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**Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011**

**Chapter 18**

Multiple-Choice Problems

1. A surface will be an equipotential surface if

A. the electric field is zero at all points on it.

B. the electric field is tangent to the surface at all points.

C. the electric field is perpendicular to the surface at all points.

2. In Figure 18.34, point P is equidistant from both point charges. At that point

A. the electric field points directly to the right.

B. the electric field is zero.

C. the potential (relative to infinity) is zero.

D. the potential (relative to infinity) points upward.

3. For the capacitor network shown in Figure 18.35, a constant potential difference of 50 V is maintained across points a and b by a battery. Which of the following statements about this network is correct?

A. The 10 μC and 20 μC capacitors have equal charges.

B. The charge on the 20 μC capacitor is twice the charge on the 10 μC capacitor.

C. The potential difference across the 10 μC capacitor is the same as the potential difference across the 10 μC capacitor.

D. The equivalent capacitance of the network is 60 μC.

4. A parallel-plate capacitor having circular plates of radius R and separation d is held at a fixed potential difference by a battery. If the plates are moved closer together while they are held at the same potential difference

A. the amount of charge on each of them will increase.

B. the amount of charge on each of them will decrease.

C. the amount of charge on each of them will stay the same.

D. the energy stored in the capacitor increases.

5. A parallel-plate capacitor having circular plates of radius R and separation d is charged to a potential difference by a battery. It is then removed from the battery. If the plates are moved closer together

A. the amount of charge on each of them will increase.

B. the amount of charge on each of them will decrease.

C. the amount of charge on each of them will stay the same.

D. the energy stored in the capacitor increases.

6. Two electrons close to each other are released from rest and are completely free to move. After being released

A. their kinetic energies gradually decrease to zero as they move apart.

B. their kinetic energies increase as they move apart.

C. their electrical potential energy gradually decreases to zero as they move apart.

D. their electrical potential energy increases as they move apart.

E. their speeds gradually decrease to zero as they move apart.

7. The capacitor network shown in Figure 18.36 is connected across a fixed potential difference of 25 V. Which statements about this network must be true?

A. The potential difference is the same across each capacitor.

B. The charge is the same on each capacitor.

C. The equivalent capacitance of the network is 30 μC

D. The equivalent capacitance of the network is less than 30 μC

8. If the potential (relative to infinity) due to a point charge is V at a distance R from this charge, the distance at which the potential (relative to infinity) is 2V is

A. 4R. B. 2R. C. R/2 D. R/4

9. If the electrical potential energy of two point charges is U when they are a distance d apart, their potential energy when they are twice as far apart will be

A. U/4 B. U/2 C. 2U D. 4U

10. An electron is released between the plates of a charged parallel-plate capacitor very close to the right-hand plate. Just as it reaches the left-hand plate, its speed is v. If the distance between the plates were halved without changing the electric potential difference between them, then the speed of the electron when it reached the left-hand plate would be

A. 2v B. v√2 C. v D. v/√2 E. v/2

11. The plates of a parallel-plate capacitor are connected across a battery of fixed potential difference and that produces a uniform electric field E between the plates. If the plates are pulled twice as far apart, but are kept connected to the battery, the electric field between the plates will be

A. 4E B. 2E C. E D. E/2 E. E/4

12. At a point P a distance d from a point charge, the potential relative to infinity is V and the electric-field magnitude is E. If you now move to a point S at which the potential is V/2, the electric-field magnitude at S will be

A. E/4 B. E/2 C. 2E D. 4E

13. When a certain capacitor carries charge of magnitude Q on each of its plates, it stores energy U. In order to store twice as much energy, how much charge should it have on its plates?

A. √2Q B. 2Q C. 4Q D. 8Q

14. Two large metal plates carry equal and opposite charges spread over their surfaces, as shown in Figure 18.37. Which statements about these plates are correct?

A. The electrical potential at point a is higher than the potential at point b.

B. The electrical potential at point a is equal to the potential at point b.

C. The electric-field strength at point a is equal to the field strength at point b.

D. If a positive point charge is released at point a, it will move with constant velocity toward point b.

15. The electric potential (relative to infinity) due to a single point charge Q is +400 V at a point that is 0.90 m to the right of Q. The electric potential (relative to infinity) at a point 0.90 m to the left of Q is

A. -400 V B. +200 V C. + 400 V

**Problems**

1. A charge of 28.0 nC is placed in a uniform electric field that is directed vertically upward and that has a magnitude of 4.00 × 104 N/C. What work is done by the electric force when the charge moves (a) 0.450 m to the right; (b) 0.670 m upward; (c) 2.60 m at an angle of 45.0° downward from the horizontal?

2. Two very large charged parallel metal plates are 10.0 cm apart and produce a uniform electric field of 2.80 × 106 N/C between them. A proton is fired perpendicular to these plates with an initial speed of 5.20 km/s starting at the middle of the negative plate and going toward the positive plate. How much work has the electric field done on this proton by the time it reaches the positive plate?

3. How far from a -7.20 μC point charge must a +2.30 μC point charge be placed in order for the electric potential energy of the pair of charges to be -0.400 J? (Take the energy to be zero when the charges are infinitely far apart.)

4. A point charge q1 = + 2.40 μC is held stationary at the origin. A second point charge q2 = -4.30 μC moves from the point x = 0.150 m, y = 0, to the point x = 0.250 m, y = 0.250 m. How much work is done by the electric force q2?

5. Two stationary point charges of +3.00 nC and +2.00 nC are separated by a distance of 50.0 cm. An electron is released from rest at a point midway between the charges and moves along the line connecting them. What is the electric potential energy for the electron when it is (a) at the midpoint and (b) 10.0 cm from the +3.00 nC charge?

6. Energy of DNA base pairing, I. (See Problem 24 in Chapter 17; see also Figure 17.43.) (a) Calculate the electric potential energy of the adenine–thymine bond, using the same combinations of molecules (O – H – N and N – H – N) as in Problem 17.24. (b) Compare this energy with the potential energy of the proton–electron pair in the hydrogen atom.

7. Energy of DNA base pairing, II. (See Problem 25 in Chapter 17; see also Figure 17.44.). Calculate the electric potential energy of the guanine–cytosine bond, using the same combinations of molecules (O – H – O, N – H – N, and O – H – N) as in Problem 17.25.

8. (a) A set of point charges is held in place at the vertices of an equilateral triangle of side 10.0 cm, as shown in Figure 18.38(a). Find the maximum amount of total kinetic energy that will be produced when the charges are released from rest in the friction-less void of outer space. (b) If the charges at the vertices of the right triangle in Figure 18.38(b) are released, how much total kinetic energy will they gain? When will this maximum kinetic energy be achieved, just following the release of the charges or after a very long time?

9. Three equal point 1.20-μC charges are placed at the corners of an equilateral triangle whose sides are 0.500 m long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

10. When two point charges are a distance R apart, their potential energy is -2.0 J. How far (in terms of R) should they be from each other so that their potential energy is -6.0 J?

11. Two large metal parallel plates carry opposite charges of equal magnitude. They are separated by 45.0 mm, and the potential difference between them is 360 V. (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the force this field exerts on a particle with charge +2.40 nC?

12. A potential difference of 4.75 kV is established between parallel plates in air. If the air becomes ionized (and hence electrically conducting) when the electric field exceeds 3.00 × 106 V/m what is the minimum separation the plates can have without ionizing the air?

13. Oscilloscope. Oscilloscopes are found in most science laboratories. Inside, they contain deflecting plates consisting of more-or-less square parallel metal sheets, typically about 2.5 cm on each side and 2.0 mm apart. In many experiments, the maximum potential across these plates is about 25 V. For this maximum potential, (a) what is the strength of the electric field between the plates, and (b) what magnitude of acceleration would this field produce on an electron midway between the plates?

14. Axons. Neurons are the basic units of the nervous system. They contain long tubular structures called axons that propagate electrical signals away from the ends of the neurons. The axon contains a solution of potassium ions K+ and large negative organic ions. The axon membrane prevents the large ions from leaking out, but the smaller K+ ions are able to penetrate the membrane to some degree. This leaves an excess negative charge on the inner surface of the axon membrane and an excess of positive charge on the outer surface, resulting in a potential difference across the membrane that prevents further K+ ions from leaking out. Measurements show that this potential difference is typically about 70 mV. The thickness of the axon membrane itself varies from about 5 to 10 nm, so we’ll use an average of 7.5 nm. We can model the membrane as a large sheet having equal and opposite charge densities on its faces. (a) Find the electric field inside the axon membrane, assuming (not too realistically) that it is filled with air. Which way does it point, into or out of the axon? (b) Which is at a higher potential, the inside surface or the outside surface of the axon membrane?

15. Electrical sensitivity of sharks. Certain sharks can detect an electric field as weak as 1.0 μV/m. To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary 1.5 V AA battery across these plates, how far apart would the plates have to be?

16. A particle with a charge of +4.20 nC is in a uniform electric field directed to the left. It is released from rest and moves to the left; after it has moved 6.00 cm, its kinetic energy is found to be +1.50 × 10-6 J. (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the endpoint? (c) What is the magnitude of E?

17. Two very large metal parallel plates are 20.0 cm apart and carry equal, but opposite, surface charge densities. Figure 18.40 shows a graph of the potential, relative to the negative plate, as a function of x. For this case, x is the distance from the inner surface of the negative plate, measured perpendicular to the plates, and points from the negative plate toward the positive plate. Find the electric field between the plates.

18. A uniform electric field has magnitude E and is directed in the negative x-direction. The potential difference between point a (at x = 0.60 m) and point b (at x = 0.90 m) is 240 V. (a) Which point, a or b, is at the higher potential? (b) Calculate the value of E. (c) A negative point charge q = -0.200 μC is moved from b to a. Calculate the work done on the point charge by the electric field.

19. A point charge has a charge of 2.50 × 10-11 C. At what distance from the point charge is the electric potential (a) 90.0 V? (b) 30.0 V? Take the potential to be zero at an infinite distance from the charge.

20. (a) An electron is to be accelerated from 3.00 × 106 m/s to 8.00 × 106 m/s. Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from 8.00 × 106 m/s to a halt?

21. A small particle has charge -5.00 μC and mass 2.00 × 10-4 kg. It moves from point A, where the electric potential is VA = +200 V, to point B, where the electric potential is VB = +800 V. The electric force is the only force acting on the particle. The particle has speed 5.00 m/s at point A. What is its speed at point B? Is it moving faster or slower at B than at A?

22. Two point charges q1 = +2.40 nC and q2 = -6.50 nC are 0.100 m apart. Point A is midway between them; point B is 0.080 m from q1 and 0.060 m from q2. Take the electric potential to be zero at infinity. Find (a) the potential at point A; (b) the potential at point B; (c) the work done by the electric field on a charge of 2.50 nC that travels from point B to point A.

23. A point charge Q = +4.60 μC is held fixed at the origin. A second point charge q = +1.20 μC with mass of 2.80 × 10-4 kg is placed on the x axis, 0.250 m from the origin. (a)What is the electric potential energy U of the pair of charges? (Take U to be zero when the charges have infinite separation.) (b) The second point charge is released from rest. What is its speed when its distance from the origin is (i) 0.500 m; (ii) 5.00 m; (iii) 50.0 m?

24. Two protons are released from rest when they are 0.750 nm apart. (a) What is the maximum speed they will reach? When does this speed occur? (b) What is the maximum acceleration they will achieve? When does this acceleration occur?

