ADDITIONS TO Y&F PROBLEMS AND PROBLEMS FROM OTHER TEXTS JULY 6 - JULY 14 YOUNG - CHAPTER 21 60. The potassium chloride molecule (KCl) has a dipole moment of 8.9 x 10^-30 Cm. (a) Assuming that this dipole moment arises from two charges, each of magnitude 1.6 x 10^-19 C, separated by distance d, calculate d. (b) What is the maximum magnitude of the torque that a uniform electric field with magnitude 6.0 x 10^5 N/C can exert on a KCl molecule? Sketch the relative orientations of the electric dipole moment p (p is a vector) and the electric field E (E is a vector) when the torque is a maximum. (c) Suppose that the electric field of part (b) points in the +y direction. If the dipole swings from an initial orientation pointing in a direction 20 degrees CCW away from the -x axis to a final orientation pointing in a direction 20 degrees CW away from the -x axis, what is the change in the dipole's electric potential energy? 78. (a) Suppose all the electrons in 20.0 g of carbon atoms were located at the North Pole of the earth and all the protons at the South Pole. What would be the total force of attraction exerted on each group of charges by the other? The atomic number of carbon is 6, and the atomic mass of carbon is 12 g/mol. (b) What would be the magnitude and direction of the force exerted by the charges in part (a) on a third charge that is equal to the charge at the South Pole, and located at a point of the surface of the earth at the equator? Draw a diagram showing the locations of the charges and the forces on the charge at the equator. Young and Freedman - 12th Ed - CHAPTER 22 30. Do not use Gauss' Law for any part of this problem. We have already found the result for the E field due to a infinitely thin sheet of charge in two different ways; you may simply use that result here. First consider each of the four infinitely thin sheets of charge separately. Then at each of the locations A, B, and C, add the vector contributions to the field from each of the four infinitely thin sheets. Show your work carefully. GBA 54. Omit part (a) of this problem. To find the E field strength within the slab, use a Gaussian pillbox which is centered on the center of the slab, and which has its parallel faces equidistant from the central plane of the slab. With such a Gaussian pillbox, the symmetry argument in part (a) is not required; instead a simpler symmetry argument about the E field direction within the slab will suffice. You must give your symmetry argument clearly for this problem. GBA YOUNG - CHAPTER 22 30. This is essentially problem 22:32 in Y&F 12e but with a nonuniform electric field. The x-component of the E field is given by (-5.00 N/(Cm))x. The y-component of the E field is zero. The z-component of the E field is given by (+3.00 N/(Cm))z. The value of L is 0.30 m. (a) Find the electric flux through each of the six cube faces S_1 through S_6. (b) In the uniform field case, the electric flux through the entire cube would have been zero (what goes in must come out); but, in this case you will find that the electric flux through the entire cube is not zero. Find the total electric charge inside the cube. 44. A small, INSULATING, spherical shell with inner radius a and outer radius b is concentric with a larger INSULATING spherical shell with inner radius c and outer radius d (Figure 22.39 in Y&F 12e). The inner shell has total charge +q distributed uniformly over its volume, and the outer shell has charge -q distributed uniformly over its volume. (a) Calculate the charge densities in the inner shell and the outer shell. (b) Calculate the electric field (magnitude and direction) in terms of q and the distance r from the common center of the two shells for (i) r<a; (ii) a<r<b; (iii) b<r<c; (iv) c<r<d; (v) r>d. (c) Show your results in a graph of the radial component of E (E is a vector) as a function of the distance r. 45. A long coaxial cable consists of an inner cylindrical conductor with radius a and an outer coaxial cylinder (also conducting) with inner radius b and outer radius c. The outer cylinder is mounted on insulating supports and has no net charge. The inner cylinder has a uniform positive charge per unit length lambda. Calculate the electric field (a) at any point between the cylinders, a distance r from the axis; (b) at any point outside the outer cylinder. (c) Sketch the magnitude of the electric field as a function of the distance r from the axis of the cable, from r=0 to r=2c. (d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder. 48. A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume