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All Textbook Solutions for Physics for Scientists and Engineers with Modern Physics

Suppose a point charge is located at the center of a spherical surface. The electric field at the surface of the sphere and the total flux through the sphere are determined. Now the radius of the sphere is halved. What happens to the flux through the sphere and the magnitude of the electric field at the surface of the sphere? (a) The flux and field both increase. (b) The flux and field both decrease. (c) The flux increases, and the field decreases. (d) The flux decreases, and the field increases, (e) The flux remains the same, and the field increases, (f) The flux decreases, and the field remains the same.If the net flux through a gaussian surface is zero, the following four statements could be true. Which of the statements must be true? (a) There are no charges inside the surface. (b) The net charge inside the surface is zero. (c) The electric field is zero everywhere on the surface. (d) The number of electric field lines entering the surface equals the number leaving the surface.A negatively charged rod of finite length carries charge with a uniform charge per unit length. Sketch the electric field lines in a plane containing the rod.A positively charged disk has a uniform charge per unit area as described in Example 23.3. Sketch the electric field lines in a plane perpendicular to the plane of the disk passing through its center.A uniformly charged ring of radius 10.0 cm has a total charge of 75.0 C. Find the electric field on the axis of the ring at (a) 1.00 cm, (b) 5.00 cm, (c) 30.0 cm, and (d) 100 cm from the center of the ring.The electric field along the axis of a uniformly charged disk of radius R and total charge Q was calculated in Example 23.3. Show that the electric field at distances x that are large compared with R approaches that of a particle with charge Q = R2. Suggestion: First show that x/(x2 + R2)1/2 = (1 + R2/x2)1/2 and use the binomial expansion (1 + )n = 1 + n, when 1.Example 23.3 derives the exact expression for the electric field at a point on the axis of a uniformly charged disk. Consider a disk of radius R = 3.00 cm having a uniformly distributed charge of +5.20 C. (a) Using the result of Example 23.3, compute the electric field at a point on the axis and 3.00 mm from the center. (b) What If? Explain how the answer to part (a) compares with the field computed from the near-field approximation E = /20. (We derived this expression in Example 23.3.) (c) Using the result of Example 23.3, compute the electric field at a point on the axis and 30.0 cm from the center of the disk. (d) What If? Explain how the answer to part (c) compares with the electric field obtained by treating the disk as a +5.20-C charged particle at a distance of 30.0 cm.A uniformly charged rod of length L and total charge Q lies along the x axis as shown in Figure P23.6. (a) Find the components of the electric field at the point P on the y axis a distance d from the origin. (b) What are the approximate values of the field components when d L? Explain why you would expect these results. Figure P23.6 A continuous line of charge lies along the x axis, extending from x = +x0 to positive infinity. The line carries positive charge with a uniform linear charge density 0. What are (a) the magnitude and (b) the direction of the electric field at the origin?A thin rod of length and uniform charge per unit length lies along the x axis as shown in Figure P23.8. (a) Show that the electric field at P, a distance d from the rod along its perpendicular bisector, has no x component and is given by E = 2k sin 0/d. (b) What If? Using your result to part (a), show that the field of a rod of infinite length is E = 2k/d. Figure P23.8 (a) Consider a uniformly charged, thin-walled, right circular cylindrical shell having total charge Q, radius R, and length . Determine the electric field at a point a distance d from the right side of the cylinder as shown in Figure P23.9. Suggestion: Use the result of Example 23.2 and treat the cylinder as a collection of ring charges. (b) What If? Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly distributed through its volume. Use the result of Example 23.3 to find the field it creates at the same point. Figure P23.9 A vertical electric field of magnitude 2.00 104 N/C exists above the Earths surface on a day when a thunderstorm is brewing. A car with a rectangular size of 6.00 m by 3.00 m is traveling along a dry gravel roadway sloping downward at 10.0. Determine the electric flux through the bottom of the car.A flat surface of area 3.20 m2 is rotated in a uniform electric field of magnitude E = 6.20 105 N/C. Determine the electric flux through this area (a) when the electric field is perpendicular to the surface and (b) when the electric field is parallel to the surface.A nonuniform electric field is given by the expression E=ayi+bzj+cxk where a, b, and c are constants. Determine the electric flux through a rectangular surface in the xy plane, extending from x = 0 to x = w and from y = 0 to y = h.An uncharged, nonconducting, hollow sphere of radius 10.0 cm surrounds a 10.0-C charge located at the origin of a Cartesian coordinate system. A drill with a radius of 1.00 mm is aligned along the z axis, and a hole is drilled in the sphere. Calculate the electric flux through the hole.Find the net electric flux through the spherical closed surface shown in Figure P23.14. The two charges on the right are inside the spherical surface. Figure P23.14 Four closed surfaces, S1 through S4 together with the charges 2Q, Q, and Q are sketched in Figure P23.15. (The colored lines are the intersections of the surfaces with the page.) Find the electric flux through each surface. Figure P23.15 A charge of 170 C is at the center of a cube of edge 80.0 cm. No other charges are nearby. (a) Find the flux through each face of the cube. (b) Find the flux through the whole surface of the cube. (c) What If? Would your answers to either part (a) or part (b) change if the charge were not at the center? Explain.(a) Find the net electric flux through the cube shown in Figure P23.17. (b) Can you use Gausss law to find the electric field on the surface of this cube? Explain. Figure P23.17 A particle with charge of 12.0 C is placed at the center of a spherical shell of radius 22.0 cm. What is the total electric flux through (a) the surface of the shell and (b) any hemispherical surface of the shell? (c) Do the results depend on the radius? Explain.A particle with charge Q = 5.00 C is located at the center of a cube of edge L = 0.100 m. In addition, six other identical charged particles having q = 1.00 C are positioned symmetrically around Q as shown in Figure P23.19. Determine the electric flux through one face of the cube. Figure P23.19 Problems 19 and 20.20P 21P Find the net electric flux through (a) the closed spherical surface in a uniform electric field shown in Figure P23.22a and (b) the closed cylindrical surface shown in Figure P23.22b. (c) What can you conclude about the charges, if any, inside the cylindrical surface? Figure P23.22 Figure P23.23 represents the top view of a cubic gaussian surface in a uniform electric field E oriented parallel to the top and bottom faces of the cube. The field makes an angle with side , and the area of each face is A. In symbolic form, find the electric flux through (a) face , (b) face , (c) face , (d) face , and (e) the top and bottom faces of the cube. (f) What is the net electric flux through the cube? (g) How much charge is enclosed within the gaussian surface? Figure P23.23 Determine the magnitude of the electric field at the surface of a lead-208 nucleus, which contains 82 protons and 126 neutrons. Assume the lead nucleus has a volume 208 times that of one proton and consider a proton to be a sphere of radius 1.20 1015 m.25P 26P A large, flat, horizontal sheet of charge has a charge per unit area of 9.00 C/m2. Find the electric field just above the middle of the sheet.A nonconducting wall carries charge with a uniform density of 8.60 C/cm2. (a) What is the electric field 7.00 cm in front of the wall if 7.00 cm is small compared with the dimensions of the wall? (b) Does your result change as the distance from the wall varies? Explain.A uniformly charged, straight filament 7.00 m in length has a total positive charge of 2.00 C. An uncharged cardboard cylinder 2.00 cm in length and 10.0 cm in radius surrounds the filament at its center, with the filament as the axis of the cylinder. Using reasonable approximations, find (a) the electric field at the surface of the cylinder and (b) the total electric flux through the cylinder.You are working on a laboratory device that includes a small sphere with a large electric charge Q. Because of this charged sphere, there is a strong electric field surrounding your device. Other researchers in your laboratory are complaining that your electric field is affecting their equipment. You think about how you can obtain the large electric field that you need close to the sphere but prohibit the field from reaching your colleagues. You decide to surround your device with a spherical transparent plastic shell. The nonconducting shell is given a uniform charge distribution. (a) The shell is placed so that the small sphere is at the exact center of the shell. Determine the charge that must he placed on the shell to completely eliminate the electric field outside of the shell. (b) What if the shell moves? Does the small sphere have to be at the center of the shell for this scheme to work?Consider a long, cylindrical charge distribution of radius R with a uniform charge density . Find the electric field at distance r from the axis, where r R.Assume the magnitude of the electric field on each face of the cube of edge L = 1.00 m in Figure P23.32 is uniform and the directions of the fields on each face are as indicated. Find (a) the net electric flux through the cube and (b) the net charge inside the cube. (c) Could the net charge he a single point charge? Figure P23.32 A solid sphere of radius 40.0 cm has a total positive charge of 26.0 C uniformly distributed throughout its volume. Calculate the magnitude of the electric field (a) 0 cm, (b) 10.0 cm, (c) 40.0 cm, and (d) 60.0 cm from the center of the sphere.A cylindrical shell of radius 7.00 cm and length 2.40 m has its charge uniformly distributed on its curved surface. The magnitude of the electric field at a point 19.0 cm radially outward from its axis (measured from the midpoint of the shell) is 36.0 kN/C. Find (a) the net charge on the shell and (b) the electric field at a point 4.00 cm from the axis, measured radially outward from the midpoint of the shell.You are working for the summer at a research laboratory. Your research director has devised a scheme for holding small charged particles at fixed positions. The scheme is shown in Figure P23.35. A large insulating sphere of radius a carries a total positive charge Q with a uniform volume charge density. A very thin tunnel is drilled through a diameter of the sphere and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on either side of the center of the sphere. Your research director has had great success with this