25. Cathode-ray tube. A cathode-ray tube (CRT) is an evacuated glass tube. Electrons are produced at one end, usually by the heating of a metal. After being focused electromagnetically into a beam, they are accelerated through a potential difference, called the accelerating potential. The electrons then strike a coated screen, where they transfer their energy to the coating through collisions, causing it to glow. CRTs are found in oscilloscopes and computer monitors, as well as in earlier versions of television screens. (a) If an electron of mass m and charge –e is accelerated from rest through an accelerating potential V, show that the speed it gains is v = √(2eV/m). (b) If the accelerating potential is 95 V, how fast will the electrons be moving when they hit the screen?

26. X-ray tube. An X-ray tube is similar to a cathode-ray tube. (See previous problem.) Electrons are accelerated to high speeds at one end of the tube. If they are moving fast enough when they hit the target at the other end, they give up their energy as X-rays (a form of nonvisible light). (a) through what potential difference should electrons be accelerated so that their speed is 1.0% of the speed of light when they hit the target? (b) What potential difference would be needed to give protons the same kinetic energy as the electrons? (c) What speed would this potential difference give to protons? Express your answer in m/s and as a percent of the speed of light.

27. A gold nucleus has a radius of 7.3 × 10-15 m and a charge of +79e. Through what voltage must an α-particle with its charge of +2e, be accelerated so that it has just enough energy to reach a distance of 2.0 × 10-14 m from the surface of a gold nucleus? (Assume the gold nucleus remains stationary and can be treated as a point charge.)

28. A parallel-plate capacitor having plates 6.0 cm apart is connected across the terminals of a 12 V battery. (a) Being as quantitative as you can, describe the location and shape of the equipotential surface that is at a potential of relative to the potential of the negative plate. Avoid the edges of the plates. (b) Do the same for the equipotential surface that is at +2.0 V relative to the negative plate. (c) What is the potential gradient between the plates?

29. Two very large metal parallel plates that are 25 cm apart, oriented perpendicular to a sheet of paper, are connected across the terminals of a 50.0 V battery. (a) Draw to scale the lines where the equipotential surfaces due to these plates intersect the paper. Limit your drawing to the region between the plates, avoiding their edges, and draw the lines for surfaces that are 10.0 V apart, starting at the low-potential plate. (b) These surfaces are separated equally in potential. Are they also separated equally in distance? (c) In words, describe the shape and orientation of the surfaces you just found.

30. (a) A +5.00 pC charge is located on a sheet of paper. (a) Draw to scale the curves where the equipotential surfaces due to these charges intersect the paper. Show only the surfaces that have a potential (relative to infinity) of 1.00 V, 2.00 V, 3.00 V, 4.00 V, and 5.00 V. (b) The surfaces are separated equally in potential. Are they also separated equally in distance? (c) In words, describe the shape and orientation of the surfaces you just found.

31. A metal sphere carrying an evenly distributed charge will have spherical equipotential surfaces surrounding it. Suppose the sphere’s radius is 50.0 cm and it carries a total charge of +1.50 μC (a) Calculate the potential of the sphere’s surface. (b) You want to draw equipotential surfaces at intervals of 500 V outside the sphere’s surface. Calculate the distance between the first and the second equipotential surfaces, and between the 20th and 21st equipotential surfaces. (c) What does the changing spacing of the surfaces tell you about the electric field?

32. Figure 18.42 shows a set of electric-field lines for a particular distribution of charges. Use these lines to draw a series of equipotential surfaces for this system. Limit yourself to the plane of the paper.

33. Dipole. A dipole is located on a sheet of paper. (a) In the plane of that paper, carefully sketch the electric field lines for this dipole. (b) Use your field lines in part (a) to sketch the equipotential curves where the equipotential surfaces intersect the paper.

34. In a particular Millikan oil-drop apparatus, the plates are 2.25 cm apart. The oil used has a density of and the atomizer that sprays the oil drops produces drops of diameter (a) What strength of electric field is needed to hold such a drop stationary against gravity if the drop contains five excess electrons? (b) What should be the potential difference across the plates to produce this electric field? (c) If another drop of the same oil requires a plate potential of 73.8 V to hold it stationary, how many excess electrons did it contain?

35. (a) If an electron and a proton each have a kinetic energy of 1.00 eV, how fast is each one moving? (b) What would be their speeds if each had a kinetic energy of 1.00 keV? (c) If they were each traveling at 1.00% the speed of light, what would be their kinetic energies in keV?

36. (a) You find that if you place charges of ±1.25 μC on two separated metal objects, the potential difference between them is 11.3 V. What is their capacitance? (b) A capacitor has a capacitance of 7.28 μF. What amount of excess charge must be placed on each of its plates to make the potential difference between the plates equal to 25.0 V?

37. The plates of a parallel-plate capacitor are 3.28 mm apart, and each has an area of 12.2 cm2. Each plate carries a charge of magnitude 4.35 × 10-8 C. The plates are in vacuum. (a) What is the capacitance? (b) What is the potential difference between the plates? (c) What is the magnitude of the electric field between the plates?

38. The plates of a parallel-plate capacitor are 2.50 mm apart, and each carries a charge of magnitude 80.0 nC. The plates are in vacuum. The electric field between the plates has a magnitude of 4.00 × 106 V/m. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the capacitance?

39. A parallel-plate air capacitor has a capacitance of 500.0 pF and a charge of magnitude 0.200 μC on each plate. The plates are 0.600 mm apart. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the electric-field magnitude between the plates? (d) What is the surface charge density on each plate?

40. Capacitance of an oscilloscope. Oscilloscopes have parallel metal plates inside them to deflect the electron beam. These plates are called the deflecting plates. Typically, they are squares 3.0 cm on a side and separated by 5.0 mm, with vacuum in between. What is the capacitance of these deflecting plates and hence of the oscilloscope?

41. A 10.0 μF parallel-plate capacitor with circular plates is connected to a 12.0 V battery. (a) What is the charge on each plate? (b) How much charge would be on the plates if their separation were doubled while the capacitor remained connected to the battery? (c) How much charge would be on the plates if the capacitor were connected to the 12.0 V battery after the radius of each plate was doubled without changing their separation?

42. A 10.0 μF parallel-plate capacitor is connected to a 12.0 V battery. After the capacitor is fully charged, the battery is disconnected without loss of any of the charge on the plates. (a) A voltmeter is connected across the two plates without discharging them. What does it read? (b) What would the voltmeter read if (i) the plate separation were doubled; (ii) the radius of each plate was doubled, but the separation between the plates was unchanged?

43. You make a capacitor by cutting the 15.0-cm-diameter bottoms out of two aluminum pie plates, separating them by 3.50 mm, and connecting them across a 6.00-V battery. (a) What’s the capacitance of your capacitor? (b) If you disconnect the battery and separate the plates to a distance of 3.50 cm without discharging them, what will be the potential difference between them?

44. A 5.00 pF parallel-plate air-filled capacitor with circular plates is to be used in a circuit in which it will be subjected to potentials of up to 1.00 × 102 V. The electric field between the plates is to be no greater than 1.00 × 104 N/C. As a budding electrical engineer for Live-Wire Electronics, your tasks are to (a) design the capacitor by finding what its physical dimensions and separation must be and (b) find the maximum charge these plates can hold.

45. How far apart would parallel pennies have to be to make a 1.00-pF capacitor? Does your answer suggest that you are justified in treating these pennies as infinite sheets? Explain.

46. A parallel-plate capacitor C is charged up to a potential V0 with a charge of magnitude Q0 on each plate. It is then disconnected from the battery, and the plates are pulled apart to twice their original separation. (a) What is the new capacitance in terms of C? (b) How much charge is now on the plates in terms of Q0? (c) What is the potential difference across the plates in terms of V0?

47. For the system of capacitors shown in Figure 18.43, find the equivalent capacitance (a) between b and c, (b) between a and c.

48. Electric eels. Electric eels and electric fish generate large potential differences that are used to stun enemies and prey. These potentials are produced by cells that each can generate 0.10 V. We can plausibly model such cells as charged capacitors. (a) How should these cells be connected (in series or in parallel) to produce a total potential of more than 0.10 V? (b) Using the connection in part (a), how many cells must be connected together to produce the 500 V surge of the electric eel?

49. In Figure 18.44, C1 = 6.00 μF, C2 = 3.00 μF, and C3 = 5.00 μF. The capacitor network is connected to an applied potential Vab. After the charges on the capacitors have reached their final values, the charge on C2 is 40.0 μC. (a) What are the charges on capacitors C1 and C3 (b) What is the applied voltage Vab?

50. You are working on an electronics project requiring a variety of capacitors, but have only a large supply of 100 nF capacitors available. Show how you can connect these capacitors to produce each of the following equivalent capacitances: (a) 50 nF, (b) 450 nF, (c) 25 nF, (d) 75 nF.

51. In Figure 18.44, C1 = 3.00 μF and Vab = 120 V. The charge on capacitor C1 is 150 μC. Calculate the voltage across the other two capacitors.

52. A 4.00 μF and a 6.00 μF capacitor are connected in series, and this combination is connected across a 48.0 V potential difference. Calculate (a) the charge on each capacitor and (b) the potential difference across each of them.

53. In the circuit shown in Figure 18.45, the potential difference across ab is +24.0 V. Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor.

54. In Figure 18.46, each capacitor has C = 4.00 μF and Vab = + 28.0 V. Calculate (a) the charge on each capacitor and (b) the potential difference across each capacitor.

55. Figure 18.47 shows a system of four capacitors, where the potential difference across ab is 50.0 V. (a) Find the equivalent capacitance of this system between a and b. (b) How much charge is stored by this combination of capacitors? (c) How much charge is stored in each of the 10.0 μF and the 9.0 μF capacitors?

56. For the system of capacitors shown in Figure 18.48, a potential difference of 25 V is maintained across ab. (a) What is the equivalent capacitance of this system between a and b? (b) How much charge is stored by this system? (c) How much charge does the 6.5 nF capacitor store? (d) What is the potential difference across the 7.5 nF

57. How much charge does a 12 V battery have to supply to fully charge a 2.5 μF capacitor and a 5.0 μF capacitor when they’re (a) in parallel, (b) in series? (c) How much energy does the battery have to supply in each case?

58. A 5.80 μF parallel-plate air capacitor has a plate separation of 5.00 mm and is charged to a potential difference of 400 V. Calculate the energy density in the region between the plates, in units of J/m3.

59. (a) How much charge does a battery have to supply to a 5.0 μF capacitor to create a potential difference of 1.5 V across its plates? How much energy is stored in the capacitor in this case? (b) How much charge would the battery have to supply to store 1.0 J of energy in the capacitor? What would be the potential across the capacitor in that case?

60. In the text, it was shown that the energy stored in a capacitor C charged to a potential V is U = QV/2. Show that this energy can also be expressed as (a) U = Q2/(2C) and (b) U = CV2/2.

61. A parallel-plate vacuum capacitor has 8.38 J of energy stored in it. The separation between the plates is 2.30 mm. If the separation is decreased to 1.15 mm, what is the energy stored (a) if the capacitor is disconnected from the potential source so the charge on the plates remains constant, and (b) if the capacitor remains connected to the potential source so the potential difference between the plates remains constant?

