111 The largest bell that is in use today is in Mandalay, Myanmar, formerly called Burma. Its mass is about 92 x 10 3 kg. Suppose the bell is supported in equilibrium as shown in the figure. Calculate the value for the charge q. 108 The nucleus in an iron atom has a radius of about 4.0 x m and contains 26 protons. (a) What is the magnitude of the repulsive electrostatic force between two of the protons that are separated by 4.0 x m? (b) What is the magnitude of the gravitational force between those same two protons? 8.9 x 10-3 C 109 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 these data: mass of Earth = 5.98 x mass of Moon = 7.35 x kg, radius of the orbit = 3.84 x 10 8 m. Treat the Moon and Earth as point particle. 112 Identical thin rods of length 2a carry equal charges +Q uniformly distributed along their lengths. The rods lie along the x axis with their centers separated by a distance b > 2a (shown in the figure. Show that the magnitude of the force exerted by the left rod on the right one is given by the equation above the figure.

17 113 The figure shows a long, nonconducting, massless rod of length L, pivoted at its center and balanced with a block of weight W at a distance x from the left end. At the left and right ends of the rod are attached small conducting spheres with positive charges q and 2q, respectively. A distance h directly beneath each of these spheres is a fixed sphere with positive charge Q. (a) Find the distance x when the rod is horizontal and balanced. (b) What value should h have so that the rod exerts no vertical force on the bearing when the rod is horizontal and balanced?. 116 A small lead sphere is encased in insulating plastic and suspended vertically from an ideal spring (k = 126 N m) above a lab table, shown in the figure. The total mass of the coated sphere is kg, and its center lies 15.0 cm above the 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. (a) (L/2)(1 + kqq/wh 2 ); (b) (3kqQ/W) 114 A DNA molecule (deoxyribonucleic acid) is 2.17 mm long. The ends of the molecule become singly ionized so that there is an increase of 1.00 percent upon becoming charged. Find the effective spring constant of the molecule. 117 In 1994, a group of British and Canadian athletes performed a rope slide off the top of Mount Gibraltar, in Canada. The speed of the sliders at times exceeded 160 km/h. The total length of the slide was 1,747 m. Suppose several sliders are located on the rope as shown. Due to friction, they acquire the electric charges shown. Find the magnitude and direction of the resultant electric force acting on the athlete at the far right of the diagram x 10-9 N/m 115 Two identical metallic blocks resting on a frictionless horizontal surface are connected by a light metallic spring having a spring constant k as shown in Figure a and an unstretched length L i. A total charge Q is slowly placed on the system, causing the spring to stretch to an equilibrium length L, as shown in Figure b. Determine the value of Q, assuming that all the charge resides on the blocks and modeling the blocks as point charges. 3.2 x N, ut the rope

18 118 In 1995, a single diamond was sold for more than 16 million. It was not the largest diamond in the world, but its mass was an impressive 20.0 g. Consider such a diamond resting on a horizontal surface. It is known that if the diamond is given a charge of 2.0 µc and a charge of at least -8.0 µc is placed on that surface at a distance of 1.7 m from it, then the diamond will barely keep from sliding. Calculate the coefficient of static friction between the diamond and the surface. 121 Inez is putting up decorations for her sister s quinceañera (fifteenth birthday party). She ties three light silk ribbons together to the top of a gateway and hangs from each ribbon a rubber balloon shown in the figure. To include the effects of the gravitational and buoyant forces on it, each balloon can be modeled as a particle of mass 2.00 g, with its center 50.0 cm from the point of support. To show off the colors of the balloons, Inez rubs the whole surface of each balloon with her woolen scarf, to make them hang separately with gaps between them. The centers of the hanging balloons form a horizontal equilateral triangle with sides 30.0 cm long. What is the common charge each balloon carries? Mycoplasma is the smallest living organism known. Its mass has an estimated value of 1.0 x g. (a) If two specimens of this organism are placed 1.0 m apart and one electron is placed on each, what is their initial acceleration due to the electric force? (b) If the medium through which the Allycoplasma move exerts a resistive force on the organisms, how large must that force be to balance the force of electrostatic repulsion? µc 120 (a) 2.3 x 10-9 m/s 2 (b) 2.3 x N The parasitic wasp Carapractus cinctus has a mass of 5.0 x 10-6 kg, which makes it one of the smallest insects in the world. If two such wasps are given equal and opposite charges with an absolute value of 2.0 x C and are placed 1.00 m from each other on a horizontal smooth surface, what extra horizontal force must be applied to each wasp to keep it from sliding? Take into account both gravitational and electric forces between the wasps. 122 A line of positive charge is formed into a semicircle of radius R = 60.0 cm as shown in the figure. The charge per unit length along the semicircle is described by the expression λ = λ 0 cos θ. The total charge on the semicircle is 12.0 µc. Calculate the total force on a charge of 3.00 µc placed at the center of curvature x N.707 downward

19 123 Two identical beads each have a mass m and charge q. When placed in a hemispherical bowl of radius R with frictionless, nonconducting walls, the beads move, and at equilibrium they are a distance R apart shown in the figure. Determine the charge on each bead. 126 In the figure, three identical conducting spheres initially have the following charges: sphere A, 4Q; sphere B, -6Q; and sphere C, 0. Spheres A and B are fixed in place, with a center-tocenter separation that is much larger than the spheres. Two experiments are conducted. In experiment 1, sphere C is touched to sphere A and then (separately) to sphere B, and then it is removed. In experiment 2, starting with the same initial states, the procedure is reversed: Sphere C is touched to sphere B and then (separately) to sphere A, and then it is removed. What is the ratio of the electrostatic force between A and B at the end of experiment 2 to that at the end of experiment 1? 124 A nonconducting spherical shell, with an inner radius of 4.0 cm and an outer radius of 6.0 cm, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density ρ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell ρ = b/r, where r is the distance in meters from the center of the shell and b = 3.0 µc/m 2. What is the net charge in the shell? The figure shows three situations involving a charged particle and a uniformly charged spherical shell. The charges are given, and the radii of the shells are indicated. Rank the situations according to the magnitude of the force on the particle due to the presence of the shell, greatest first. 3.8 x 10-8 C 125 A charged nonconducting rod, with a length of 2.00 m and a cross-sectional area of 4.00 cm 2, lies along the positive side of an x axis with one end at the origin. The volume charge density ρ is charge per unit volume in coulombs per cubic meter. (a) How many excess electrons are on the rod if ρ is uniform, with a value of µc/m 3, with a value given by ρ = bx 2, where b = µc/m 5? (b) How many excess electrons are on the rod if ρ is nonuniform, with a value given by ρ = bx 2, where b = μc/m 5? b and c tie, then a (zero) (a) 2.00 x electrons (b) 1.33 x electrons

20 128 In figure a, two identical, electrically isolated conducting spheres A and B are separated by a (center-to-center) distance a that is large compared to the spheres. Sphere A has a positive charge of +Q, and sphere B is electrically neutral. Initially, there is no electrostatic force between the spheres. (Assume that there is no induced charge on the spheres because of their large separation.) (a) Suppose the spheres are connected for a moment by a conducting wire. The wire is thin enough so that any net charge on it is negligible. What is the electrostatic force between the spheres after the wire is removed? (b) Next, suppose sphere A is grounded momentarily, and then the ground connection is removed. What now is the electrostatic force between the spheres? (a) (b) 129 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 next makes contact with sphere A and is then moved far away. (a) F AB = kq 2 /R 2 away from B (b) F AB = kq 2 /2R 2 away from B (c) F AB = 3kQ 2 /8R 2 away from B