scheme. (a) Determine the specific value of r at which equilibrium exists. (b) Your research director asks you to see if he can extend the system as follows. Determine if it is possible to add transparent plastic tubes as extensions of the tunnel and have the small spheres be in equilibrium at a position for which r a. Figure P23.35 You are working for the summer at a research laboratory. Your research director has devised a scheme for holding small charged particles at fixed positions. The scheme is shown in Figure P23.36. An insulating cylinder of radius a and length L a is positively charged and carries a uniform volume charge density . A very thin tunnel is drilled through a diameter of the cylinder and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on opposite sides of the axis of the cylinder. Your research director has had great success with this scheme. (a) Determine the specific value of rat which equilibrium exists. (b) Your research director asks you see if he can extend the system as follows. Determine if it is possible to add transparent plastic tubes as extensions of the tunnel and have the small spheres be in equilibrium at a position for which r a. Figure P23.36 Find the electric flux through the plane surface shown in Figure P23.37 if = 60.0, E = 350 N/C, and d = 5.00 cm. The electric field is uniform over the entire area of the surface. Figure P23.37 38AP 39AP Show that the maximum magnitude Emax of the electric field along the axis of a uniformly charged ring occurs at x=a/2 (see Fig. 23.3) and has the value Q/(630a2). Figure 23.3 (Example 23.2) A uniformly charged ring of radius c. (a) The field at P on the x axis due to an element of charge dq. (b) The perpendicular component of the field at P due to segment 1 is canceled by the perpendicular component due to segment 2.A line of positive charge is formed into a semicircle of radius R = 60.0 cm as shown in Figure P23.41. The charge per unit length along the semicircle is given by the expression = 0 cos . The total charge on the semicircle is 12.0 C. Calculate the total total on a charge of 3.00 C placed at the center of curvature P. Figure P23.41 42AP A sphere of radius R = 1.00 m surrounds a particle with charge Q = 50.0 C located at its center as shown in Figure P23.43. Find the electric flux through a circular cap of half-angle = 45.0. Figure P23.43 A sphere of radius R surrounds a particle with charge Q located at its center as shown in Figure P23.43. Find the electric flux through a circular cap of half-angle . Figure P23.43 A slab of insulating material has a nonuniform positive charge density = Cx2, where x is measured from the center of the slab as shown in Figure P23.45 and C is a constant. The slab is infinite in the y and z directions. Derive expressions for the electric field in (a) the exterior regions (|x| d/2) and (b) the interior region of the slab (d/2 x d/2). Figure P23.45 A sphere of radius 2a is made of a nonconducting material that has a uniform volume charge density . Assume the material does not affect the electric field. A spherical cavity of radius a is now removed from the sphere as shown in Figure P23.46. Show that the electric field within the cavity is uniform and is given by Ex 0 and Ey a/3c0. Figure P23.46 47CP 48CP Review. A slab of insulating material (infinite in the y and z directions) has a thickness d and a uniform positive charge density . An edge view of the slab is shown in Figure P23.45. (a) Show that the magnitude of the electric field a distance x from its center and inside the slab is E = x/0. (b) What If? Suppose an electron of charge e and mass me can move freely within the slab. It is released from rest at a distance x from the center. Show that the electron exhibits simple harmonic motion with a frequency f=12eme0 Figure P23.45 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 (Fig. P23.30). Show that the magnitude of the force exerted by the left rod on the right one is F=(keQ24a2)ln(b2b24a2) Figure P23.50 A solid insulating sphere of radius R has a nonuniform charge density that varies with r according to the expression = Ar2, where A is a constant and r R is measured from the center of the sphere. (a) Show that the magnitude of the electric field outside (r R) the sphere is E = AR5/50r2. (b) Show that the magnitude of the electric field inside (r R) the sphere is E = Ar3/50. Note: The volume element dV for a spherical shell of radius r and thickness dr is equal to 4r2dr.two points and are located within a region in which there is an electric field. (i) How would you describe the potential difference V = V V? (a) It is positive. (b) It is negative. (c) It is zero. (ii) A negative charge is placed at and then moved to . How would you describe the change in potential energy of the chargefield system for this process? Choose from the same possibilities. Figure 24.1 (Quick Quiz 24.1) Two points in an electric field.QUICK QUIZ 24.2 The labeled points in Figure 24.4 are on a series of equipotential surfaces associated with an electric field. Rank (from greatest to least) the work done by the electric field on a positively charged particle that moves from to , from to , from to , and from to .In Figure 24.8b, take q2, to be a negative source charge and q1 to be a second charge whose sign can be changed. (i) If q1 is initially positive and is changed to a charge of the same magnitude but negative, what happens to the potential at the position of q1 due to q2? (a) It increases. (b) It decreases. (c) It remains the same. (ii) When q1 is changed from positive to negative, what happens to the potential energy of the two-charge system? Choose from the same possibilities.In a certain region of space, the electric potential is zero everywhere along the x axis. (i) From this information, you can conclude that the x component of the electric field in this region is (a) zero, (b) in the positive x direction, or (c) in the negative x direction, (ii) Suppose the electric potential is +2 V everywhere along the x axis. From the same choices, what can you conclude about the x component of the electric field now?How much work is done (by a battery, generator, or some other source of potential difference) in moving Avogadros number of electrons from an initial point where the electric potential is 9.00 V to a point where the electric potential is 5.00 V? (The potential in each case is measured relative to a common reference point.)(a) Find the electric potential difference Ve required to stop an electron (called a stopping potential) moving with an initial speed of 2.85 107 m/s. (b) Would a proton traveling at the same speed require a greater or lesser magnitude of electric potential difference? Explain. (c) Find a symbolic expression for the ratio of the proton stopping potential and the electron stopping potential. Vp/Ve.Oppositely charged parallel plates are separated by 5.33 mm. A potential difference of 600 V exists between the plates. (a) What is the magnitude of the electric field between the plates? (b) What is the magnitude of the force on an electron between the plates? (c) How much work must be done on the electron to move it to the negative plate if it is initially positioned 2.90 mm from the positive plate?Starting with the definition of work, prove that at every point on an equipotential surface, the surface must be perpendicular to the electric field there.An insulating rod having linear charge density = 40.0 C/m and linear mass density = 0.100 kg/m is released from rest in a uniform electric field E = 100 V/m directed perpendicular to the rod (Fig. P24.5). (a) Determine the speed of the rod after it has traveled 2.00 m. (b) What If? How does your answer to part (a) change if the electric field is not perpendicular to the rod? Explain. Figure P24.5 Review. A block having mass m and charge + Q is connected to an insulating spring having a force constant k. The block lies on a frictionless, insulating, horizontal track, and the system is immersed in a uniform electric field of magnitude E directed as shown in Figure P24.6. The block is released from rest when the spring is unstretched (at x = 0). We wish to show that the ensuing motion of the block is simple harmonic. (a) Consider the system of the block, the spring, and the electric field. Is this system isolated or nonisolated? (b) What kinds of potential energy exist within this system? (c) Call the initial configuration of the system that existing just as the block is released from rest. The final configuration is when the block momentarily comes to rest again. What is the value of x when the block comes to rest momentarily? (d) At some value of x we will call x = x0, the block has zero net force on it. What analysis model describes the particle in this situation? (c) What is the value of x0? (f) Define a new coordinate system x such that x = x x0. Show that x satisfies a differential equation for simple harmonic motion. (g) Find the period of the simple harmonic motion. (h) How does the period depend on the electric field magnitude? Figure P24.6 Three positive charges are located at the corners of an equilateral triangle as in Figure P24.7. Find an expression for the electric potential at the center of the triangle. Figure P24.7 Two point charges Q1 = +5.00 nC and Q2 = 3.00 nC are separated by 35.0 cm. (a) What is the electric potential at a point midway between the charges? (b) What is the potential energy of the pair of charges? What is the significance of the algebraic sign of your answer?You are working on a laboratory device that includes a small sphere with a large electric charge Q. Because of this charged sphere, there is a strong electric field surrounding your device. Other researchers in your laboratory are complaining that your electric field is affecting their equipment. You think about how you can obtain the large electric field that you need close to the sphere but prohibit the field from reaching your colleagues. You decide to surround your device with a spherical transparent plastic shell of radius R. The plastic has a very thin coating of conducting material on the outside that only minimally reduces the transparency of the material. The shell is placed so that the small sphere is at the exact center of the shell. Determine to what electric potential the outer shell must be raised to completely eliminate the electric field outside of the shell.Your roommate is having trouble understanding why solids form. He asks, Why would atoms bond into solids rather than just floating freely with respect to each other? To help him understand at least one type of bonding in solids, you decide to embark on an energy explanation. You show him a drawing of a primitive cell of a sodium chloride crystal, NaCl, or simple table salt. The drawing is shown in Figure P24.10, where the orange spheres are Na+ ions and the blue spheres are Cl ions. Each ion has a charge of magnitude equal to the elementary charge e. The ions lie on the comers of a cube of side d. You explain to your roommate that the electrical potential energy is defined as zero when all eight charges are infinitely far apart from each other. Then you bring them together to form the crystal structure shown. (a) Evaluate the electric potential energy of the crystal as shown and (b) show that it is