rho. (a) Derive the expression for the electric field inside the volume at a distance r from the axis of the cylinder in terms of the charge density rho. (b) What is the electric field at a point outside the volume in terms of the charge per unit length labmda in the cylinder? (c) Compare the answers to parts (a) and (b) for r=R. (d) Graph the electric field magnitude as a function of r from r=0 to r=3R. (P prefixes) HRW Problem Supplement #1 - CHAPTER 22 26. Calculate the number of coulombs of positive charge in 250 cm^3 of (neutral) water (about a glassful). Young and Freedman - 12th Ed - CHAPTER 23 50. To get any credit for this problem, you MUST prove any result that you take from problem 23.49. Please do so by first using Gauss' Law to find the electric field at all points in space, then integrating the electric field over the appropriate displacement to find the desired potential or potential difference. 74. In part (a) of this question, Y&F ask "What will a voltmeter read ....". This is not a serious question as asked. The shell is insulating and all voltmeters measure current (as you will learn in section 26.3). No current will flow in this case, so the voltmeter will always read zero. So please change the wording of this phrase to "What is the absolute value of the potential difference between the following points?" For parts (b) and (c), disregard the book and follow the instructions here; there is an additional part (d) here. (b) Sketch a graph of electric potential versus radial position, setting V=0 at infinity, and labeling the positions a, b, and c on your position axis. (c) Redraw the graph for the case when the surface charge is negative instead of positive. (d) Redraw the graph for the case when the insulating material is a solid ball (instead of a shell) of radius 60 cm, and the +150 microC is distributed uniformly throughout the volume of the insulating material rather than only on the outer surface. YOUNG - CHAPTER 23 48. Three small spheres with charge 2.00 microC are arranged in a line, with sphere 2 in the middle. Adjacent spheres are initially 8.0 cm apart. The spheres have masses m1=20.0 g, m2=85.0 g, and m3=20.0 g, and their radii are much smaller than their separation. The three spheres are released from rest. (a) What is the acceleration of sphere 1 just after it is released? (b) What is the speed of each sphere when they are far apart? WOLFSON - CHAPTER 23 16. A charge 3q is at the origin, and a charge -2q is on the positive x axis at x = a. Where would you place a third charge so it would experience no net electric force? 22. Three identical charges +q and a fourth charge -q form a square of side a. (a) Find the magnitude of the electric force on a charge Q placed at the center of the square. (b) Describe the direction of this force. 32. A 1.0 microC charge and a 2.0 microC charge are 10 cm apart. (a) Find a point where the net electric field is zero. (b) Sketch the net electric field lines qualitatively. 34. A positive charge +2q lies on the x axis at x = -a, and a charge -q lies at x = +a. (a) Find an expression for the electric field as a function of x for points to the right of the charge -q. (b) Taking q = 1.0 microC and a = 1.0 m, plot the field as a function of position for x = 5.0 m to x = 25 m. 50. A seimcircular loop of radius a carries positive charge Q distributed uniformly over its length (the loop begins at the point (0,a) and extends counterclockwise to the point (0,-a)). Find the electric field at the center of the loop. Show all the steps in your work. 54. A thin rod of length L carries charge Q distributed uniformly over its length (the rod begins at the point (-L/2,0) and extends rightward to the point (+L/2,0)). (a) What is the line charge density on the rod? (b) What must be the electric field direction on the rod's perpendicular bisector (i.e. the y axis). (c) Find an expression for the electric field at a point P at the location (0,+y). (d) Show that your result for (c) reduces to the field of a point charge Q for y >> L. 56. How strong an electric field is needed to accelerate electrons in an old-style TV tube from rest to one-tenth the speed of light in a distance of 5.0 cm? 58. An oscilloscope display requires that a beam of electrons moving at 8.2 Mm/s be deflected through an angle of 22 degrees by a uniform electric field that occupies a region 5.0 cm long. What should be the field strength? (P prefixes) HRW Problem Supplement #1 - CHAPTER 23 25. A semi-infinite nonconducting rod (i.e. infinite in one