62. (a) How many excess electrons must be added to one plate and removed from the other to give a 5.00 nF parallel-plate capacitor 25.0 μJ of stored energy? (b) How could you modify the geometry of this capacitor to get it to store 50.0 μJ of energy without changing the charge on its plates?

63. For the capacitor network shown in Figure 18.49, the potential difference across ab is 36 V. Find (a) the total charge stored in this network, (b) the charge on each capacitor, (c) the total energy stored in the network, (d) the energy stored in each capacitor, and (e) the potential difference across each capacitor.

64. For the capacitor network shown in Figure 18.50, the potential difference across ab is 220 V. Find (a) the total charge stored in this network, (b) the charge on each capacitor, (c) the total energy stored in the network, (d) the energy stored in each capacitor, and (e) the potential difference across each capacitor.

65. A 20.0 μC capacitor is charged to a potential difference of 800 V. The terminals of the charged capacitor are then connected to those of an uncharged 10.0 μC capacitor. Compute (a) the original charge of the system, (b) the final potential difference across each capacitor, (c) the final energy of the system, and (d) the decrease in energy when the capacitors are connected.

66. For the capacitor network shown in Figure 18.51, the potential difference across ab is 12.0 V. Find (a) the total energy stored in this network and (b) the energy stored in the 4.80 μF capacitor.

67. A parallel-plate air capacitor has a capacitance of 920 pF. The charge on each plate is 2.55 μC (a) What is the potential difference between the plates? (b) If the charge is kept constant, what will be the potential difference between the plates if the separation is doubled? (c) How much work is required to double the separation?

68. A parallel-plate capacitor has capacitance C0 = 5.00 pF when there is air between the plates. The separation between the plates is 1.50 mm. (a) What is the maximum magnitude of charge Q that can be placed on each plate if the electric field in the region between the plates is not to exceed 3.00 × 104 V/m? (b) A dielectric with K = 2.70 is inserted between the plates of the capacitor, completely filling the volume between the plates. Now what is the maximum magnitude of charge on each plate if the electric field between the plates is not to exceed 3.00 × 104 V/m?

69. Cell membranes. Cell membranes (the walled enclosure around a cell) are typically about 7.5 nm thick. They are partially permeable to allow charged material to pass in and out, as needed. Equal but opposite charge densities build up on the inside and outside faces of such a membrane, and these charges prevent additional charges from passing through the cell wall. We can model a cell membrane as a parallel-plate capacitor, with the membrane itself containing proteins embedded in an organic material to give the membrane a dielectric constant of about 10. (a) What is the capacitance per square centimeter of such a cell wall? (b) In its normal resting state, a cell has a potential difference of 85 mV across its membrane. What is the electric field inside this membrane?

70. A parallel-plate capacitor is to be constructed by using, as a dielectric, rubber with a dielectric constant of 3.20 and a dielectric strength of 20.0 MV/m. The capacitor is to have a capacitance of 1.50 nF and must be able to withstand a maximum potential difference of 4.00 kV. What is the minimum area the plates of this capacitor can have?

71. A 12.5 μC capacitor is connected to a power supply that keeps a constant potential difference of 24.0 V across the plates. A piece of material having a dielectric constant of 3.75 is placed between the plates, completely filling the space between them. (a) How much energy is stored in the capacitor before and after the dielectric is inserted? (b) By how much did the energy change during the insertion? Did it increase or decrease?

72. The paper dielectric in a paper-and-foil capacitor is 0.0800 mm thick. Its dielectric constant is 2.50, and its dielectric strength is 50.0 MV/m. Assume that the geometry is that of a parallel-plate capacitor, with the metal foil serving as the plates. (a) What area of each plate is required for a 0.200 μF capacitor? (b) If the electric field in the paper is not to exceed one-half the dielectric strength, what is the maximum potential difference that can be applied across the capacitor?

73. A constant potential difference of 12 V is maintained between the terminals of a 0.25 μF parallel-plate, air capacitor. (a) A sheet of Mylar is inserted between the plates of the capacitor, completely filling the space between the plates. When this is done, how much additional charge flows onto the positive plate of the capacitor (see Table 18.1)? (b) What is the total induced charge on either face of the Mylar sheet? (c) What effect does the Mylar sheet have on the electric field between the plates? Explain how you can reconcile this with the increase in charge on the plates, which acts to increase the electric field.

74. (a) If a spherical raindrop of radius 0.650 mm carries a charge of -1.20 pC uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infinite distance from the raindrop.) (b) Two identical raindrops, each with radius and charge specified in part (a), collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?

75. At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 V and 12.0 V/m, respectively. (Take the potential to be zero at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric field directed toward or away from the point charge?

76. Two oppositely charged identical insulating spheres, each 50.0 cm in diameter and carrying a uniform charge of magnitude 175 μC are placed 1.00 m apart center to center (Fig. 18.53). (a) If a voltmeter is connected between the nearest points (a and b) on their surfaces, what will it read? (b) Which point, a or b, is at the higher potential? How can you know this without any calculations?

77. Potential in human cells. Some cell walls in the human body have a layer of negative charge on the inside surface and a layer of positive charge of equal magnitude on the outside surface. Suppose that the charge density on either surface is ±0.50 × 10-3 C/m2, the cell wall is 5.0 nm thick, and the cell-wall material is air. (a) Find the magnitude of E in the wall between the two layers of charge. (b) Find the potential difference between the inside and the outside of the cell. Which is at the higher potential? (c) A typical cell in the human body has a volume of Estimate the total electric-field energy stored in the wall of a cell of this size. (d) In reality, the cell wall is made up, not of air, but of tissue with a dielectric constant of 5.4. Repeat parts (a) and (b) in this case.

78. An alpha particle with a kinetic energy of 10.0 MeV makes a head-on collision with a gold nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the gold nucleus remains stationary and that it may be treated as a point charge. The atomic number of gold is 79, and an alpha particle is a helium nucleus consisting of two protons and two neutrons.)

79. In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius r. Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron’s speed. (b) Obtain an expression for the electron’s kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using r = 5.29 × 10-11 m. Give your numerical result in joules and in electron volts.

80. A proton and an alpha particle are released from rest when they are 0.225 nm apart. The alpha particle (a helium nucleus) has essentially four times the mass and two times the charge of a proton. Find the maximum speed and maximum acceleration of each of these particles. When do these maxima occur, just following the release of the particles or after a very long time?

81. A parallel-plate air capacitor is made from two plates 0.200 m square, spaced 0.800 cm apart. It is connected to a 120-V battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of 1.60 cm, what are the answers to parts (a), (b), (c), and (d)?

82. In the previous problem, suppose the battery remains connected while the plates are pulled apart. What are the answers then to parts (a), (b), (c), and (d) after the plates have been pulled apart?

83. A capacitor consists of two parallel plates, each with an area of 16.0 cm2, separated by a distance of 0.200 cm. The material that fills the volume between the plates has a dielectric constant of 5.00. The plates of the capacitor are connected to a 300-V battery. (a) What is the capacitance of the capacitor? (b) What is the charge on either plate? (c) How much energy is stored in the charged capacitor?

84. Electronic flash units for cameras contain a capacitor for storing the energy used to produce the flash. In one such unit, the flash lasts for 1/675 s with an average light power output of 2.70 × 105 W (a) If the conversion of electrical energy to light is 95% efficient (the rest of the energy goes to thermal energy), how much energy must be stored in the capacitor for one flash? (b) The capacitor has a potential difference between its plates of 125 V when the stored energy equals the value calculated in part (a). What is the capacitance?

85. In Figure 18.54, each capacitance C1 is 6.9 μF and each capacitance C2 is 4.6 μF (a) Compute the equivalent capacitance of the network between points a and b. (b) Compute the charge on each of the three capacitors nearest a and b when Vab = 420 V.

86. A parallel-plate capacitor is made from two plates 12.0 cm on each side and 4.50 mm apart. Half of the space between these plates contains only air, but the other half is filled with Plexiglas® of dielectric constant 3.40. An 8.0 V battery is connected across the plates. (a) What is the capacitance of this combination? (b) How much energy is stored in the capacitor? (c) If we remove the Plexiglas®, but change nothing else, how much energy will be stored in the capacitor?

87. A parallel-plate capacitor with plate separation d has the space between the plates filled with two slabs of dielectric, one with constant and the other with constant and each having thickness d/2 (a) Show that the capacitance is given by C = 2εA/d(K1K2/(K1+K2)) (b) To see if your answer is reasonable, check it in the following cases: (i) There is only one dielectric, with constant K, and it completely fills the space between the plates. (ii) The plates have nothing but air, which we can treat as vacuum, between them.

**Chapter 19**

Multiple-Choice Problems

1. A cylindrical metal rod has a resistance R. If both its length and its diameter are tripled, its new resistance will be:

A. R B. 9R C. R/3 D. 3R

2. A resistor R and another resistor 2R are connected in series across a battery. If heat is produced at a rate of 10 W in R, then in 2R it is produced at a rate of

A. 40 W B. 20 W C. 10 W D. 5 W

3. A resistor R and another resistor 2R are connected in parallel across a battery. If heat is produced at a rate of 10 W in R, in 2R it is produced at a rate of

A. 40 W B. 20 W C. 10 W D. 5 W

4. Which statements about the circuit shown in Figure 19.39 are correct? All meters are considered to be ideal, the connecting leads have no resistance, and the battery has no internal resistance.

A. The reading in ammeter A1 is greater than the reading in A2 because current is lost in the resistor.

B. The two ammeters have exactly the same readings.

C. The voltmeter reads less than 25 V because some voltage is lost in the resistor.

D. The voltmeter reads exactly 25 V.

5. When the switch S in Figure 19.40 is closed, the reading of the voltmeter V will

A. increase

B. decrease

C. stay the same

6. When the switch S in the circuit in the previous question is closed, the reading of the ammeter A will

A. increase B. decrease C. stay the same

7. Three identical lightbulbs are connected in the circuit shown in Figure 19.41. After the switch S is closed, what will be true about the brightness of these bulbs?

A. B1 will be brightest and B3 will be dimmest.

B. B3 will be brightest and B1 will be dimmest.

C. All three bulbs will have the same brightness.

8. A cylindrical metal rod of length L and diameter D is connected across a battery having no internal resistance. An ammeter in the circuit measures the current to be I. If we now double the diameter of the rod, but change nothing else, the ammeter will read

A. 4I B. 2I C. I/2 D. I/4

9. Two identical metal rods are welded together end to end. If each rod has a length L and resistivity ρ, the resistivity of the combination will be

A. 4 ρ B. 2 ρ C. ρ D. ρ/2

10. In the circuit shown in Figure 19.42, resistor A has three times the resistance of resistor B. Therefore,

A. the current through A is three times the current through B.

B. the current through B is three times the current through A.

C. the potential difference across A is three times the potential difference across B.

D. the potential difference is the same across both resistors.

11. In which of the two circuits shown in Figure 19.43 will the capacitors charge more rapidly when the switch is closed?