energetically favorable for such crystals to form. Figure P24.10 Four point charges each having charge Q are located at the corners of a square having sides of length a. Find expressions for (a) the total electric potential at the center of the square due to the four charges and (b) the work required to bring a fifth charge q from infinity to the center of the square.The two charges in Figure P24.12 are separated by a distance d = 2.00 cm, and Q = +5.00 nC. Find (a) the electric potential at A, (b) the electric potential at B, and (c) the electric potential difference between B and A. Figure P24.12 Show that the amount of work required to assemble four identical charged particles of magnitude Q at the corners of a square of side s is 5.41keQ2/s.Two charged particles of equal magnitude are located along the y axis equal distances above and below the x axis as shown in Figure P24.14. (a) Plot a graph of the electric potential at points along the x axis over the interval 3a x 3a. You should plot the potential in units of keQ/a. (b) Let the charge of the particle located at y = a be negative. Plot the potential along the y axis over the interval 4a y 4a. Figure P24.14 Three particles with equal positive charges q are at the corners of an equilateral triangle of side a as shown in Figure P24.15. (a) At what point, if any, in the plane of the particles is the electric potential zero? (b) What is the electric potential at the position of one of the particles due to the other two particles in the triangle? Figure P24.15 16P 17P 18P How much work is required to assemble eight identical charged particles, each of magnitude q, at the corners of a cube of side s?Four identical particles, each having charge q and mass m, are released from rest at the vertices of a square of side L. How fast is each particle moving when their distance from the center of the square doubles?It is shown in Example 24.7 that the potential at a point P a distance a above one end of a uniformly charged rod of length lying along the x axis is V=keQlln(l+a2+l2a) Use this result to derive an expression for the y component of the electric field at P.Figure P24.22 represents a graph of the electric potential in a region of space versus position x, where the electric field is parallel to the x axis. Draw a graph of the x component of the electric field versus x in this region. Figure P24.22 Figure P24.23 shows several equipotential lines, each labeled by its potential in volts. The distance between the lines of the square grid represents 1.00 cm. (a) Is the magnitude of the field larger at A or at B? Explain how you can tell. (b) Explain what you can determine about E at B. (c) Represent what the electric field looks like by drawing at least eight field lines. Figure P24.23 An electric field in a region of space is parallel to the x axis. The electric potential varies with position as shown in Figure P24.24. Graph the x component of the electric field versus position in this region of space. Figure P24.24 A rod of length L (Fig. P24.25) lies along the x axis with its left end at the origin. It has a nonuniform charge density = x, where is a positive constant. (a) What are the units of ? (b) Calculate the electric potential at A. Figure P24.25 Problems 25 and 26.For the arrangement described in Problem 25, calculate the electric potential at point B, which lies on the perpendicular bisector of the rod a distance b above the x axis.A wire having a uniform linear charge density is bent into the shape shown in Figure P24.27. Find the electric potential at point O. Figure P24.27 You are a coach for the Physics Olympics team participating in a competition overseas. You are given the following sample problem to present to your team of students, which you need to solve very quickly: A person is standing on the midline of a soccer field. At one end of the field, as shown in Figure P24.28, is a letter D, consisting of a semicircular metallic ring of radius R and a long straight metal rod of length 2R, the diameter of the ring. The plane of the ring is perpendicular to the ground and perpendicular to the midline of the field shown by the broken line in Figure P24.28. Because of an approaching lightning storm, the semicircular ring and the rod become charged. The ring and the rod each attain a charge Q. What is the electric potential at point P, which is at a position x along the midline of the field, measured from the center of the rod, due to the letter D? Think quickly and use all resources available to you, which include your physics textbook: yon are in competition! Figure P24.28 The electric field magnitude on the surface of an irregularly shaped conductor varies from 56.0 kN/C to 28.0 kN/C. Can you evaluate the electric potential on the conductor? If so, find its value. If not, explain why not.Why is the following situation impossible? A solid copper sphere of radius 15.0 cm is in electrostatic equilibrium and carries a charge of 40.0 nC. Figure P24.30 shows the magnitude of the electric field as a function of radial position r measured from the center of the sphere. Figure P24.30 A solid metallic sphere of radius a carries total charge Q. No other charges are nearby. The electric field just outside its surface is keQ/a2 radially outward. At this close point, the uniformly charged surface of the sphere looks exactly like a uniform flat sheet of charge. Is the electric field here given by /0 or by /20?32P A very large, thin, flat plate of aluminum of area A has a total charge Q uniformly distributed over its surfaces. Assuming the same charge is spread uniformly over the upper surface of an otherwise identical glass plate, compare the electric fields just above the center of the upper surface of each plate.34P 35P A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of , and the cylinder has a net charge per unit length of 2. From this information, use Gausss law to find (a) the charge per unit length on the inner surface of the cylinder, (b) the charge per unit length on the outer surface of the cylinder, and (c) the electric field outside the cylinder a distance r from the axis.37AP 38AP 39AP Why is the following situation impossible? You set up an apparatus in your laboratory as follows. The x axis is the symmetry axis of a stationary, uniformly charged ring of radius R = 0.500 m and charge Q = 50.0 C (Fig. P24.40). You place a particle with charge Q = 50.0 C and mass m = 0.100 kg at the center of the ring and arrange for it to be constrained to move only along the x axis. When it is displaced slightly, the particle is repelled by the ring and accelerates along the x axis. The particle moves faster than you expected and strikes the opposite wall of your laboratory at 40.0 m/s. Figure R24.40 The thin, uniformly charged rod shown in Figure P24.41 has a linear charge density . Find an expression for the electric potential at P.A GeigerMueller tube is a radiation detector that consists of a closed, hollow, metal cylinder (the cathode) of inner radius ra and a coaxial cylindrical wire (the anode) of radius rb (Fig. P24.42a). The charge per unit length on the anode is , and the charge per unit length on the cathode is . A gas fills the space between the electrodes. When the tube is in use (for example, in measuring radioactivity from fruit in Fig. P24.42b) and a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the electric potential difference between the wire and the cylinder is V2keln(rarb) (b) Show that the magnitude of the electric field in the space between cathode and anode is E=Vln(ra/rb)(1r) where r is the distance from the axis of the anode to the point where the field is to be calculated. Figure P24.42 Review. Two parallel plates having charges of equal magnitude but opposite sign are separated by 12.0 cm. Each plate has a surface charge density of 36.0 nC/m2. A proton is released from rest at the positive plate. Determine (a) the magnitude of the electric field between the plates from the charge density, (b) the potential difference between the plates, (c) the kinetic energy of the proton when it reaches the negative plate, (d) the speed of the proton just before it strikes the negative plate, (e) the acceleration of the proton, and (f) the force on the proton. (g) From the force, find the magnitude of the electric field. (h) How does your value of the electric field compare with that found in part (a)?When an uncharged conducting sphere of radius a is placed at the origin of an xyz coordinate system that lies in an initially uniform electric field E=E0k, the resulting electric potential is V(x, y, z) = V0 for points inside the sphere and V(x,y,z)=V0E0z+E0a3z(x2+y2+z2)3/2 for points outside the sphere, where V0 is the (constant) electric potential on the conductor. Use this equation to determine the x, y, and z components of the resulting electric field (a) inside the sphere and (b) outside the sphere.A solid, insulating sphere of radius a has a uniform charge density throughout its volume and a total charge Q. Concentric with this sphere is an uncharged, conducting, hollow sphere whose inner and outer radii are b and e as shown in Figure P24.45. We wish to understand completely the charges and electric fields at all locations. (a) Find the charge contained within a sphere of radius r a. (b) From this value, find the magnitude of the electric field for r a. (c) What charge is contained within a sphere of radius r when a r b? (d) From this value, find the magnitude of the electric field for r when a r b. (e) Now consider r when b r c. What is the magnitude of the electric field for this range of values of r? (f) From this value, what must be the charge on the inner surface of the hollow sphere? (g) From part (f), what must be the charge on the outer surface of the hollow sphere? (h) Consider the three spherical surfaces of radii a, b, and c. Which of these surfaces has the largest magnitude of surface charge density? Figure P24.45 Problems 43 and 47.46AP For the configuration shown in Figure P24.45, suppose a = 5.00 cm, b = 20.0 cm, and e = 25.0 cm. Furthermore, suppose the electric field at a point 10.0 cm from the center is measured to be 3.60 103 N/C radially inward and the electric field at a point 50.0 cm from the center is of magnitude 200 N/C and points radially outward. From this information, find (a) the charge on the insulating sphere, (b) the net charge on the hollow conducting sphere, (c) the charge on the inner surface of the hollow conducting sphere, and (d) the charge on the outer surface of the hollow conducting sphere. Figure P24.45 An electric dipole is located along the y axis as shown in Figure P24.48. The magnitude of its electric dipole moment is defined as p = 2aq. (a) At a point P, which is far from the dipole (r a), show that the electric potential is V=kepcosr2 (b) Calculate the radial component Er and the perpendicular component E of the associated electric field. Note that E = (1/r)(V/). Do these results seem reasonable for (c) = 90 and 0? (d) For r = 0? (e) For the dipole arrangement shown in Figure P24.48, express V in terms of Cartesian coordinates using r = (x2 + y2)1/2 and cos=y(x2+y2)1/2 (f) Using these results and again taking r a, calculate the field components Ex and Ey. Figure P24.48 A disk of radius R (Fig. P24.49) has