direction only) has uniform positive linear charge density lambda. Show that the electric field at point P makes an angle of 45 degrees with the rod and that this result is independent of the distance R. (HINT: Separately find the parallel and perpendicular (to the rod) components of the electric field at P, and then compare those components.) rod starts here _ +++++++++++++++++++++++++ =====> very long | | | R | | | | - P <== this point is a distance R from the end of the rod 37. In Millikan's experiment, an oil drop of radius 1.64 microns and density 0.851 g/cm^3 is suspended in the experimental chamber when a downward-directed electric field of 1.92 X 10^5 N/C is applied. Find the charge on the drop, in terms of e. 55. 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. (P prefixes) HRW Problem Supplement #1 - CHAPTER 24 24. A charge of uniform linear density 2.0 nC/m is distributed along a long, thin, nonconducting rod. The rod is coaxial with a long, hollow, conducting cylinder (inner radius = 5.0 cm, outer radius = 10 cm). The net charge on the conductor is zero. (a) What is the magnitude of the electric field 15 cm from the axis of the cylinder? What is the surface charge density on (b) the inner surface and (c) the outer surface of the conductor? 66. A metallic spherical shell of inner radius a1 and outer radius a2 has charge Qa. Concentric with it is another metallic spherical shell of inner radius b1 (where b1 > a2) and outer radius b2; shell b has charge Qb. Both Qa and Qb are positive. Find the electric field strength at points a distance r from the common center, where (a) r < a1, (b) a2 < r < b1, and (c) r > b2. (d) Determine how the charges are distributed on the inner and outer surfaces of the shells (give the surface charge densities). (e) Sketch the electric field strength as a function of r over the range r=0 to r=2*b2. GIANCOLI - CHAPTER 23 11. A uniform electric field of strength 4200 N/C points in the negative x direction. Point A is at location (4.00 m, 1.00 m), B is at (4.00 m, 4,00 m), and C is at (-3.00 m, 4.00 m). A charge of 28.0 nC is moving within this uniform field. What work is done by the electric force when the charge moves (a) from A to B, (b) from B to C, (c) from C to A. (d) If point A is assigned as the location where the electric potential is zero, what is the electric potential at location B and at location C? (e) Location D is 3.68 m from location A in a direction of 210 degrees measured counterclockwise from the positive x direction. What work is done by the electric force when the 28.0 nC charge moves from A to D, and what is the electric potential at point D? WOLFSON - CHAPTER 24 24. A thin spherical shell of radius 15 cm carries 4.8 microC, distributed uniformly over its surface. At the center of the shell is a point charge. (a) If the electric field at the surface of the sphere is 750 kN/C and points outward, what is the charge of the point charge? (b) What is the field just inside the shell? 56. A point charge -q is at the center of a insulating thin spherical shell carrying charge -(3/2)q. That shell, in turn, is concentric with a larger shell (also insulating and also thin) carrying charge +2q. (a) Draw a cross section of this structure, and sketch the electric field lines using the convention that 8 lines correspond to a charge of magnitude q. (b) If the shells were metallic instead of insulating, how would the field lines that you have drawn be different, and what would be the total charge on each of the inner and outer surfaces of each of the two shells? FISHBANE - CHAPTER 24 38. Two large, thin, metallic plates are placed parallel to each other, separated by 15 cm. The top plate carries a uniform charge density of 24 microC/m^2, while the bottom plate carries a uniform charge density of -38 microC/m^2. What is the electric field (magnitude and direction) (a) halfway between the plates? (b) above the two plates? (c) below the two plates? (d) What are the surface charge densities on the top and bottom surfaces of both plates? 