A. Circuit (a)

B. Circuit (b)

C. The capacitors will charge at the same rate in the two circuits.

12. The battery shown in the circuit in Figure 19.44 has some internal resistance. When we close S, the reading of the voltmeter V will

A. increase. B. stay the same. C. decrease.

13. A battery with no internal resistance is connected across identical lightbulbs as shown

in Figure 19.45. When you close the switch S, bulbs B1 and B2 will be

A. brighter than before. B. dimmer than before. C. just as bright as before.

14. The battery shown in the circuit in Figure 19.46 has no internal resistance. After you close the switch S, the brightness of bulb will

A. increase. B. decrease. C. remain the same.

15. Three identical light bulbs, A, B, and C, are connected in the circuit shown in Figure 19.47. When the switch S is closed,

A. the brightness of A and B remains the same as it was, but C goes out.

B. the brightness of A and B remains the same as it was, but C will be about half as bright as it was.

C. the brightness of A and B decreases, and C goes out.

D. the brightness of A and B increases, and C will be about half as bright as it was.

E. the brightness of A and B increases, but C goes out.

**Problems**

1. Typical household currents are on the order of a few amperes. If a 1.50 A current flows through the leads of an electrical appliance, (a) how many electrons per second pass through it, (b) how many coulombs pass through it in 5.0 min, and (c) how long does it take for 7.50 C of charge to pass through?

2. Lightning strikes. During lightning strikes from a cloud to the ground, currents as high as 25,000 A can occur and last for about 40 μs. How much charge is transferred from the cloud to the earth during such a strike?

3. Transmission of nerve impulses. Nerve cells transmit electric signals through their long tubular axons. These signals propagate due to a sudden rush of Na+ ions, each with charge +e into the axon. Measurements have revealed that typically about 5.6 × 1011 Na+ ions enter each meter of the axon during a time of 10 ms. What is the current during this inflow of charge in a meter of axon?

4. In an ionic solution, a current consists of Ca2+ ions (of charge +2e) and Cl- ions (of charge –e) traveling in opposite directions. If 5.11 × 1018 Cl- ions go from A to B every 0.50 min, while 3.24 × 1018 Ca2+ ions move from B to A, what is the current (in mA) through this solution, and in which direction (from A to B or from B to A) is it going?

5. Copper has 8.5 × 1028 electrons per cubic meter. (a) How many electrons are there in a 25.0 cm length of 12-gauge copper wire (diameter 2.05 mm)? (b) If a current of 1.55 A is flowing in the wire, what is the average drift speed of the electrons along the wire?

6. A 14 gauge copper wire of diameter 1.628 mm carries a current of 12.5 mA. (a) What is the potential difference across a 2.00 m length of the wire? (b) What would the potential difference in part (a) be if the wire were silver instead of copper, but all else was the same?

7. You want to precut a set of 1.00 Ω strips of 14 gauge copper wire (of diameter 1.628 mm). How long should each strip be?

8. A wire 6.50 m long with diameter of 2.05 mm has a resistance of 0.0290 Ω. What material is the wire most likely made of?

9. A tightly coiled spring having 75 coils, each 3.50 cm in diameter, is made of insulated metal wire 3.25 mm in diameter. An ohmmeter connected across opposite ends of the spring reads 1.74 Ω. What is the resistivity of the metal?

10. What diameter must a copper wire have if its resistance is to be the same as that of an equal length of aluminum wire with diameter 3.26 mm?

11. An aluminum bar 3.80 m long has a rectangular cross section 1.00 cm by 5.00 cm. (a) What is its resistance? (b) What is the length of a copper wire 1.50 mm in diameter having the same resistance?

12. If you triple the length of a cable and at the same time double its diameter, what will be its resistance if its original resistance was R?

13. A ductile metal wire has resistance R. What will be the resistance of this wire in terms of R if it is stretched to three times its original length, assuming that the density and resistivity of the material do not change when the wire is stretched.

14. What is the resistance of a Nichrome™ wire at 0.0°C if its resistance is 100.00 Ω at 11.5°C? The temperature coefficient of resistivity for Nichrome™ is 0.00040 (C°)-1.

15. A 1.50-m cylindrical rod of diameter 0.500 cm is connected to a power supply that maintains a constant potential difference of 15.0 V across its ends, while an ammeter measures the current through it. You observe that at room temperature (20.0 °C) the ammeter reads 18.5 A, while at 92.0 °C it reads 17.2 A. You can ignore any thermal expansion of the rod. Find (a) the resistivity and (b) the temperature coefficient of resistivity at 20°C for the material of the rod.

16. A carbon resistor having a temperature coefficient of resistivity of 0.00050 (C°)-1 is to be used as a thermometer. On a winter day when the temperature is 4.0 °C, the resistance of the carbon resistor is 217.3 Ω. What is the temperature on a spring day when the resistance is 215.8 Ω.

17. In a laboratory experiment, you vary the current through an object and measure the resulting potential difference across it in each case. Figure 19.48 shows a graph of this potential V as a function of the current I. (a) Does Ohm’s law apply to this object? Why do you say so? (b) How is the resistance of the object related to the slope of the graph? Show why. (c) Use the slope of the graph to find the resistance of the object.

18. The following measurements of current and potential difference were made on a resistor constructed of Nichrome™ wire, where is the potential difference across the wire and I is the current through it:

I(A) 0.50 1.00 2.00 4.00

Vab(V) 1.94 3.88 7.76 15.52

(a) Graph Vab as a function of I. (b) Does Ohm’s law apply to Nichrome™? How can you tell? (c) What is the resistance of the resistor in ohms?

19. A battery-powered light bulb has a tungsten filament. When the switch connecting the bulb to the battery is first turned on and the temperature of the bulb is 20 °C, the current in the bulb is 0.860 A. After the bulb has been on for 30 s, the current is 0.220 A. What is then the temperature of the filament?

20. When you connect an unknown resistor across the terminals of a 1.50 V AAA battery having negligible internal resistance, you measure a current of 18.0 mA flowing through it. (a) What is the resistance of this resistor? (b) If you now place the resistor across the terminals of a 12.6 V car battery having no internal resistance, how much current will flow? (c) You now put the resistor across the terminals of an unknown battery of negligible internal resistance and measure a current of 0.453 A flowing through it. What is the potential difference across the terminals of the battery?

21. Current in the body. The resistance of the body varies from approximately 500 kΩ (when it is very dry) to about 1 kΩ (when it is wet). The maximum safe current is about 5.0 mA. At 10 mA or above, muscle contractions can occur that may be fatal. What is the largest potential difference that a person can safely touch if his body is wet? Is this result within the range of common household voltages?

22. “Current Baba.” According to a July 20, 2004, newspaper article, a Hindu holy man known as “Current Baba” touches an electric wire three times daily to become “intoxicated.” According to a doctor quoted in the article, “The human body can absorb currents up to 12 volts. In this case, however, repeated exposure to electricity seems to have built up [“Current Baba’s”] body’s tolerance levels to as much as 16 volts.” (a) What is wrong with the doctor’s statement? What do you think he really meant to say? (b) Since “Current Baba” was after the maximum “intoxication,” he should have his body wet. In that case, how much current would he get with each jolt? Is this enough to be dangerous?

23. A copper transmission cable 100 km long and 10.0 cm in diameter carries a current of 125 A. What is the potential drop across the cable?

24. A gold wire 6.40 m long and of diameter 0.840 mm carries a current of 1.15 A. Find (a) the resistance of this wire and (b) the potential difference between its ends.

25. When a solid cylindrical rod is connected across a fixed potential difference, a current I flows through the rod. What would be the current (in terms of I) if (a) the length were doubled, (b) the diameter were doubled, (c) both the length and the diameter were doubled?

26. A 6.00 V lantern battery is connected to a 10.5 Ω lightbulb, and the resulting current in the circuit is 0.350 A. What is the internal resistance of the battery?

27. When switch S in Figure 9.49 is open, the voltmeter V across the battery reads 3.08 V. When the switch is closed, the voltmeter reading drops to 2.97 V and the ammeter A reads 1.65 A. Find the emf, the internal resistance of the battery, and the circuit resistance R. Assume that the two meters are ideal, so that they don’t affect the circuit.

28. A complete series circuit consists of a 12.0 V battery, a 4.70 Ω resistor, and a switch. The internal resistance of the battery is 0.30 Ω. The switch is open. What does an ideal voltmeter read when placed (a) across the terminals of the battery, (b) across the resistor, (c) across the switch? (d) Repeat parts

(a), (b), and (c) for the case when the switch is closed.

29. With a 1500 MΩ resistor across its terminals, the terminal voltage of a certain battery is 2.50 V. With only a 5.00 Ω resistor across its terminals, the terminal voltage is 1.75 V. (a) Find the internal emf and the internal resistance of this battery. (b) What would be the terminal voltage if the 5.00 Ω resistor were replaced by a 7.00 Ω resistor?

30. An automobile starter motor is connected to a 12.0 V battery. When the starter is activated it draws 150 A of current, and the battery voltage drops to 7.0 V. What is the battery’s internal resistance?

31. Consider the circuit shown in Figure 19.50. The terminal voltage of the 24.0 V battery is 21.2 V. What is (a) the internal resistance r of the battery; (b) the resistance R of the circuit resistor?

32. When switch S in Fig. 19.51 is open, the voltmeter V of the battery reads 3.08 V. When the switch is closed, the voltmeter reading drops to 2.97 V, and the ammeter A reads 1.65 A. Find the emf, the internal resistance of the battery, and the circuit resistance R. Assume that the two meters are ideal, so they don’t affect the circuit.

33. A resistor with a 15.0 V potential difference across its ends develops thermal energy at a rate of 327 W. (a) What is the current in the resistor? (b) What is its resistance?

34. Power rating of a resistor. The power rating of a resistor is the maximum power it can safely dissipate without being damaged by overheating. (a) If the power rating of a certain 15 kΩ resistor is 5.0 W, what is the maximum current it can carry without damage? What is the greatest allowable potential difference across the terminals of this resistor? (b) If a 9.0 kΩ resistor is to be connected across a 120 V potential difference, what power rating is required for that resistor?

35. An idealized voltmeter is connected across the terminals of a 15.0 V battery, and a 75.0 Ω appliance is also connected across its terminals. If the voltmeter reads 11.3 V: (a) how much power is being dissipated by the appliance, and (b) what is the internal resistance of the battery?

36. Treatment of heart failure. A heart defibrillator is used to enable the heart to start beating if it has stopped. This is done by passing a large current of 12 A through the body at 25 V for a very short time, usually about 3.0 ms. (a) What power does the defibrillator deliver to the body, and (b) how much energy is transferred?

37. Lightbulbs. The wattage rating of a lightbulb is the power it consumes when it is connected across a 120 V potential difference. For example, a 60 W lightbulb consumes 60.0 W of electrical power only when it is connected across a 120 V potential difference. (a) What is the resistance of a 60 W lightbulb? (b) Without doing any calculations, would you expect a 100 W bulb to have more or less resistance than a 60 W bulb? Calculate and find out.

38. Electrical safety. This procedure is not recommended! You’ll see why after you work the problem. You are on an aluminum ladder that is standing on the ground, trying to fix an electrical connection with a metal screwdriver having a metal handle. Your body is wet because you are sweating from the exertion; therefore, it has a resistance of 1.0 kΩ. (a) If you accidentally touch the “hot” wire connected to the 120 V line, how much current will pass through your body? Is this amount enough to be dangerous? (The maximum safe current is about 5 mA.) (b) How much electrical power is delivered to your body?

39. Electric eels. Electric eels generate electric pulses along their skin that can be used to stun an enemy when they come into contact with it. Tests have shown that these pulses can be up to 500 V and produce currents of 80 mA (or even larger). A typical pulse lasts for 10 ms. What power and how much energy are delivered to the unfortunate enemy with a single pulse, assuming a steady current?