a nonuniform surface charge density = Cr, where C is a constant and r is measured from the center of the disk to a point on the surface of the disk. Find (by direct integration) the electric potential at P. Figure P24.49 50CP (a) A uniformly charged cylindrical shell with no end caps has total charge Q, radius R, and length h. Determine the electric potential at a point a distance d from the right end of the cylinder as shown in Figure P24.51. Suggestion: Use the result of Example 24.5 by treating the cylinder as a collection of ring charges. (b) What If? Use the result of Example 24.6 to solve the same problem for a solid cylinder. Figure P24.51 A capacitor stores charge Q at a potential difference V. What happens if the voltage applied to the capacitor by a battery is doubled to 2 V? (a) The capacitance falls to half its initial value, and the charge remains the same. (b) The capacitance and the charge both fall to half their initial values. (c) The capacitance and the charge both double. (d) The capacitance remains the same, and the charge doubles.Many computer keyboard buttons are constructed of capacitors as shown in Figure 25.3. When a key is pushed down, the soft insulator between the movable plate and the fixed plate is compressed. When the key is pressed, what happens to the capacitance? (a) It increases. (b) It decreases. (c) It changes in a way you cannot determine because the electric circuit connected to the keyboard button may cause a change in V.Two capacitors are identical. They can be connected in series or in parallel. If you want the smallest equivalent capacitance for the combination, how should you connect them? (a) in series (b) in parallel (c) either way because both combinations have the same capacitance You have three capacitors and a battery. In which of the following combinations of the three capacitors is the maximum possible energy stored when the combination is attached to the battery? (a) series (b) parallel (c) no difference because both combinations store the same amount of energy If you have ever tried to hang a picture or a mirror, you know it can be difficult to locate a wooden stud in which to anchor your nail or screw. A carpenters stud finder is a capacitor with its plates arranged side by side instead of facing each other as shown in Figure 25.14. When the device is moved over a stud, does the capacitance (a) increase or (b) decrease?(a) When a battery is connected to the plates of a 3.00-F capacitor, it stores a charge of 27.0 C. What is the voltage of the battery? (b) If the same capacitor is connected to another batten and 36.0 C of charge is stored on the capacitor, what is the voltage of the battery?Two conductors having net charges of +10.0 C and 10.0 C have a potential difference of 10.0 V between them. (a) Determine the capacitance of the system. (b) What is the potential difference between the two conductors if the charges on each are increased to +100 C and 100 C?When a potential difference of 150 V is applied to the plates of a parallel-plate capacitor, the plates carry a surface charge density of 30.0 nC/cm2. What is the spacing between the plates?An air-filled parallel-plate capacitor has plates of area 2.30 cm2 separated by 1.50 mm. (a) Find the value of its capacitance. The capacitor is connected to a 12.0-V battery. (b) What is the charge on the capacitor? (c) What is the magnitude of the uniform electric field between the plates?5P Review. A small object of mass m carries a charge q and is suspended by a thread between the vertical plates of a parallel-plate capacitor. The plate separation is d. If the thread makes an angle with the vertical, what is the potential difference between the plates?Find the equivalent capacitance of a 4.20-F capacitor and an 8.50-F capacitor when they are connected (a) in series and (b) in parallel.8P A group of identical capacitors is connected first in series and then in parallel. The combined capacitance in parallel is 100 times larger than for the series connection. How many capacitors are in the group?Three capacitors are connected to a battery as shown in Figure P25.10. Their capacitances are C1 = 3C, C2 = C, and C3 = 5C. (a) What is the equivalent capacitance of this set of capacitors? (b) State the ranking of the capacitors according to the charge they store from largest to smallest. (c) Rank the capacitors according to the potential differences across them from largest to smallest. (d) What If? Assume C3 is increased. Explain what happens to the charge stored by each capacitor. Figure P25.10 Four capacitors are connected as shown in Figure P25.11. (a) Find the equivalent capacitance between points a and b. (b) Calculate the charge on each capacitor, taking Vab = 15.0 V. Figure P25.11 (a) Find the equivalent capacitance between points a and b for the group of capacitors connected as shown in Figure P25.12 (page 686). Take C1 = 5.00 F, C2 = 10.0 F, and C3 = 2.00 F. (b) What charge is stored on C3 if the potential difference between points a and b is 60.0 V? Figure P25.12 Find the equivalent capacitance between points a and b in the combination of capacitors shown in Figure P25.13. Figure P25.13 You are working at an electronics fabrication shop. Your current project is on the team producing capacitors for the timer circuit that delays the closing of an elevator door. According to its design specification, the timer circuit is to have a capacitance of 32.0 F between two points A and B. As sour capacitors come off the assembly line, you find that they have a variation of 5.00% from this value. After a team meeting to evaluate this situation, the team decides that capacitances in the range 32.0 0.5 F are acceptable and do not need modification. For capacitances outside this range, the director does not wish to discard the capacitors, but rather to add extra capacitors in series or parallel with the main capacitor to bring the total equivalent capacitance to the exact design value of 32.0 F. You are put in charge of procuring the extra capacitors. What range of capacitances for these extra capacitors do you need to cover the entire range of variation of 5.00%? All capacitances can be measured to three significant figures.Two capacitors give an equivalent capacitance of 9.00 pF when connected in parallel and an equivalent capacitance of 2.00 pF when connected in series. What is the capacitance of each capacitor?16P A 3.00-F capacitor is connected to a 12.0-V battery. How much energy is stored in the capacitor? (b) Had the capacitor been connected to a 6.00-V battery, how much energy would have been stored?18P 19P Two identical parallel-plate capacitors, each with capacitance C, are charged to potential difference V and then disconnected from the battery. They are then connected to each other in parallel with plates of like sign connected. Finally, the plate separation in one of the capacitors is doubled. (a) Find the total energy of the system of two capacitors before the plate separation is doubled. (b) Find the potential difference across each capacitor after the plate separation is doubled. (c) Find the total energy of the system after the plate separation is doubled. (d) Reconcile the difference in die answers to parts (a) and (c) with the law of conservation of energy.Two capacitors, C1 = 25.0 F and C2 = 5.00 F, are connected in parallel and charged with a 100-V power supply. (a) Draw a circuit diagram and (b) calculate the total energy stored in the two capacitors. (c) What If? What potential difference would be required across the same two capacitors connected in series for the combination to store the same amount of energy as in part (b)? (d) Draw a circuit diagram of the circuit described in part (c).A parallel-plate capacitor has a charge Q and plates of area A. What force acts on one plate to attract it toward the other plate? Because the electric field between the plates is E = Q/A0, you might think the force is F = QE = Q2/A0. This conclusion is wrong because the field E includes contributions from both plates, and the field created by the positive plate cannot exert any force on the positive plate. Show that the force exerted on each plate is actually F = Q2/2A0. Suggestion: Let C = 0A/x for an arbitrary plate separation x and note that the work done in separating the two charged plates is W=Fdx.23P 24P Determine (a) the capacitance and (b) the maximum potential difference that can be applied to a Teflon-filled parallel-plate capacitor haring a plate area of 1.75 cm2 and a plate separation of 0.040 0 mm.The voltage across an air-filled parallel-plate capacitor is measured to be 85.0 V as shown in Figure P25.26a. When a dielectric is inserted and completely fills the space between the plates as in Figure P25.26b, the voltage drops to 25.0 V. (a) What is the dielectric constant of the inserted material? (b) Can you identify the dielectric? If so, what is it? (c) If the dielectric does not completely fill the space between the plates, what could you conclude about the voltage across the plates? Figure P25.26 27P Each capacitor in the combination shown in Figure P25.28 has a breakdown voltage of 15.0 V. What is the breakdown voltage of the combination between points a and b?29P An infinite line of positive charge lies along the y axis, with charge density = 2.00 C/m. A dipole is placed with its center along the x axis at x = 25.0 cm. The dipole consists of two charges 10.0 C separated by 2.00 cm. The axis of the dipole makes an angle of 35.0 with the x axis, and the positive charge is farther from the line of charge than the negative charge. Find the net force exerted on the dipole.31P 32P 33AP Four parallel metal plates P1, P2, P3, and P4, each of area 7.50 cm2, are separated successively by a distance d = 1.19 mm as shown in Figure P25.34. Plate P1 is connected to the negative terminal of a battery, and P2 is connected to the positive terminal. The battery maintains a potential difference of 12.0 V. (a) If P3 is connected to the negative terminal, what is the capacitance of the three-plate system P1P2P3? (b) What is the charge on P2? (c) If P4 is now connected to the positive terminal, what is the capacitance of the four-plate system P1P2P3P4? (d) What is the charge on P4?A uniform electric field E = 3 000 V/m exists within a certain region. What volume of space contains an energy equal to 1.00 107 J? Express your answer in cubic meters and in liters.Two large, parallel metal plates, each of area A, are oriented horizontally and separated by a distance 3d. A grounded conducting wire joins them, and initially each plate carries no charge. Now a third identical plate carrying charge Q is inserted between the two plates, parallel to them and located a distance d from the upper plate as shown in Figure P25.36. (a) What induced charge appears on each of the two original plates? (b) What potential difference appears between the middle plate and each of the other plates?A parallel-plate capacitor with vacuum between its horizontal plates has a capacitance of 25.0 F. A nonconducting liquid with dielectric constant 6.50 is poured into the space between the plates, filling up a fraction f of its volume. (a) Find the new capacitance as a function of f. (b) What should you expect the capacitance to be when f = 0? Does your expression