45. A metal sphere of radius a is surrounded by a metal shell of inner radius b and outer radius R. The flux through a spherical Gaussian surface located between a and b is Q/epsilon0, and the flux through a spherical Gaussian surface just outside radius R is 2Q/epsilon0. (a) What are the total charges on the inner sphere and on the shell? (b) Where are the charges located, and (c) what are the charge densities? Halliday and Resnick - 2nd Edition - CHAPTER 25 1. Water in an irrigation ditch of width w = 3.22 m and depth d = 1.04 m flows with a speed of 0.207 m/s. The mass flux of the flowing water through an imaginary surface is the product of the water's density (1000 kg/m^3) and its volume flux through that surface. Find the mass flux through the following imaginary surfaces: (a) a surface of area wd, entirely in the water, perpendicular to the flow; (b) a surface with area 3wd/2, of which wd is in the water, perpendicular to the flow; (c) a surface of area wd/2, entirely in the water, perpendicular to the flow; (d) a surface of area wd, half in the water and half out, perpendicular to the flow; (e) a surface of area wd, entirely in the water, with its normal 34 degrees from the direction of flow. (P prefixes) HRW Problem Supplement #1 - CHAPTER 25 58. Consider a flat, nonconducting ring of outer radius R and inner radius r = 0.200R; the ring has a uniform charge per unit area of sigma. With V = 0 at infinity, find an expression for the electric potential at point P on the central axis of the ring, at a distance z = 2.00R from the center of the ring. 88. Three particles with the same charge q and same mass m are initially fixed in place to form an equilateral triangle with edge lengths d. (a) If the particles are released simultaneously, what are their speeds when they have traveled a large distance (effectively an infinite distance) from each other? (Measure the speeds in the original rest frame of the particles.) Suppose, instead, the particles are released one at a time: The first one is released, and then, when the first one is at a large distance, a second one is released, and then, when that second one is at a large distance, the last one is released. What then are the final speeds of (b) the first particle, (c) the second particle, and (d) the last particle? WOLFSON - CHAPTER 25 26. A solid insulating sphere of radius R carries a net charge Q distributed uniformly throughout its volume. (a) Show how to use Gauss' Law to find the electric field strength (as a function of radius) inside the sphere. (b) Find the absolute value of the potential difference from the sphere's surface to its center. (c) Which is at the higher potential, the surface or the center, if Q is positive? If Q is negative? (d) If Q is positive, what is the potential at the surface if zero potential is chosen to be at the center of the sphere? If zero potential is chosen to be at infinity? 35. A hollow, spherical conducting shell of inner radius b and outer radius c surrounds and is concentric with a solid conducting sphere of radius a (a < b). The sphere carries a net charge -Q and the shell carries a net charge +3Q. Both conductors are in electrostatic equilibrium. (a) In terms of the radial distance r as measured from the center of the sphere, find the electric potential at all points in space. Set V=0 at r=infinity. (b) Make a graph of the electric potential versus r. 50. A charge +4q is located at the origin and a charge -q is on the x axis at x = a. (a) Write an expression for the potential on the x axis for x > a. (b) Find a point in this region where V = 0. (c) Use the result of (a) to find the electric field on the x axis for x > a, and (d) find a point where E = 0. 58. Two small metal spheres are located 2.0 m apart. One has radius 0.50 cm and carries 0.20 microC. The other has radius 1.0 cm and carries 0.080 microC. (a) What is the potential difference between the spheres? (b) If they were connected by a thin wire, how much charge would move along it, and in what direction? WOLFSON - CHAPTER 26 3. Four 50 microC charges are brought from far apart onto a line where they are spaced at 2.0 cm intervals. How much work does it take to assemble this charge distribution? Halliday and Resnick - 2nd Ed - CHAPTER 26 19. A positive charge per unit length lambda is distributed uniformly along a straight-line segment of length L. (a) Determine the potential (chosen to be zero at infinity) at a point P a distance y from one end of the charged segment and in line with it. (b) Use the result of (a) to compute the component of the electric field at P in the y-direction (along the line). (c) Determine the component of the electric field at P in a direction perpendicular to the straight line. - . P | | y | | - + Figure for H 26:19 | + | + | + the line of plus marks L + stands for the line of | + positive charge | + | + - + 20. On a thin rod of length L lying along the x-axis with one end at the origin (x=0), there is distributed a positive charge per unit length given by lambda = cx, where c is a constant. (a) Taking