40. Electric space heater. A “540 W” electric heater is designed to operate from 120 V lines. (a) What is its resistance, and (b) what current does it draw? (c) At 7.4 ¢ per how much does it cost to operate this heater for an hour? (d) If the line voltage drops to 110 V, what power does the heater take, in watts? (Assume that the resistance is constant, although it actually will change because of the change in temperature.)

41. The battery for a certain cell phone is rated at 3.70 V. According to the manufacturer it can produce 3.15 × 104 J of electrical energy, enough for 5.25 h of operation, before needing to be recharged. Find the average current that this cell phone draws when turned on.

42. For the circuit in Fig. 19.52, find (a) the rate of conversion of internal (chemical) energy to electrical energy within the battery, (b) the rate of dissipation of electrical energy in the battery, (c) the rate of dissipation of electrical energy in the external resistor.

43. 540-W electric heater is designed to operate from 120 V lines. (a) What is its resistance? (b) What current does it draw? (c) If the line voltage drops to 110 V, what power does the heater take? (Assume that the resistance is constant. Actually, it will change because of the change in temperature.) (d) The heater coils are metallic, so that the resistance of the heater decreases with decreasing temperature. If the change of resistance with temperature is taken into account, will the electrical power consumed by the heater be larger or smaller than what you calculated in part (c)? Explain.

44. Electricity through the body, I. A person with a body resistance of 10 kΩ between his hands accidentally grasps the terminals of a 14 kV power supply. (a) If the internal resistance of the power supply is 2000 Ω what is the current through the person’s body? (b) What is the power dissipated in his body? (c) If the power supply is to be made safe by increasing its internal resistance, what should the internal resistance be for the maximum current in the situation just described to be 1.00 mA or less?

45. Electricity through the body, II. The average bulk resistivity of the human body (apart from surface resistance of the skin) is about 5.0 Ω·m. The conducting path between the hands can be represented approximately as a cylinder 1.6 m long and 0.10 m in diameter. The skin resistance can be made negligible by soaking the hands in salt water. (a) What is the resistance between the hands if the skin resistance is negligible? (b) What potential difference between the hands is needed for a lethal shock current of 100 mA? (Note that your result shows that small potential differences produce dangerous currents when the skin is damp.) (c) With the current in part (b), what power is dissipated in the body?

46. Find the equivalent resistance of each combination shown in Figure 19.53.

47. Calculate the (a) maximum and (b) minimum values of resistance that can be obtained by combining resistors of 36 Ω, 47 Ω, and 51 Ω.

48. Each of two identical uniform metal bars has a resistance R. If they are welded together along one-third of their lengths, what is the resistance of this combination in terms of R?

49. A resistor and a resistor are connected in parallel, and the combination is connected across a 120-V dc line. (a) What is the resistance of the parallel combination? (b) What is the total current through the parallel combination? (c) What is the current through each resistor?

50. Three resistors having resistances of 1.60 Ω, 2.40 Ω, and 4.80 Ω, respectively, are connected in parallel to a 28.0 V battery that has negligible internal resistance. Find (a) the equivalent resistance of the combination, (b) the current in each resistor, (c) the total current through the battery, (d) the voltage across each resistor, and (e) the power dissipated in each resistor. (f) Which resistor dissipates the most power, the one with the greatest resistance or the one with the least resistance? Explain why this should be.

51. Now the three resistors of the previous problem are connected in series to the same battery. Answer the same questions for this situation.

52. Compute the equivalent resistance of the network in Figure 19.55, and find the current in each resistor. The battery has negligible internal resistance.

53. Compute the equivalent resistance of the network in Figure 19.56, and find the current in each resistor. The battery has negligible internal resistance.

54. Lightbulbs in series, I. The power rating of a lightbulb is the power it consumes when connected across a 120 V outlet. (a) If you put two 100 W bulbs in series across a 120 V outlet, how much power would each consume if its resistance were constant? (b) How much power does each one consume if you connect them in parallel across a 120 V outlet?

55. You absentmindedly solder a 69.8 kΩ resistor into a circuit where a 36.5 kΩ should be. How can you get the proper resistance without replacing the bigger resistor or removing anything from the circuit?

56. You need to connect a 68 kΩ resistor and one other resistor to a 110 V power line. If you want the two resistors to use 4 times as much power when connected in parallel as they use when connected in series, what should be the value of the unknown resistor?

57. The batteries shown in the circuit in Figure 19.57 have negligibly small internal resistances. Find the current through (a) the 30.0 Ω resistor, (b) the 20.0 Ω resistor, and (c) the 10.0 V battery.

58. Find the emf’s E1 and E2 in the circuit shown in Figure 19.58.

59. In the circuit shown in Figure 19.59, ammeter A1 reads 10.0 A and the batteries have no appreciable internal resistance. (a) What is the resistance of R? (b) Find the readings in the other ammeters.

60. In the circuit shown in Figure 19.60, find (a) the current in resistor R, (b) the value of the resistance R, and (c) the unknown emf E.

61. In the circuit shown in Figure 19.61, current flows through the 5.00 Ω resistor in the direction shown, and this resistor is measured to be consuming energy at a rate of 20.0 W. The batteries have negligibly small internal resistance. What current does the ammeter A read?

62. In the circuit shown in Fig. 19.62, the 6.0 Ω resistor is consuming energy at a rate of 24 J/s when the current through it flows as shown. (a) Find the current through the ammeter A. (b) What are the polarity and emf of the battery E, assuming it has negligible internal resistance?

63. A 500.0 Ω resistor is connected in series with a capacitor. What must be the capacitance of the capacitor to produce a time constant of 2.00 s?

64. A fully charged 6.0 μF capacitor is connected in series with a 1.5 × 105 Ω resistor. What percentage of the original charge is left on the capacitor after 1.8 s of discharging?

65. A 12.4 μF capacitor is connected through a 0.895 MΩ resistor to a constant potential difference of 60.0 V. (a) Compute the charge on the capacitor at the following times after the connections are made: 0, 5.0 s, 10.0 s, 20.0 s, and 100.0 s. (b) Compute the charging currents at the same instants. (c) Graph the results of parts (a) and (b) for t between 0 and 20 s.

66. A 6.00 μF capacitor that is initially uncharged is connected in series with a 4500 Ω resistor and a 500 V emf source with negligible internal resistance. Just after the circuit is completed, what are (a) the voltage drop across the capacitor, (b) the voltage drop across the resistor, (c) the charge on the capacitor, and (d) the current through the resistor? (e) A long time after the circuit is completed (after many time constants), what are the values of the preceding four quantities?

67. A capacitor is charged to a potential of 12.0 V and is then connected to a voltmeter having an internal resistance of 3.40 MΩ. After a time of 4.00 s the voltmeter reads 3.0 V. What are (a) the capacitance and (b) the time constant of the circuit?

68. When a capacitor is being charged up, (a) how many time constants are required for it to receive 95% of its maximum charge, and (b) what is the current in the circuit at that time?

69. In the circuit shown in Figure 19.63, the capacitors are all initially uncharged and the battery has no appreciable internal resistance. After the switch S is closed, find (a) the maximum charge on each capacitor, (b) the maximum potential difference across each capacitor, (c) the maximum reading of the ammeter A, and (d) the time constant for the circuit.

70. Charging and discharging a capacitor. A 1.50 μF capacitor is charged through a 125 Ω resistor and then discharged through the same resistor by short-circuiting the battery. While the capacitor is being charged, find (a) the time for the charge on its plates to reach 1 – 1/e of its maximum value and (b) the current in the circuit at that time. (c) During the discharge of the capacitor, find the time for the charge on its plates to decrease to 1/e of its initial value. Also, find the time for the current in the circuit to decrease to 1/e of its initial value.

71. Charging and discharging a capacitor. An initially uncharged capacitor C charges through a resistor R for many time constants and then discharges through the same resistor. Call Cmax the maximum charge on its plates and Imax the maximum current in the circuit. (a) Sketch clear graphs of the charge on the plates and the current in the circuit as functions of time for the charging process. (b) During the discharging process, the charge on the capacitor and the current both decrease exponentially from their maximum values. Use this fact to sketch graphs of the current in the circuit and the charge on the capacitor as functions of time.

72. The circuit shown in Figure 19.64 contains two batteries, each with an emf and an internal resistance, and two resistors. Find (a) the current in the circuit (magnitude and direction) and (b) the terminal voltage Vab of the 16.0 V battery.

73. If an ohmmeter is connected between points a and b in each of the circuits shown in Fig. 19.65, what will it read?

74. A refrigerator draws 3.5 A of current while operating on a 120 V power line. If the refrigerator runs 50% of the time and electric power costs $0.12 per kWh, how much does it cost to run this refrigerator for a 30-day month?

75. A toaster using a Nichrome™ heating element operates on 120 V. When it is switched on at 20 °C, the heating element carries an initial current of 1.35 A. A few seconds later, the current reaches the steady value of 1.23 A. (a) What is the final temperature of the element? The average value of the temperature coefficient of resistivity for Nichrome™ over the temperature range from 20°C to the final temperature of the element is 4.5 × 10-4 (ºC)-1 (b) What is the power dissipated in the heating element (i) initially; (ii) when the current reaches a steady value?

76. A piece of wire has a resistance R. It is cut into three pieces of equal length, and the pieces are twisted together parallel to each other. What is the resistance of the resulting wire in terms of R?

77. Flashlight batteries. A typical small flashlight contains two batteries, each having an emf of 1.5 V, connected in series with a bulb having resistance 17 Ω. (a) If the internal resistance 17 V. of the batteries is negligible, what power is delivered to the bulb? (b) If the batteries last for 5.0 h, what is the total energy delivered to the bulb? (c) The resistance of real batteries increases as they run down. If the initial internal resistance is negligible, what is the combined internal resistance of both batteries when the power to the bulb has decreased to half its initial value?

78. In the circuit of Figure 19.66, find (a) the current through the 8.0 Ω resistor and (b) the total rate of dissipation of electrical energy in the 8.0 Ω resistor and in the internal resistance of the batteries. (c) In one of the batteries, chemical energy is being converted into electrical energy. In which one is this happening and at what rate?

79. Struck by lightning. Lightning strikes can involve currents as high as 25,000 A that last for about 40 μs. If a person is struck by a bolt of lightning with these properties, the current will pass through his body. We shall assume that his mass is 75 kg, that he is wet (after all, he is in a rainstorm) and therefore has a resistance of and that his body is all water (which is reasonable for a rough, but plausible, approximation). (a) By how many degrees Celsius would this lightning bolt increase the temperature of 75 kg of water? (b) Given that the internal body temperature is about 37 °C, would the person’s temperature actually increase that much? Why not? What would happen first?

80. Navigation of electric fish. Certain fish, such as the Nile fish (Gnathonemus), concentrate charges in their head and tail, thereby producing an electric field in the water around them. This field creates a potential difference of a few volts between the head and tail, which in turn causes current to flow in the conducting seawater. As the fish swims, it passes near objects that have resistivities different from that of seawater, which in turn causes the current to vary. Cells in the skin of the fish are sensitive to this current and can detect changes in it. The changes in the current allow the fish to navigate. (In the next chapter, we shall investigate how the fish might detect this current.). Since the electric field is weak far from the fish, we shall consider only the field running directly from the head to the tail. We can model the seawater through which that field passes as a conducting tube of area 1.0 cm2 and having a potential difference of 3.0 V across its ends. The length of a Nile fish is about 20 cm, and the resistivity of seawater is 0.13 Ω·m (a) How large is the current through the tube of seawater? (b) Suppose the fish swims next to an object that is 10 cm long and 1.0 cm2 in cross-sectional area and has half the resistivity of seawater. This object replaces the seawater for half the length of the tube. What is the current through the tube now? How large is the change in the current that the fish must detect?