from part (a) agree with your answer? (c) What capacitance should you expect when f = 1? Does the expression from part (a) agree with your answer?Why is the following situation impossible? A 10.0-F capacitor has plates with vacuum between them. The capacitor is charged so that it stores 0.050 0 J of energy. A particle with charge 3.00 C is fired from the positive plate toward the negative plate with an initial kinetic energy equal to 1.00 104 J. The particle arrives at the negative plate with a reduced kinetic energy.Two square plates of sides are placed parallel to each other with separation d as suggested in Figure P25.39. You may assume d is much less than . The plates carry uniformly distributed static charges +Q0 and Q0. A block of metal has width , length , and thickness slightly less than d. It is inserted a distance x into the space between the plates. The charges on the plates remain uniformly distributed as the block slides in. In a static situation, a metal prevents an electric field from penetrating inside it. The metal can he thought of as a perfect dielectric, with . (a) Calculate the stored energy in the system as a function of x. (b) Find the direction and magnitude of the force that acts on the metallic block. (c) The area of the advancing front face of the block is essentially equal to d. Considering the force on the block as acting on this face, find the stress (force per area) on it. (d) Express the energy density in the electric field between the charged plates in terms of Q0, , d, and 0. (e) Explain how the answers to parts (c) and (d) compare with each other. Figure P25.39 (a) Two spheres have radii a and b, and their centers are a distance d apart. Show that the capacitance of this system is C=401a+1b2d provided d is large compared with a and b. Suggestion: Because the spheres are far apart, assume the potential of each equals the sum of the potentials due to each sphere. (b) Show that as d approaches infinity, the above result reduces to that of two spherical capacitors in series.41AP A parallel-plate capacitor of plate separation d is charged to a potential difference V0. A dielectric slab of thickness d and dielectric constant is introduced between the plates while the battery remains connected to the plates. (a) Show that the ratio of energy stored after the dielectric is introduced to the energy stored in the empty capacitor is U/U0 = . (b) Give a physical explanation for this increase in stored energy. (c) What happens to the charge on the capacitor? Note: This situation is not the same as in Example 25.5, in which the battery was removed from the circuit before the dielectric was introduced.To repair a power supply for a stereo amplifier, an electronics technician needs a 100-F capacitor capable of withstanding a potential difference of 90 V between the plates. The immediately available supply is a box of five 100-F capacitors, each basing a maximum voltage capability of 50 V. (a) What combination of these capacitors has the proper electrical characteristics? Will the technician use all the capacitors in the box? Explain your answers. (b) In the combination of capacitors obtained in part (a), what will be the maximum voltage across each of the capacitors used?44AP You are part of a team working in a machine parts mechanics shop. An important customer has asked your company to provide springs with a very precise force constant k. You dense the electrical circuit shown in Figure P25.45 to measure the spring constant of each of the springs to be provided to the customer. The circuit consists of two identical, parallel metal plates free to move, other than being connected to identical metal springs, a switch, and a battery with terminal voltage V. With the switch open, the plates are uncharged, are separated by a distance d, and have a capacitance C. When the switch is closed, the plates become charged and attract each other. The distance between the plates changes by a factor f, after which the plates are in equilibrium between the spring forces and the attractive electric force between the plates. To keep the plates from going into oscillations, you hold each plate with insulating gloves as the switch is closed and apply a force on the plates that allows them to move together at a slow constant speed until they are at the equilibrium separation, at which point you can release the plates. You determine an expression for the spring constant in terms of C, d, V, and f. Figure P25.45 Problems 45 and 50.Consider two long, parallel, and oppositely charged wires of radius r with their centers separated by a distance D that is much larger than r. Assuming the charge is distributed uniformly on the surface of each wire, show that the capacitance per unit length of this pair of wires is Cl=0ln(D/r)Some physical systems possessing capacitance continuously distributed over space can be modeled as an infinite array of discrete circuit elements. Examples are a microwave waveguide and the axon of a nerve cell. To practice analysis of an infinite array, determine the equivalent capacitance C between terminals X and Y of the infinite set of capacitors represented in Figure P25.47. Each capacitor has capacitance C0. Suggestions: Imagine that the ladder is cut at the line AB and note that the equivalent capacitance of the infinite section to the right of AB is also C.A parallel-plate capacitor with plates of area LW and plate separation t has the region between its plates filled with wedges of two dielectric materials as shown in Figure P25.48. Assume t is much less than both L and W. (a) Determine its capacitance. (b) Should the capacitance be the same if the labels 1 and 2 are interchanged? Demonstrate that your expression does or does not have this property. (c) Show that if 1 and 2 approach equality to a common value , your result becomes the same as the capacitance of a capacitor containing a single dielectric: C = 0LW/t. Figure P25.48 A capacitor is constructed from two square, metallic plates of sides and separation d. Charges +Q and Q are placed on the plates, and the power supply is then removed. A material of dielectric constant is inserted a distance x into the capacitor as shown in Figure P25.49 (page 690). Assume d is much smaller than x. (a) Find the equivalent capacitance of the device. (b) Calculate the energy stored in the capacitor. (c) Find the direction and magnitude of the force exerted by the plates on the dielectric. (d) Obtain a numerical value for the force when x = /2, assuming = 5.00 cm, d = 2.0 mm, the dielectric is glass ( = 4.50), and the capacitor was charged to 2.00 103 V before the dielectric was inserted. Suggestion: The system can be considered as two capacitors connected in parallel. Figure P25.49 This problem is a continuation of Problem 45. You are part of a team working in a machine parts mechanics shop. An important customer has asked your company to provide springs with a very precise force constant k. You devise the electrical circuit shown in Figure P25.45 to measure the spring constant of each of the springs to be provided to the customer. The circuit consists of two identical, parallel metal plates connected to identical metal springs, a switch, and a battery with emf V. With the switch open, the plates are uncharged, are separated by a distance d, and have a capacitance C. To provide a comparison value for the spring constant that you found in Problem 45, you slide a slab of material with dielectric constant and thickness t between the plates, so that it is in contact with one of the plates as shown in Figure P25.50. When the switch is closed, the plates become charged and attract each other. The distance between the plates changes by a factor f, after which the plates are in equilibrium between the spring forces and the attractive electric force between the plates. To keep the plates from going into oscillations, you hold each plate with insulating gloves as the switch is closed and apply a force on the plates that allows them to move together at a slow constant speed until they are at the equilibrium separation, at which point you can release the plates. Find an expression for the spring constant in terms of C, d, V, k, t, and f. Figure P25.50 Consider positive and negative charges of equal magnitude moving horizontally through the four regions shown in Figure 26.4. Rank the current in these four regions from highest to lowest. Figure 26.4 (Quick Quiz (26.1) Charges move through four regions.26.2QQ 26.3QQ When does an incandescent lightbulb carry more current, (a) immediately after it is turned on and the glow of the metal filament is increasing or (b) after it has been on for a few milliseconds and the glow is steady?1P A small sphere that carries a charge q is whirled in a circle at the end of an insulating string. The angular frequency of revolution is . What average current does this revolving charge represent?3P 4P 5P Figure P26.6 represents a section of a conductor of nonuniform diameter carrying a current of I = 5.00 A. The radius of cross-section A1 is r1 = 0.400 cm. (a) What is the magnitude of the current density across A1? The radius r2 at A2 is larger than the radius r1 at A1. (b) Is the current at A2 larger, smaller, or the same? (c) Is the current density at A2 larger, smaller, or the same? Assume A2 = 4A1. Specify the (d) radius, (e) current, and (f) current density at A2. Figure P26.6 The quantity of charge q (in coulombs) that has passed through a surface of area 2.00 cm2 varies with time according to the equation q = 4t3 + 5t + 6, where t is in seconds. (a) What is the instantaneous current through the surface at t = 1.00 s? (b) What is the value of the current density?8P 9P A wire 50.0 m long and 2.00 mm in diameter is connected to a source with a potential difference of 9.11 V, and the current is found to be 36.0 A. Assume a temperature of 20.0C and, using Table 26.2, identify the metal out of which the wire is made.11P 12P 13P 14P 15P 16P 17P 18P An aluminum wire with a diameter of 0.100 mm has a uniform electric field of 0.200 V/m imposed along its entire length. The temperature of the wire is 50.0C. Assume one free electron per atom. (a) Use the information in Table 26.2 to determine the resistivity of aluminum at this temperature. (b) What is the current density in the wire? (c) What is the total current in the wire? (d) What is the drift speed of the conduction electrons? (e) What potential difference must exist between the ends of a 2.00-m length of the wire to produce the stated electric field?20P At what temperature will aluminum have a resistivity that is three times the resistivity copper has at room temperature?You are working in a laboratory that studies the effects of currents in various crystals. One of the experiments involves a requirement for a steady current of I = 0.500 A in a wire that delivers the current to the crystal. Both the wire and the crystal are in a chamber whose interior temperature T will vary from 40.0C to 150C. The wire is made of tungsten and is of length L = 25.0 cm and radius r = 1.00 mm. A test run is being made before the crystal is added to the circuit. Your supervisor asks you to determine the range of voltages that must be supplied to the wire in the test run to maintain its current at 0.500 