the electrostatic potential at infinity to be zero, find V at the point P on the y-axis. (b) Determine the vertical component E_y of the electric field at P from the result of part (a). (c) Why cannot E_x, the horizontal component of the electric field at P, be found using the result of part (a)? y axis | | Figure for H 26:20 P is on the | y axis, a . P distance y | from the | the line of plus marks origin | stands for the line of | positive charge | --------+++++++++++++++++++------ x axis (0,0)|<------ L ------> | | A Prefixes A1: An isolated copper sphere of radius 16 cm is found to have an electric field at its surface which is radially inward (i.e. pointing towards the center of the sphere) and which has a strength 1150 N/C. (a) How many excess electrons are on this copper sphere? (b) A square made up of nine copper atoms (3 atoms by 3 atoms) takes up an area of about one square nanometer. Approximately how many copper atoms are on the surface of this copper sphere? (c) On the surface of the copper sphere, there are about how many copper atoms per excess electron? A2: Consider a thin disk, of negligible thickness, of radius R which is oriented perpendicular to the x axis such that the x axis runs through the center of the disk. The disk is centered at x=0, and has positive charge density sigma. (a) Show that the electric field on the positive x axis is given by E_x = (sigma/(2epsilon0))*(1-1/sqrt(1+(R/x)^2)) (b) Show how to apply the binomial expansion to your result for (a) in order to demonstrate that the commonly-used expression, E=sigma/(2epsilon0), is valid to within 1.0% for all values of x which are 1.0% of R or less. A3: Assume that water flowing through a pipe with a circular cross section flows most rapidly at the center of the pipe and least rapidly near the pipe walls (this might be reasonable because of friction between the water and the pipe walls). Assume that the water speed at the walls is half as great as the speed at the center, and assume that the decrease is linear (i.e. the graph of water speed versus radius is a straight line). If the radius of the pipe is 10 cm and if the volume flux of water is 5.0 liters per second, find the speed of the water at the center of the pipe. (One liter is 1000 cubic centimeters or 0.001 cubic meters). HINT: Draw the graph of water speed versus distance from the centerline of the pipe; you must do the integral for volume flux symbolically. A4: A thin wooden stick of length 12 cm has a tiny metal sphere glued to each end. A charge of +3 microC is placed on one sphere and a charge of -2 microC is placed on the other. The center of mass is located 7 cm from the positively- charged sphere. The system is mounted on a fixed horizontal E-W axle passing through the center of mass about which the system is free to rotate with no friction. When the system is then placed in a horizontal uniform southward electric field of 800 N/C, the resulting equilibrium position of the system is horizontal with the positive charge due S of the negative charge. (a) When the system is in its equilibrium position, what is the horizontal force on the system by the axle? (b) What amount of torque (about the axle) is required to hold the system at an angular displacement of 25 degrees away from the equilibrium position? (c) What minimum amount of work must be done to move the system from its equilibrium position to an angular displacement of 25 degrees? (HINT: write the needed torque as a function of displacement, then integrate from zero to 25 degrees to find the needed work.) (d) What work would be done by the electric field during the displacement described in part (c)? A5: Two conductors, A and B, are each in the shape of a tetrahedron, but of different sizes. They are charged in the following manner: 1. Tetrahedron A is charged from an electrostatic generator to charge q. 2. Tetrahedron A is briefly touched to tetrahedron B. 3. Steps 1 and 2 are repeated until the charge on tetrahedron B reaches a maximum value; i.e. the process is repeated until further repetitions will no longer result in the transfer of additional charge from tetrahedron A to tetrahedron B. If the charge on tetrahedron B was q/5 after the first time it touched tetrahedron A, what is the final charge on tetrahdedron B? HINT: the way to make progress is to think carefully about the first and second transfers, and then about the final attempted transfer (in which no charge is actually transferred). Be sure to express your logic clearly and thoroughly when writing up this question.