81. Each of three resistors in Figure 19.68 has a resistance of 2.00 Ω and can dissipate a maximum of 32.0 W without becoming excessively heated. What is the maximum power the circuit can dissipate?

82. Leakage in a dielectric. Two parallel plates of a capacitor have equal and opposite charges Q. The dielectric has a dielectric constant K and a resistivity ρ. Show that the “leakage” current I carried by the dielectric is given by I = Q/Kε0ρ.

83. Energy use of home appliances. An 1800 W toaster, a 1400 W electric frying pan, and a 75 W lamp are plugged into the same electrical outlet in a 20 A, 120 V circuit. (Note: When plugged into the same outlet, the three devices are in parallel with each other across the 120 V outlet.) (a) What current is drawn by each device? (b) Will this combination blow the circuit breaker?

84. Two identical 1.00 Ω wires are laid side by side and soldered together so that they touch each other for half of their lengths. What is the equivalent resistance of this combination?

85. Three identical resistors are connected in series. When a certain potential difference is applied across the combination, the total power dissipated is 27 W. What power would be dissipated if the three resistors were connected in parallel across the same potential difference?

86. (a) Calculate the equivalent resistance of the circuit of Figure 19.70 between x and y. (b) If a voltmeter is connected between points a and x when the current in the 8.0 Ω resistor is 2.4 A in the direction from left to right in the figure, what will it read?

87. A power plant transmits 150 kW of power to a nearby town, through wires that have total resistance of 0.25 Ω. What percentage of the power is dissipated as heat in the wire if the power is transmitted at (a) 220 V and (b) 22 kV?

88. What must the emf E in Figure 19.71 be in order for the current through the resistor to be 1.80 A? Each emf source has negligible internal resistance.

89. For the circuit shown in Figure 19.72, if a voltmeter is connected across points a and b, (a) what will it read, and (b) which point is at a higher potential, a or b?

90. A 4600 Ω resistor is connected across a charged 0.800 nF capacitor. The initial current through the resistor, just after the connection is made, is measured to be 0.250 A. (a) What magnitude of charge was initially on each plate of this capacitor? (b) How long after the connection is made will it take before the charge is reduced to 1/e of its maximum value?

91. A capacitor that is initially uncharged is connected in series with a resistor and a 400.0 V emf source with negligible internal resistance. Just after the circuit is completed, the current through the resistor is 0.800 mA and the time constant for the circuit is 6.00 s. What are (a) the resistance of the resistor and (b) the capacitance of the capacitor?

92. In the circuit shown in Fig. 19.73, R is a variable resistor whose value can range from 0 to ∞, and a and b are the terminals of a battery having an emf E = 15.0 V and an internal resistance of 4.00 Ω. The ammeter and voltmeter are both idealized meters. As R varies over its full range of values, what will be the largest and smallest readings of (a) the voltmeter and (b) the ammeter? (c) Sketch qualitative graphs of the readings of both meters as functions of R, as R ranges from 0 to ∞.

**Chapter 20**

Multiple-Choice Problems

1. An electron traveling at a high speed enters a uniform magnetic field directed perpendicular to its path. Which of the following quantities will change while the electron travels through the field?

A. Its speed.

B. Its velocity.

C. Its acceleration.

D. Its kinetic energy.

E. Its potential energy.

2. A negatively charged particle shoots into a uniform magnetic field directed out of the paper, as shown in Figure 20.48. A possible path of this particle is

A. 1. B. 2. C. 3. D. 4.

3. A beam of protons is directed horizontally into the region between two bar magnets, as shown in Figure 20.49. The magnetic field in this region is horizontal. What is the effect of the magnetic field on the protons?

A. The protons are accelerated to the left, toward the S magnetic pole.

B. The protons are accelerated to the right, toward the N magnetic pole.

C. The protons are accelerated upward.

D. The protons are accelerated downward.

E. The protons are not accelerated, since the magnetic field does not change their speed.

4. A wire carrying a current in the direction shown in Fig. 20.50 passes between the poles of two bar magnets. What is the direction of the magnetic force on this wire due to the magnet?

A. out of the paper.

B. into the paper.

C. toward the N pole of the magnet.

D. toward the S pole of the magnet.

5. A solenoid is connected to a battery as shown in Fig. 20.51, and a bar magnet is placed nearby. What is the direction of the magnetic force that this solenoid exerts on the bar magnet?

A. upward.

B. downward.

C. to the right, away from the solenoid.

D. to the left, toward the solenoid

6. Two very long, straight parallel wires carry currents of equal magnitude, but opposite direction, perpendicular to the paper in the directions shown in Figure 20.52. The direction of the net magnetic field due to these two wires at point a, which is equidistant from both wires, is

A. directly to the right. B. directly to the left.

C. straight downward. D. straight upward.

7. A light circular wire suspended by a thin silk thread in a uniform magnetic field carries a current in the direction shown in Figure 20.53. The magnetic field is perpendicular to the plane of the paper, and the wire is held at rest in that plane. If the wire is suddenly released so that it is free to rotate,

A. It will rotate so that point a goes into the paper.

B. It will rotate so that point a goes out of the paper.

C. It will not rotate.

8. An electron is moving directly toward you in a horizontal path when it suddenly enters a uniform magnetic field that is either vertical or horizontal. If the electron begins to curve upward in its motion just after it enters the field, you can conclude that the direction of the magnetic field is

A. upward. B. downward.

C. to your left. D. to your right.

9. The two coils shown in Figure 20.54 are parallel to each other and are connected to batteries. Coil A is held in place, but coil C is free to move. After the switch S is closed, coil C will initially move

A. toward coil A.

B. away from coil A.

C. upward.

D. downward.

10. A loose, floppy coil of wire is carrying current I. The loop of wire is placed on a horizontal table in a uniform magnetic field B perpendicular to the plane of the table. This causes the loop to expand into a circular shape while still lying on the table. What orientation of the current and magnetic field could cause this to happen?

A. Current clockwise, B upward.

B. Current clockwise, B downward.

C. Current counterclockwise, B upward.

D. Current counterclockwise, B downward.

11. A metal bar connected by metal leads to the terminals of a battery hangs between the poles of a horseshoe magnet, as shown in Figure 20.55. Just after the switch S is closed, what will happen problem to the bar?

A. It will be pushed upward, decreasing the tension in the leads.

B. It will be pushed downward, increasing the tension in the leads.

C. It will swing outward, away from the magnet.

D. It will swing inward, into the magnet.

12. A certain current produces a magnetic field B near the center of a solenoid. If the current is doubled, the field near the center will be

A. 4B. B. 2B. C. B√2 D. B.

13. A coil is connected to a battery as shown in Figure 20.56. A bar magnet is suspended with its N pole just above the center of the coil. What will happen to the bar magnet just after the switch S is closed?

A. It will be pulled toward the coil.

B. It will be pushed away from the coil.

C. It will be pushed out of the paper.

D. It will be pushed into the paper.

14. The force exerted by a constant uniform magnetic field on a current-carrying wire of length L produces a force per unit length of FL on the wire. If a wire twice as long and carrying the same current is placed in the same field with the same orientation, the force per unit length would be

A. 2FL B. FL C. FL/2

15. A particle enters a uniform magnetic field initially traveling perpendicular to the field lines and is bent in a circular arc of radius R. If this particle were traveling twice as fast, the radius of its circular arc would be

A. 2R. B. R√2 C. R/√2 D. R/2

**Problems**

1. In a 1.25 T magnetic field directed vertically upward, a particle having a charge of magnitude and initially moving northward at is deflected toward the east. (a) What is the sign of the charge of this particle? Make a sketch to illustrate how you found your answer. (b) Find the magnetic force on the particle.

2. An ion having charge +6e is traveling horizontally to the left at 8.50 km/s when it enters a magnetic field that is perpendicular to its velocity and deflects it downward with an initial magnetic force of 6.94 × 10-15 N. What are the direction and magnitude of this field? Illustrate your method of solving this problem with a diagram.

3. A proton traveling at 3.60 km/s suddenly enters a uniform magnetic field of 0.750 T, traveling at an angle of 55.0° with the field lines. (a) Find the magnitude and direction of the force this magnetic field exerts on the proton. (b) If you can vary the direction of the proton’s velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

4. A particle having a mass of 0.195 g carries a charge of -2.50 × 10-8 C. The particle is given an initial horizontal northward velocity of 4.00 × 104 m/s. What are the magnitude and direction of the minimum magnetic field that will balance the earth’s gravitational pull on the particle?

5. At a given instant, a particle with a mass of 5.00 × 10-3 kg and a charge of 3.50 × 10-8 C has a velocity with a magnitude of 2.00 × 105 m/s in the +y direction. It is moving in a uniform magnetic field that has magnitude 0.8 T and is in the –x direction. What are (a) the magnitude and direction of the magnetic force on the particle and (b) its resulting acceleration?

6. If the magnitude of the magnetic force on a proton is F when it is moving at 15.0° with respect to the field, what is the magnitude of the force (in terms of F) when this charge is moving at 30.0° with respect to the field?

7. A 9Be nucleus containing four protons and five neutrons has a mass of 1.50 × 10-26 kg and is traveling vertically upward at 1.35 km/s. If this particle suddenly enters a horizontal magnetic field of 1.12 T pointing from west to east, find the magnitude and direction of its acceleration vector the instant after it enters the field.

8. A particle with a charge of -2.50 × 10-8 C is moving with an instantaneous velocity of magnitude 40.0 km/s in the xy-plane at an angle of 50° counterclockwise from the axis. What are the magnitude and direction of the force exerted on this particle by a magnetic field with magnitude 2.00 T in the (a) –x direction, and (b) +z direction?

9. A particle with mass 1.81 × 10-3 kg and charge of +1.22 × 10-8 C has, at a given instant, a velocity of 3.00 × 104 m/s along the +y-axis, as shown in Figure 20.58. What are the magnitude and direction of the particle’s acceleration produced by a magnetic field of magnitude 1.25 T in the xy-plane, directed at an angle of 45.0° counterclockwise from the x-axis?

10. A 150 V battery is connected across two parallel metal plates of area 28.5 cm2 and separation 8.20 mm. A beam of alpha particles (charge +2e, mass 6.64 × 10-27 kg) is accelerated from rest through a potential difference of 1.75 kV and enters the region between the plates perpendicular to the electric field. What magnitude and direction of magnetic field are needed so that the alpha particles emerge undeflected from between the plates?

11. A velocity selector having uniform perpendicular electric and magnetic fields is shown in Figure 20.59. The electric field is provided by a 150 V DC battery connected across two large parallel metal plates that are 4.50 cm apart. (a) What must be the magnitude of the magnetic field so that charges having a velocity of 3.25 km/s perpendicular to the fields will pass through undeﬂected? (b) Show how the magnetic field should point in the region between the plates.

12. An electron moves at 2.50 × 106 m/s through a region in which there is a magnetic field of unspecified direction and magnitude 7.40 × 10-2 T. (a) What are the largest and smallest possible magnitudes of the acceleration of the electron due to the magnetic field? (b) If the actual acceleration of the electron is one-fourth of the largest magnitude in part (a), what is the angle between the electron velocity and the magnetic field?