A.Assume that global lightning on the Earth constitutes a constant current of 1.00 kA between the ground and an atmospheric layer at potential 300 kV. (a) Find the power of terrestrial lightning. (b) For comparison, find the power of sunlight falling on the Earth. Sunlight has an intensity of 1 370 W/m2 above the atmosphere. Sunlight falls perpendicularly on the circular projected area that the Earth presents to the Sun.The Van de Graaff generator, diagrammed in Figure P26.24, is an electrostatic device that can raise the metal dome to a high voltage. The dome of such a generator is seen on the left in Figure 22.1a. In the device, charge is delivered continuously to the high-potential dome by means of a moving belt of insulating material. The belt is charged at point by means of a discharge between comb-like metallic needles and a grounded grid. The needles are maintained at a positive electric potential of typically 104 V. The positive charge on the moving belt is transferred to the dome by a second comb of needles at point . Because the electric field inside the dome is negligible, the positive charge on the belt is easily transferred to the dome from its interior regardless of its potential. Suppose the generator is operating so that the potential difference between the high potential dome and the charging needles at is 15.0 kV. Calculate the power required to drive the belt against electrical forces at an instant when the effective current delivered to the dome is 500 A. Figure P26.24 25P The potential difference across a resting neuron in the human body is about 75.0 mV and carries a current of about 0.200 mA. How much power does the neuron release?27P 28P 29P 30P 31P 32P 33P 34AP 35AP You are working with an oceanographer who is studying how the ion concentration in seawater depends on depth. She shows you the device that she uses to measure the resistivity of water from a boat. It consists of a pair of concentric metallic cylinders at the end of a cable (Fig. P26.36). Seawater flows freely between the two cylindrical shells. She makes a measurement by lowering the device into the water and applying a potential difference V between the inner and outer cylinders. This produces an outward radial current I in the seawater between the shells. She shows you the current and voltage data for the water at a particular depth and is then called away to answer a long call on her cellphone about a laboratory issue back on the mainland. As she leaves, she says, Have the resistivity of the water calculated when I get back. She forgot to show you any tables or formulas to use to determine the resistivity, so you are on your own. Quick! Find an expression for the resistivity in terms of I and V before she finishes her phone call! Figure P26.36 A charge Q is placed on a capacitor of capacitance C. The capacitor is connected into the circuit shown in Figure P26.37, with an open switch, a resistor, and an initially uncharged capacitor of capacitance 3C. The switch is then closed, and the circuit comes to equilibrium. In terms of Q and C, find (a) the final potential difference between the plates of each capacitor, (b) the charge on each capacitor, and (c) the final energy stored in each capacitor. (d) Find the internal energy appearing in the resistor. Figure P26.37 38AP 39AP 40AP Review. An office worker uses an immersion heater to warm 250 g of water in a light, covered, insulated cup from 20.0C to 100C in 4.00 min. The heater is a Nichrome resistance wire connected to a 120-V power supply. Assume the wire is at 100C throughout the 4.00-min time interval. (a) Specify a relationship between a diameter and a length that the wire can have. (b) Can it be made from less than 0.500 cm3 of Nichrome?42AP A close analogy exists between the flow of energy by heat because of a temperature difference (see Section 19.6) and the flow of electric charge because of a potential difference. In a metal, energy dQ and electrical charge dq are both transported by free electrons. Consequently, a good electrical conductor is usually a good thermal conductor as well. Consider a thin conducting slab of thickness dx, area A, and electrical conductivity , with a potential difference dV between opposite faces. (a) Show that the current I = dq/dt is given by the equation on the left: ChargeconductionThermalconductiondqdt=A|dVdx|dQdt=kA|dTdx| In the analogous thermal conduction equation on the right (Eq. 19.17), the rate dQ/dt of energy flow by heat (in SI units of joules per second) is due to a temperature gradient dT/dx in a material of thermal conductivity k. (b) State analogous rules relating the direction of the electric current to the change in potential and relating the direction of energy flow to the change in temperature.The dielectric material between the plates of a parallel-plate capacitor always has some nonzero conductivity . Let A represent the area of each plate and d the distance between them. Let represent the dielectric constant of the material. (a) Show that the resistance R and the capacitance C of the capacitor are related by RC=0 (b) Find the resistance between the plates of a 14.0-nF capacitor with a fused quartz dielectric.Review. A parallel-plate capacitor consists of square plates of edge length that are separated by a distance d, where d . A potential difference V is maintained between the plates. A material of dielectric constant fills half the space between the plates. The dielectric slab is withdrawn from the capacitor as shown in Figure P26.45. (a) Find the capacitance when the left edge of the dielectric is at a distance x from the center of the capacitor. (b) If the dielectric is removed at a constant speed v, what is the current in the circuit as the dielectric is being withdrawn? Figure P26.45 46AP 47AP 48CP 49CP Material with uniform resistivity is formed into a wedge as shown in Figure P26.50. Show that the resistance between face A and face B of this wedge is R=Lw(y2y1)lny2y1 Figure P26.50 To maximize the percentage of the power from the emf of a battery that is delivered to a device external to the battery, what should the internal resistance of the battery be? (a) It should be as low as possible. (b) It should be as high as possible. (c) The percentage does not depend on the internal resistance.With the switch in the circuit of Figure 27.4a closed, there is no current in R2 because the current has an alternate zero-resistance path through the switch. There is current in R1, and this current is measured with the ammeter (a device for measuring current) at the bottom of the circuit. If the switch is opened (Fig. 27.4b), there is current in R2. What happens to the reading on the ammeter when the switch is opened? (a) The reading goes up. (b) The reading goes down. (c) The reading does not change. Figure 27.4 (Quick Quiz 27.2) What happens when the switch is opened?With the switch in the circuit of Figure 27.6a open, there is no current in R2. There is current in R1, however, and it is measured with the ammeter at the right side of the circuit. If the switch is closed (Fig. 27.6b), there is current in R2. What happens to the reading on the ammeter when the switch is closed? (a) The reading increases. (b) The reading decreases. (c) The reading does not change. Figure 27.6 (Quick Quiz 27.3) What happens when the switch is closed?27.4QQ Consider the circuit in Figure 27.17 and assume the battery has no internal resistance. (i) Just after the switch is closed, what is the current in the battery? (a) 0 (b) /2R (c) 2/R (d) /R (e) impossible to determine (ii) After a very long time, what is the current in the batten? Choose from the same choices. Figure 27.17 (Quick Quiz 27.5) How does the current vary after the switch is closed?Two 1.50-V batterieswith their positive terminals in the same directionare inserted in series into a flashlight. One battery has an internal resistance of 0.255 , and the other has an internal resistance of 0.153 . When the switch is closed, the bulb carries a current of 600 mA. (a) What is the bulbs resistance? (b) What fraction of the chemical energy transformed appears as internal energy in the batteries?As in Example 27.2, consider a power supply with fixed emf and internal resistance r causing current in a load resistance R. In this problem, R is fixed and r is a variable. The efficiency is defined as the energy delivered to the load divided by the energy delivered by the emf. (a) When the internal resistance is adjusted for maximum power transfer, what is the efficiency? (b) What should be the internal resistance for maximum possible efficiency? (c) When the electric company sells energy to a customer, does it have a goal of high efficiency or of maximum power transfer? Explain. (d) When a student connects a loudspeaker to an amplifier, does she most want high efficiency or high power transfer? Explain.Figure P27.3 shows the interior of a three-way incandescent lightbulb, which provides three levels of light intensity. The socket of the lamp is equipped with a four-position switch for selecting different light intensities, with the positions described as follows: (1) off (switches S1 and S2 both open), (2) switch S1 closed, (3) switch S2 closed, and (4) switches S1 and S2 both closed. The lightbulb contains two filaments. When the lamp is connected to a 120-V source, one filament receives 100 W of power and the other receives 75 W. What is the total power input to the light bulb when (a) only switch S1 is closed, (b) only switch S2 is closed, and (c) both switches are closed? (d) What If? Suppose the 75-W filament breaks and no longer is able to carry a current. How many switch positions will result in light leaving the bulb and what will be the power input to the bulb in those positions? Figure P27.3 4P Consider the two circuits shown in Figure P27.5 in which the batteries are identical. The resistance of each lightbulb is R. Neglect the internal resistances of the batteries. (a) Find expressions for the currents in each lightbulb. (b) How does the brightness of B compare with that of C? Explain. (c) How does the brightness of A compare with that of B and of C? Explain. Figure P27.5 Consider strings of incandescent lights that are used for many ornamental purposes, such as decorating Christmas trees. Over the years, both parallel and series connections have been used for strings of lights, and the bulbs have varied in design. Because series-wired lightbulbs operate with less energy per bulb and at a lower temperature, they are safer than parallel-wired lightbulbs, where each bulb operates at 120 V. To prevent the failure of one lightbulb from causing the entire string to go out for the bulbs wired in series, a new design was developed. Figure P27.6a shows one of these types of miniature lightbulb designed to operate in a series connection. When the filament breaks in one of these lightbulbs, the break in the filament represents the largest resistance in the series, much larger than that of the intact filaments. As a result, most of the applied 120 V appears across the lightbulb with the broken filament. Inside the lightbulb, a small jumper loop covered by an insulating material is wrapped around the filament leads. When the filament fails and 120 V appears across the lightbulb, an are burns the insulation on the jumper and connects the filament leads, as shown in Figure P27.6b. This connection now provides a low-resistance path through the lightbulb, even though its filament is no longer active, and the voltage across the bulb drops to zero. All the other lightbulbs not only stay on, but they glow more brightly because the total resistance of the string is reduced and consequently the current in each remaining lightbulb increases. Suppose you have a string of 48 bulbs, each one with a resistance of 8.00 . Assume the resistance of a bulb with its filament broken drops to zero. Suppose that a bulb becomes dangerously warm, so that it could set something on fire, if it receives a power of 1.75 W. How many bulbs can fail before the string of lights becomes dangerous? Figure P27.6 You are working at an electronics fabrication shop. Your current project is on the team producing resistors for the timer circuit that delays the closing of an elevator door. According to its design specification, the timer circuit is to have a resistance of 32.0 between two points A and B. As your resistors come off the assembly line, you find that they have a variation of 5.00% from this value. After a team meeting to evaluate this situation, the team decides that resistances in the range 32.0 0.5 are acceptable and do not need modification. For resistances outside this range, the director does not wish to discard the resistors, but rather to add extra resistors in series or parallel with the main resistor to bring the total equivalent resistance to the exact design value of 32.0 . You are put in charge of procuring the extra resistors. What range of resistances for these extra resistors do you need to cover the entire range of variation of 5.00%? All resistances can be measured to three significant figures.In your new job at an engineering company, your supervisor asks you to fabricate a resistor that has a resistance of R = 0.100 and no change in resistance with temperature. She suggests making the resistor from lengths of cylindrical carbon and Nichrome wires of equal radius, placed end-to-end. She wants the combination to fit into a machine that allows for a radius of the resistor to be r = 1.50 mm. What are the lengths of the two segments of the resistor?A battery with = 6.00 V and no internal resistance supplies current to the circuit shown in Figure P27.9. When the double-throw switch S is open as shown in the figure, the current in the battery is 1.00 mA. When the switch is closed in position a, the current in the battery is 1.20 mA. When the switch is closed in position b, the current in the battery is 2.00 mA. Find the resistances (a) R1, (b) R2, and (c) R3. Figure P27.9 Problems 9 and 10.A battery with emf and no internal resistance supplies current to the circuit shown in Figure P27.9. When the double-throw switch S is open as shown in the figure, the current in the battery is I0. When the switch is closed in position a, the current in the battery is Ia. When the switch is closed in position b, the current in the battery is Ib. Find the resistances (a) R1, (b) R2, and (c) R3. Figure P27.9 Problems 9 and 10.Todays class on current and resistance is about to begin and you await your professor, who is known for unorthodox demonstrations. He walks in just at the beginning time for the class, and is carrying hot dogs! He then proceeds to set up a demonstration using an older style of hot dog cooker in which the hot dogs are directly connected across 120 V from the wall socket. He has modified the cooker so it simultaneously applies the 120 V to three combinations: across the ends of a single hot dog, across the ends of two hot dogs in parallel, and across the outer ends of two hot dogs in series. He explains that he has measured the resistance of a hot dog to be 11.0 , and that a hot dog requires 75.0 kJ of energy to cook it. He says he will give extra credit to anyone who, before any hot dog begins smoking, can determine (a) which hot dog(s) will cook first, and (b) the time interval for each hot dog to cook. Quick! Get to work!Why is the following situation impossible? A technician is testing a circuit that contains a resistance R. He realizes that a better design for the circuit would include a resistance 73R rather than R. He has three additional resistors, each with resistance R. By combining these additional resistors in a certain combination that is then placed in series with the original resistor, he achieves the desired resistance.Calculate the power delivered to each resistor in the circuit shown in Figure P27.13. Figure P27.13 For the purpose of measuring the electric resistance of shoes through the body of the wearer standing on a metal ground plate, the American National Standards Institute (ANSI) specifies the circuit shown in Figure P27.14. The potential difference V across the 1.00-M resistor is measured with an ideal voltmeter. (a) Show that the resistance of the footwear is Rshoes=50.0VVV (b) In a medical test, a current through the human body should not exceed 150 A. Can the current delivered by the ANSI-specified circuit exceed 150 A? To decide, consider a person standing barefoot on the ground plate. Figure P27.14 Four resistors are connected to a battery as shown in Figure P27.15. (a) Determine the potential difference across each resistor in terms of . (b) Determine the current in each resistor in terms of I. (c) What If? If R3 is increased, explain what happens to the current in each of the resistors. (d) In the limit that R3 , what are the new values of the current in each resistor in terms of I, the original current in the battery? Figure P27.15 You have a faculty position at a community college and are teaching a class in automotive technology. You are deep in a discussion of using jumper cables to start a car with a dead battery from a car with a fresh battery. You have drawn the circuit diagram in Figure P27.16 to explain the process. The battery on the left is the live battery in the correctly functioning car, with emf and internal resistance RL, where the L subscript refers to live. Its terminals are connected directly across those of the dead battery, in the middle of the diagram, with emf and internal resistance RD, where the D subscript refers to dead. Then, the starter in the car with the dead battery is activated by closing the ignition switch, allowing the car to start. The resistance of the starter is RS. A student raises his hand and asks, So is the dead battery being charged while the starter is operating? How do you respond? Figure P27.16 The circuit shown in Figure P27.17 is connected for 2.00 min. (a) Determine the current in each branch of the circuit. (b) Find the energy delivered by each battery. (c) Find the energy delivered to each resistor. (d) Identify the type of energy storage transformation that occurs in the operation of the circuit. (e) Find the total amount of energy transformed into internal energy in the resistors.The following equations describe an electric circuit: I1(220)+5.80VI2(370)=0+I2(370)+I3(150)3.10V=0I1+I3I2=0 (a) Draw a diagram of the circuit. (b) Calculate the unknowns and identify the physical meaning of each unknown.Taking R = 1.00 k and = 250 V in Figure P27.19, determine the direction and magnitude of the current in the horizontal wire between a and e. Figure P27.19 In the circuit of Figure P27.20, the current I1 = 3.00 A and the values of for the ideal battery and R are unknown. What are the currents (a) I2 and (b) I3? (c) Can you find the values of and R? If so, find their values. If not, explain. Figure P27.20 (a) Can the circuit shown in Figure P27.21 be reduced to a single resistor connected to a battery? Explain. Calculate the currents (b) I1, (c) I2, and (d) I3. Figure P27.21 For the circuit shown in Figure P27.22, we wish to find the currents I1, I2, and I3. Use Kirchhoffs rules to obtain equations for (a) the upper loop, (b) the lower loop, and (c) the junction on the left side. In each case, suppress units for clarity and simplify, combining the terms. (d) Solve the junction equation for I3. (e) Using the equation found in part (d), eliminate I3 from the equation found in part (b). (f) Solve the equations found in parts (a) and (e) simultaneously for the two unknowns I1 and I2. (g) Substitute the answers found in part (f) into the junction equation found in part (d), solving for I3. (h) What is the significance of the negative answer for I2? Figure P27.22 An uncharged capacitor and a resistor are connected in series to a source of emf. If = 9.00 V, C = 20.0 F, and R = 100 , find (a) the time constant of the circuit, (b) the maximum charge on the capacitor, and (c) the charge on the capacitor at a time equal to one time constant after the battery is connected.24P In the circuit of Figure P27.25, the switch S has been open for a long time. It is then suddenly closed. Take = 10.0 V, R1 = 50.0 k, R2 = 100 k, and C = 10.0 F. Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at t = 0. Determine the current in the switch as a function of time. Figure P27.25 Problems 25 and 26.In the circuit of Figure P27.25, the switch S has been open for a long time. It is then suddenly closed. Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at t = 0. Determine the current in the switch as a function of time. Figure P27.25 Problems 25 and 26.A 10.0-F capacitor is charged by a 10.0-V battery through a resistance R. The capacitor reaches a potential difference of 4.00 V in a time interval of 3.00 s after charging begins. Find R.28P 29P 30P 31P 32AP Find the equivalent resistance between points a and b in Figure P27.33. Figure P27.33 The circuit in Figure P27.34a consists of three resistors and one battery with no internal resistance. (a) Find the current in the 5.00- resistor. (b) Find the power delivered to the 5.00- resistor. (c) In each of the circuits in Figures P27.34b, P27.34c, and P27.34d, an additional 15.0-V battery has been inserted into the circuit. Which diagram or diagrams represent a circuit that requires the use of Kirchhoffs rules to find the currents? Explain why. (d) In which of these three new circuits is the smallest amount of power delivered to the 10.0- resistor? (You need not calculate the power in each circuit if you explain your answer.) Figure P27.34 The circuit in Figure P27.35 has been connected for several seconds. Find the current (a) in the 4.00-V battery, (b) in the 3.00- resistor, (c) in the 8.00-V battery, and (d) in the 3.00-V battery. (e) Find the charge on the capacitor. Figure P27.35 The resistance between terminals a and b in Figure P27.36 is 75.0 . If the resistors labeled R have the same value, determine R. Figure P27.36 (a) Calculate the potential difference between points a and b in Figure P27.37 and (b) identify which point is at the higher potential. Figure P27.37 Why is the following situation impossible? A battery has an emf of = 9.20 V and an internal resistance of r = 1.20 . A resistance R is connected across the battery and extracts from it a power of P = 21.2 W.When two unknown resistors are connected in series with a battery, the battery delivers 225 W and carries a total current of 5.00 A. For the same total current, 50.0 W is delivered when the resistors are connected in parallel. Determine the value of each