13. In a cloud chamber experiment, a proton enters a uniform 0.250 T magnetic field directed perpendicular to its motion. You measure the proton’s path on a photograph and find that it follows a circular arc of radius 6.13 cm. How fast was the proton moving?

14. An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of 6.64 ×10-27 kg) traveling horizontally at 35.6 km/s enters a uniform, vertical, 1.10 T magnetic field. (a) What is the diameter of the path followed by this alpha particle? (b) What effect does the magnetic field have on the speed of the particle? (c) What are the magnitude and direction of the acceleration of the alpha particle while it is in the magnetic field? (d) Explain why the speed of the particle does not change even though an unbalanced external force acts on it.

15. A deuteron particle (the nucleus of an isotope of hydrogen consisting of one proton and one neutron and having a mass of 3.34 ×10-27 kg moving horizontally enters a uniform, vertical, 0.500 T magnetic field and follows a circular arc of radius 55.6 cm. (a) How fast was this deuteron moving just before it entered the magnetic field and just after it came out of the field? (b) What would be the radius of the arc followed by a proton that entered the field with the same velocity as the deuteron?

16. A beam of protons traveling at 1.20 km/s enters a uniform magnetic field, traveling perpendicular to the field. The beam exits the magnetic field in a direction perpendicular to its original direction (Fig. 20.60). The beam travels a distance of 1.18 cm while in the field. What is the magnitude of the magnetic field?

17. A uniform magnetic field bends an electron in a circular arc of radius R. What will be the radius of the arc (in terms of R) if the field is tripled?

18. An electron at point A in Figure 20.61 has a speed v0 of 1.41 × 106 m/s Find (a) the magnitude and direction of the magnetic field that will cause the electron to follow the semicircular path from A to B and (b) the time required for the electron to move from A to B. (c) What magnetic field would be needed if the particle were a proton instead of an electron?

19. A beam of protons is accelerated through a potential difference of 0.745 kV and then enters a uniform magnetic field traveling perpendicular to the field. (a) What magnitude of field is needed to bend these protons in a circular arc of diameter 1.75 m? (b) What magnetic field would be needed to produce a path with the same diameter if the particles were electrons having the same speed as the protons?

20. A 3.25 g bullet picks up an electric charge of 1.65 μC as it travels down the barrel of a rifle. It leaves the barrel at a speed of traveling perpendicular to the earth’s magnetic field, which has a magnitude of 5.50 × 10-4 T. Calculate (a) the magnitude of the magnetic force on the bullet and (b) the magnitude of the bullet’s acceleration due to the magnetic force at the instant it leaves the rifle barrel.

21. An electron in the beam of a TV picture tube is accelerated through a potential difference of 2.00 kV. It then passes into a magnetic field perpendicular to its path, where it moves in a circular arc of diameter 0.360 m. What is the magnitude of this field?

22. (a) What is the speed of a beam of electrons when the simultaneous influence of an electric field of 1.56 × 104 V/m and a magnetic field of 4.62 × 10-3 T with both fields normal to the beam and to each other, produces no deflection of the electrons? (b) In a diagram, show the relative orientation of the vectors v, E and B. (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

23. Singly ionized (one electron removed) atoms are accelerated and then passed through a velocity selector consisting of perpendicular electric and magnetic fields. The electric field is and the magnetic field is 0.0315 T. The ions next enter a uniform magnetic field of magnitude 0.0175 T that is oriented perpendicular to their velocity. (a) How fast are the ions moving when they emerge from the velocity selector? (b) If the radius of the path of the ions in the second magnetic field is 17.5 cm, what is their mass?

24. Determining diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the 12C and 13C isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed and want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 cm for the 12C. The measured masses of these isotopes are 1.99 × 10-26 kg (12C) and 2.16 × 10-26 kg (13C) (a) What strength of magnetic field is required? (b) What is the diameter of the 13C semicircle? (c) What is the separation of the 13C and 12C ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?

25. Ancient meat eating. The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes nitrogen-15 to nitrogen-14 in bone from human remains. Carnivores concentrate 15N so this ratio tells archaeologists how much meat was consumed by ancient people. Use the spectrometer of the previous problem to find the separation of the 14N and 15N isotopes at the detector. The measured masses of these isotopes are 2.32 × 10-26 kg (14N) and 2.49 × 10-26 kg (15N).

26. A straight vertical wire carries a current of 1.20 A downward in a region between the poles of a large electromagnet where the field strength is 0.588 T and is horizontal. What are the magnitude and direction of the magnetic force on a 1.00 cm section of this wire if the magnetic-field direction is (a) toward the east, (b) toward the south, (c) 30.0° south of west?

27. Magnetic force on a lightning bolt. Currents during lightning strikes can be up to 50,000 A (or more!). We can model such a strike as a 50,000 A vertical current perpendicular to the earth’s magnetic field, which is about ½ gauss. What is the force on each meter of this current due to the earth’s magnetic field?

28. A horizontal rod 0.200 m long carries a current through a uniform horizontal magnetic field of magnitude 0.067 T that points perpendicular to the rod. If the magnetic force on this rod is measured to be 0.13 N, what is the current flowing through the rod?

29. A straight 2.5 m wire carries a typical household current of 1.5 A (in one direction) at a location where the earth’s magnetic field is 0.55 gauss from south to north. Find the magnitude and direction of the force that our planet’s magnetic field exerts on this wire if is oriented so that the current in it is running (a) from west to east, (b) vertically upward, (c) from north to south. (d) Is the magnetic force ever large enough to cause significant effects under normal household conditions?

30. Between the poles of a powerful magnet is a cylindrical uniform magnetic field with a diameter of 3.50 cm and a strength of 1.40 T. A wire carries a current through the center of the field at an angle of 65.0° to the magnetic field lines. If the wire experiences a magnetic force of 0.0514 N, what is the current flowing in it?

31. A rectangular 10.0 cm by 20.0 cm circuit carrying an 8.00 A current is oriented with its plane parallel to a uniform 0.750 T magnetic field (Figure 20.62). (a) Find the magnitude and direction of the magnetic force on each segment (ab, bc, etc.) of this circuit. Illustrate your answers with clear diagrams. (b) Find the magnitude of the net force on the entire circuit.

32. A long wire carrying a 6.00 A current reverses direction by means of two right angle bends, as shown in Figure 20.63. The part of the wire where the bend occurs is in a magnetic field of 0.666 T confined to the circular region of diameter 75 cm, as shown. Find the magnitude and direction of the net force that the magnetic field exerts on this wire.

33. A long wire carrying 4.50 A of current makes two 90° bends, as shown in Figure 20.64. The bent part of the wire passes through a uniform 0.240 T magnetic field directed as shown in the figure and confined to a limited region of space. Find the magnitude and direction of the force that the magnetic field exerts on the wire.

34. The 20.0 cm by 35.0 cm rectangular circuit shown in Figure 20.65 is hinged along side ab. It carries a clockwise 5.00 A current and is located in a uniform 1.20 T magnetic field oriented perpendicular to two of its sides, as shown. (a) Make a clear diagram showing the direction of the force that the magnetic field exerts on each segment of the circuit (ab, bc, etc.) (b) Of the four forces you drew in part (a), decide which ones exert a torque about the hinge ab. Then calculate only those forces that exert this torque. (c) Use your results from part (b) to calculate the torque that the magnetic field exerts on the circuit about the hinge axis ab.

35. The plane of a 5.0 cm by 8.0 cm rectangular loop of wire is parallel to a 0.19 T magnetic field, and the loop carries a current of 6.2 A. (a) What torque acts on the loop? (b) What is the magnetic moment of the loop?

36. A circular coil of wire 8.6 cm in diameter has 15 turns and carries a current of 2.7 A. The coil is in a region where the magnetic field is 0.56 T. (a) What orientation of the coil gives the maximum torque on the coil, and what is this maximum torque? (b) For what orientation of the coil is the magnitude of

the torque 71% of the maximum found in part (a)?

37. A rectangular coil of wire 22.0 cm by 35.0 cm and carrying a current of 1.40 A is oriented with the plane of its loop perpendicular to a uniform 1.50 T magnetic field, as shown in Figure 20.66. (a) Calculate the net force and torque that the magnetic field exerts on this coil. (b) The coil is now rotated through a 30.0° angle about the axis shown, the left side coming out of the plane and the right side going into the plane. Calculate the net force and torque that the magnetic field exerts on the coil.

38. A solenoid having 165 turns and a cross-sectional area of 6.75 cm2 carries a current of 1.20 A. If it is placed in a uniform 1.12 T magnetic field, find the torque this field exerts on the solenoid if its axis is oriented (a) perpendicular to the field, (b) parallel to the field, (c) at 35.0° with the field.

39. A circular coil of 50 loops and diameter 20.0 cm is lying flat on a tabletop, and carries a clockwise current of 2.50 A. A magnetic field of 0.450 T, directed to the north and at an angle of 45.0° from the vertical down through the coil and into the tabletop is turned on. (a) What is the torque on the coil, and (b) which side of the coil (north or south) will tend to rise from the tabletop?

40. You want to produce a magnetic field of magnitude 5.50 × 10-4 T at a distance of 0.040 m from a long, straight wire’s center. (a) What current is required to produce this field? (b) With the current found in part (a), how strong is the magnetic field 8.00 cm from the wire’s center?

41. Household magnetic fields. Home circuit breakers typically have current capacities of around 10 A. How large a magnetic field would such a current produce 5.0 cm from a long wire’s center? How does this field compare with the strength of the earth’s magnetic field?

42. (a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 cm from the wire is equal to 1.00 G (comparable to the earth’s northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth’s magnetic field? (c) Repeat part (b) except with the wire vertical and the current going upward.

43. Currents in the heart. The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about 1.0 μG. Although the actual currents are rather complicated, we can gain a rough understanding of their magnitude if we model them as a long, straight wire. If the surface of the chest is 5.0 cm from this current, how large is the current in the heart?

44. Magnetic sensitivity of electric fish. In a problem dealing with electric fish in Chapter 19, we saw that these fish navigate by responding to changes in the current in seawater. This current is due to a potential difference of around 3.0 V generated by the fish and is about 12 mA within a centimeter or so from the fish. Receptor cells in the fish are sensitive to the current. Since the current is at some distance from the fish, the sensitivity of these cells suggests that they might be responding to the magnetic field created by the current. To get some estimate of how sensitive the cells are, we can model the current as that of a long, straight wire with the receptor cells 2.0 cm away. What is the strength of the magnetic field at the receptor cells?

45. In a conventional cheap flashlight, a straight copper strip runs along the tube of the flashlight to connect the bulb to the negative terminal of the battery at the bottom of the tube. If this strip carries a current of 0.65 A while you’re holding the flashlight, what is the magnitude of the magnetic field at the surface of your hand, 0.30 cm from the strip? How does your answer compare to the earth’s magnetic field?

46. If the magnetic field due to a long, straight current-carrying wire has a magnitude B at a distance R from the wire’s center, how far away must you be (in terms of R) for the magnetic field to decrease to B/3.

47. A current in a long, straight wire produces a magnetic field of 8.0 μT at 2.0 cm from the wire’s center. Answer the following questions without finding the current: (a) What is the magnetic field strength 4.0 cm from the wire’s center? (b) How far from the wire’s center will the field be 1.0 μT (c) If the current were doubled, what would the field be 2.0 cm from the wire’s center?