resistor.40AP The circuit in Figure P27.41 contains two resistors, R1 = 2.00 k and R2 = 3.00 k, and two capacitors, C1 = 2.00 F and C2 = 3.00 F, connected to a battery with emf = 120 V. If there are no charges on the capacitors before switch S is closed, determine the charges on capacitors (a) C1 and (b) C2 as functions of time, after the switch is closed. Figure P27.41 42AP A power supply has an open-circuit voltage of 40.0 V and an internal resistance of 2.00 . It is used to charge two storage batteries connected in series, each having an emf of 6.00 V and internal resistance of 0.300 . If the charging current is to be 4.00 A, (a) what additional resistance should be added in series? At what rate does the internal energy increase in (b) the supply, (c) in the batteries, and (d) in the added series resistance? (e) At what rate does the chemical energy increase in the batteries?A battery is used to charge a capacitor through a resistor as shown in Figure P27.44. Show that half the energy supplied by the battery appears as internal energy in the resistor and half is stored in the capacitor. Figure P27.44 45AP (a) Determine the equilibrium charge on the capacitor in the circuit of Figure P27.46 as a function of R. (b) Evaluate the charge when R = 10.0 . (c) Can the charge on the capacitor be zero? If so, for what value of R? (d) What is the maximum possible magnitude of the charge on the capacitor? For what value of R is it achieved? (c) Is it experimentally meaningful to take R = ? Explain your answer. If so, what charge magnitude does it imply? Figure P27.46 In Figure P27.47, suppose the switch has been closed for a time interval sufficiently long for the capacitor to become fully charged. Find (a) the steady-state current in each resistor and (b) the charge Qmax on the capacitor. (c) The switch is now opened at t = 0. Write an equation for the current in R2 as a function of time and (d) find the time interval required for the charge on the capacitor to fall to one-fifth its initial value. Figure P27.47 Figure P27.48 shows a circuit model for the transmission of an electrical signal such as cable TV to a large number of subscribers. Each subscriber connects a load resistance RL between the transmission line and the ground. The ground is assumed to be at zero potential and able to carry any current between any ground connections with negligible resistance. The resistance of the transmission line between the connection points of different subscribers is modeled as the constant resistance RT. Show that the equivalent resistance across the signal source is Req=12[(4RTRL+RT2)1/2+RT] Suggestion: Because the number of subscribers is large, the equivalent resistance would not change noticeably if the first subscriber canceled the service. Consequently, the equivalent resistance of the section of the circuit to the right of the first load resistor is nearly equal to Req. Figure P27.48 The student engineer of a campus radio station wishes to verify the effectiveness of the lightning rod on the antenna mast (Fig. P27.49). The unknown resistance Rx is between points C and E. Point E is a true ground, but it is inaccessible for direct measurement because this stratum is several meters below the Earths surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance Ry. The procedure is as follows. Measure resistance R1 between points A and B, then connect A and B with a heavy conducting wire and measure resistance R2 between points A and C. (a) Derive an equation for Rx in terms of the observable resistances, R1, and R2. (b) A satisfactory ground resistance would Rx 2.00 . Is the grounding of the station adequate if measurements give R1 = 13.0 and R2 = 6.00 ? Explain. Figure P27.49 50AP The switch in Figure P27.51a closes when Vc23Vand opens when Vc13V. The ideal voltmeter reads a potential difference as plotted in Figure P27.51b. What is the period T of the waveform in terms of R1, R2, and C? Figure P27.51 An electron moves in the plane of this paper toward the top of the page. A magnetic field is also in the plane of the page and directed toward the right. What is the direction of the magnetic force on the electron? (a) toward the top of the page (b) toward the bottom of the page (c) toward the left edge of the page (d) toward the right edge of the page (e) upward out of the page (f) downward into the page 28.2QQ A wire carries current in the plane of this paper toward the top of the page. The wire experiences a magnetic force toward the right edge of the page. Is the direction of the magnetic field causing this force (a) in the plane of the page and toward the left edge, (b) in the plane of the page and toward the bottom edge, (c) upward out of the page, or (d) downward into the page?(i) Rank the magnitudes of the torques acting on the rectangular loops (a), (b), and (c) shown edge-on in Figure 28.24 (page 760) from highest to lowest. All loops are identical and carry the same current. (ii) Rank the magnitudes of the net forces acting on the rectangular loops shown in Figure 28.24 from highest to lowest. Figure 28.24 (Quick Quiz 28.4) Which current loop (seen edge-on) experiences the greatest torque, (a), (b), or (c)? Which experiences the greatest net force?At the equator, near the surface of the Earth, the magnetic field is approximately 50.0 T northward, and the electric field is about 100 N/C downward in fair weather. Find the gravitational, electric, and magnetic forces on an electron in this environment, assuming that the electron has an instantaneous velocity of 6.00 106 m/s directed to the east.Consider an electron near the Earths equator. In which direction does it tend to deflect if its velocity is (a) directed downward? (b) Directed northward? (c) Directed westward? (d) Directed southeastward?Find the direction of the magnetic field acting on a positively charged particle moving in the various situations shown in Figure P28.3 if the direction of the magnetic force acting on it is as indicated. Figure P28.3 A proton moving at 4.00 106 m/s through a magnetic field of magnitude 1.70 T experiences a magnetic force of magnitude 8.20 1013 N. What is the angle between the protons velocity and the field?A proton travels with a speed of 5.02 106 m/s in a direction that makes an angle of 60.0 with the direction of a magnetic field of magnitude 0.180 T in the positive x direction. What are the magnitudes of (a) the magnetic force on the proton and (b) the protons acceleration?6P 7P An accelerating voltage of 2.50103 V is applied to an electron gun, producing a beam of electrons originally traveling horizontally north in vacuum toward the center of a viewing screen 35.0 cm away. What are (a) the magnitude and (b) the direction of the deflection on the screen caused by the Earths gravitational field? What are (c) the magnitude and (d) the direction of the deflection on the screen caused by the vertical component of the Earths magnetic field, taken as 20.0 T down? (e) Does an electron in this vertical magnetic field move as a projectile, with constant vector acceleration perpendicular to a constant northward component of velocity? (f) Is it a good approximation to assume it has this projectile motion? Explain.A proton (charge + e, mass mp), a deuteron (charge + e, mass 2mp), and an alpha particle (charge +2e, mass 4mp) are accelerated from rest through a common potential difference V. Each of the particles enters a uniform magnetic field B, with its velocity in a direction perpendicular to B. The proton moves in a circular path of radius rp. In terms of rp, determine (a) the radius rd of the circular orbit for the deuteron and (b) the radius ra for the alpha particle.10P Review. One electron collides elastically with a second electron initially at rest. After the collision, the radii of their trajectories are 1.00 cm and 2.40 cm. The trajectories are perpendicular to a uniform magnetic field of magnitude 0.044 0 T. Determine the energy (in keV) of the incident electron.Review. One electron collides elastically with a second electron initially at rest. After the collision, the radii of their trajectories are r1 and r2. The trajectories arc perpendicular to a uniform magnetic field of magnitude B. Determine the energy of the incident electron.Review. An electron moves in a circular path perpendicular to a constant magnetic field of magnitude 1.00 mT. The angular momentum of the electron about the center of the circle is 4.00 1025 kg m2/s. Determine (a) the radius of the circular path and (b) the speed of the electron.A cyclotron designed to accelerate protons has a magnetic field of magnitude 0.450 T over a region of radius 1.20 m. What are (a) the cyclotron frequency and (b) the maximum speed acquired by the protons?15P 16P A cyclotron (Fig. 28.16) designed to accelerate protons has an outer radius of 0.350 m. The protons are emitted nearly at rest from a source at the center and are accelerated through 600 V each time they cross the gap between the dees. The dees are between the poles of an electromagnet where the field is 0.800 T. (a) Find the cyclotron frequency for the protons in this cyclotron. Find (b) the speed at which protons exit the cyclotron and (c) their maximum kinetic energy. (d) How many revolutions does a proton make in the cyclotron? (e) For what time interval does the proton accelerate?A particle in the cyclotron shown in Figure 28.16a gains energy qV from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is T=2=2mqB so the particles average rate of increase in energy is 2qVT=q2BVm Notice that this power input is constant in time. On the other hand, the rate of increase in the radius r of its path is not constant. (a) Show that the rate of increase in the radius r of the panicles path is given by drdt=1rVB (b) Describe how the path of the particles in Figure 28.16a is consistent with the result of part (a). (c) At what rate is the radial position of the protons in a cyclotron increasing immediately before the protons leave the cyclotron? Assume the cyclotron has an outer radius of 0.350 m, an accelerating voltage of V = 600 V, and a magnetic field of magnitude 0.800 T. (d) By how much does the radius of the protons path increase during their last full revolution? Figure 28.16 (a) A cyclotron consists of an ion source at P, two does D1 and D2 across which an alternating potential difference is applied, and a uniform magnetic field. (The south pole of the magnet is not shown.) (b) The first cyclotron, invented by E. O. Lawrence and M. S. Livingston in 1934.19P 20P A wire carries a steady current of 2.40 A. A straight section of the wire is 0.750 m long and lies along the x axis within a uniform magnetic field. B=1.60kT. If the current is in the positive x direction, what is the magnetic force on the section of wire?22P Review. A rod of mass 0.720 kg and radius 6.00 cm rests on two parallel rails (Fig. P28.23) that are d = 12.0 cm apart and L = 45.0 cm long. The rod carries a current of I = 48.0 A in the direction shown and rolls along the rails without slipping. A uniform magnetic field of magnitude 0.240 T is directed perpendicular to the rod and the rails. If it starts from rest, what is the speed of the rod as it leaves the rails? Figure P28.23 Problems 23 and 24.

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