48. EMF. Currents in DC transmission lines can be 100 A or more. Some people have expressed concern that the electromagnetic fields (EMFs) from such lines near their homes could cause health dangers. Using your own observations, estimate how high such lines are above the ground. Then use your estimate to calculate the strength of the magnetic field these lines produce at ground level. Express your answer in teslas and as a percent of the earth’s magnetic field (which is 0.50 gauss). Does it seem that there is cause for worry?

49. A long, straight telephone cable contains six wires, each carrying a current of 0.300 A. The distances between wires can be neglected. (a) If the currents in all six wires are in the same direction, what is the magnitude of the magnetic field 2.50 m from the cable? (b) If four wires carry currents in one direction and the other two carry currents in the opposite direction, what is the magnitude of the field 2.50 m from the cable?

50. Two insulated wires perpendicular to each other in the same plane carry currents as shown in Figure 20.67. Find the magnitude of the net magnetic field these wires produce at points P and Q if the 10.0 A current is (a) to the right or (b) to the left.

51. Two long, straight parallel wires are 10.0 cm apart and carry 4.00 A currents in the same direction (Figure 20.68). Find the magnitude and direction of the magnetic field at (a) point P1 midway between the wires, (b) point P2 25.0 cm to the right of P1.

52. Two long parallel transmission lines 40.0 cm apart carry 25.0 A and 75.0 A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction, (b) opposite directions.

53. Two high-current transmission lines carry currents of 25 A and 75 A in the same direction and are suspended parallel to each other 35 cm apart. If the vertical posts supporting these wires divide the lines into straight 15 m segments, what magnetic force does each segment exert on the other? Is this force attractive or repulsive?

54. Two long current-carrying wires run parallel to each other. Show that if the currents run in the same direction, these wires attract each other, whereas if they run in opposite directions, the wires repel.

55. A 2.0 m ordinary lamp extension cord carries a 5.0 A current. Such a cord typically consists of two parallel wires carrying equal currents in opposite directions. Find the magnitude and direction (attractive or repulsive) that the two segments of this cord exert on each other.

56. An electric bus operates by drawing current from two parallel overhead cables, at a potential difference of 600 V, and spaced 55 cm apart. When the power input to the bus’s motor is at its maximum power of 65 hp, (a) what current does it draw and (b) what is the attractive force per unit length between the cables?

57. A circular metal loop is 22 cm in diameter. (a) How large a current must flow through this metal so that the magnetic field at its center is equal to the earth’s magnetic field of 0.50 × 10-4 T. (b) Show how the loop should be oriented so that it can cancel the earth’s magnetic field at its center.

58. A closely wound circular coil with a diameter of 4.00 cm has 600 turns and carries a current of 0.500 A. What is the magnetic field at the center of the coil?

59. A closely wound circular coil has a radius of 6.00 cm and carries a current of 2.50 A. How many turns must it have if the magnetic field at its center is 6.39 × 10-4 T.

60. Currents in the brain. The magnetic field around the head has been measured to be approximately 3.0 × 10-8 gauss. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 cm (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

61. A closely wound, circular coil with radius 2.40 cm has 800 turns. What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 T?

62. Two circular concentric loops of wire lie on a tabletop, one inside the other. The inner loop has a diameter of 20.0 cm and carries a clockwise current of 12.0 A, as viewed from above, and the outer wire has a diameter of 30.0 cm. What must be the magnitude and direction (as viewed from above) of the current in the outer loop so that the net magnetic field due to this combination of loops is zero at the common center of the loops?

63. Calculate the magnitude and direction of the magnetic field at point P due to the current in the semicircular section of wire shown in Figure 20.69.

64. A solenoid contains 750 coils of very thin wire evenly wrapped over a length of 15.0 cm. Each coil is 0.800 cm in diameter. If this solenoid carries a current of 7.00 A, what is the magnetic field at its center?

65. As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150 T magnetic field near its center. You have enough wire for 4000 circular turns, and the solenoid must be 1.40 m long and 2.00 cm in diameter. What current will you need to produce the necessary field?

66. A solenoid is designed to produce a 0.0279 T magnetic field near its center. It has a radius of 1.40 cm and a length of 40.0 cm, and the wire carries a current of 12.0 A. (a) How many turns must the solenoid have? (b) What total length of wire is required to make this solenoid?

67. A single circular current loop 10.0 cm in diameter carries a 2.00 A current. (a) What is the magnetic field at the center of this loop? (b) Suppose that we now connect 1000 of these loops in series within a 500 cm length to make a solenoid 500 cm long. What is the magnetic field at the center of this solenoid? Is it 1000 times the field at the center of the loop in part (a)? Why or why not?

68. A solenoid that is 35 cm long and contains 450 circular coils 2.0 cm in diameter carries a 1.75 A current. (a) What is the magnetic field at the center of the solenoid, 1.0 cm from the coils? (b) Suppose we now stretch out the coils to make a very long wire carrying the same current as before. What is the magnetic field 1.0 cm from the wire’s center? Is it the same as you found in part (a)? Why or why not?

69. You have 25 m of wire, which you want to use to construct a 44 cm diameter coil whose magnetic field at its center will exactly cancel the earth’s field of 0.55 gauss. What current will your coil require?

70. A toroidal solenoid (see Figure 20.42) has inner radius r1 = 15.0 cm and outer radius r2 = 18.0 cm. The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) 12.0 cm; (b) 16.0 cm; (c) 20.0 cm?

71. A long, straight wire carries a current of 10.0 A, as shown in Figure 20.70. Use the law of Biot and Savart to find the magnitude and direction of the magnetic field at point P due to each of the following 2.00 mm segments of this wire: (a) segment A and (b) segment C.

72. A long wire carrying a 5.00 A current makes an abrupt right-angle bend as shown in Figure 20.71. Use the law of Biot and Savart to determine the magnitude and direction of the magnetic field at point P due to the 1.50 cm bent segment if P is 15.0 cm from the midpoint of that segment.

73. Three long, straight electrical cables, running north and south, are tightly enclosed in an insulating sheath. One of the cables carries a 23.0 A current southward; the other two carry currents of 17.5 A and 11.3 A northward. Use Ampere’s law to calculate the magnitude of the magnetic field at a distance of 10.0 m from the cables.

74. A long, straight, cylindrical wire of radius R carries a current uniformly distributed over its cross section. At what location is the magnetic field produced by this current equal to half of its largest value? Use Ampere’s law and consider points inside and outside the wire.

75. Platinum is a paramagnetic metal having a relative permeability of 1.00026. (a) What is the magnetic permeability of platinum? (b) If a thin rod of platinum is placed in an external magnetic field of 1.3500 T, with its axis parallel to that field, what will be the magnetic field inside the rod?

76. When a certain paramagnetic material is placed in an external magnetic field of 1.5000 T, the field inside the material is measured to be 1.5023 T. Find (a) the relative permeability and (b) the magnetic permeability of this material.

77. A 150 g ball containing 4.00 × 108 excess electrons is dropped into a 125 m vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal 0.250 T magnetic field directed from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enters the field.

78. Magnetic balance. The circuit shown in Figure 20.72 is used to make a magnetic balance to weigh objects. The mass m to be measured is hung from the center of the bar, which is in a uniform magnetic field of 1.50 T directed into the plane of the figure. The battery voltage can be adjusted to vary the current in the circuit. The horizontal bar is 60.0 cm long and is made of extremely lightweight material, so its mass can be neglected. It is connected to the battery by thin vertical wires that can support no appreciable tension; all the weight of the mass m is supported by the magnetic force on the bar. A 5.00 Ω resistor is in series with the bar, and the resistance of the rest of the circuit is negligibly small. (a) Which point, a or b, should be the positive terminal of the battery? (b) If the maximum terminal voltage of the battery is 175 V, what is the greatest mass m that this instrument can measure?

79. A thin 50.0-cm-long metal bar with mass 750 g rests on, but is not attached to, two metal supports in a uniform 0.450 T magnetic field, as shown in Figure 20.73. A battery and a 25.0 Ω resistor in series are connected to the supports. What is the largest terminal voltage the battery can have without breaking the circuit at the supports?

80. A long, straight wire containing a semicircular region of radius 0.95 m is placed in a uniform magnetic field of magnitude 2.20 T as shown in Figure 20.74. What is the net magnetic force acting on the wire when it carries a current of 3.40 A?

81. A singly charged ion of 7Li (an isotope of lithium containing three protons and four neutrons) has a mass of 1.16 × 10-26 kg. It is accelerated through a potential difference of 220 V and then enters a 0.723 T magnetic field perpendicular to the ion’s path. What is the radius of the path of this ion in the magnetic field?

82. An insulated circular ring of diameter 6.50 cm carries a 12.0 A current and is tangent to a very long, straight insulated wire carrying 10.0 A of current, as shown in Figure 20.75. Find the magnitude and direction of the magnetic field at the center of the ring due to this combination of wires.

83. The effect of transmission lines. Two hikers are reading a compass under an overhead transmission line that is 5.50 m above the ground and carries a current of 0.800 kA in a horizontal direction from north to south. (a) Find the magnitude and direction of the magnetic field at a point on the ground directly under the transmission line. (b) One hiker suggests that they walk 50 m away from the lines to avoid inaccurate compass readings due to the current. Considering that the earth’s magnetic field is on the order of 0.5 × 10-4 T. is the current really a problem?

84. A long, straight horizontal wire carries a current of 2.50 A directed toward the right. An electron is traveling in the vicinity of this wire. (a) At the instant the electron is 4.50 cm above the wire’s center and moving with a speed of 6.00 × 104 m/s directly toward it, what are the magnitude and direction of the force that the magnetic field of the current exerts on the electron? (b) What would be the magnitude and direction of the magnetic force if the electron were instead moving parallel to the wire in the same direction as the current?

85. Two very long, straight wires carry currents as shown in Figure 20.76. For each case shown, find all locations where the net magnetic field due to these wires is zero.

86. Bubble chamber, I. Certain types of bubble chambers are filled with liquid hydrogen. When a particle (such as an electron or a proton) passes through the liquid, it leaves a track of bubbles, which can be photographed to show the path of the particle. The apparatus is immersed in a known magnetic field, which causes the particle to curve. Figure 20.77 is a trace of a bubble chamber image showing the path of an electron. (a) How could you determine the sign of the charge of a particle from a photograph of its path? (b) How can physicists determine the momentum and the speed of this electron by using measurements made on the photograph, given that the magnetic field is known and is perpendicular to the plane of the figure? (c) The electron is obviously spiraling into smaller and smaller circles. What properties of the electron must be changing to cause this behavior? Why does this happen? (d) What would be the path of a neutron in a bubble chamber? Why?

87. A 3.00 N metal bar, 1.50 m long and having a resistance of 10.0 Ω rests horizontally on conducting wires connecting it to the circuit shown in Figure 20.78. The bar is in a uniform, horizontal, 1.60 T magnetic field and is not attached to the wires in the circuit. What is the acceleration of the bar just after the switch S is closed?

88. A pair of long, rigid metal rods, each of length L, lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant k (Figure 20.79) and negligible unstretched length. If a current I runs through this circuit, the springs will stretch. At what separation will the rods remain at rest? Assume that k is large enough so that the separation of the rods will be much less than L.

89. Atom smashers! A cyclotron particle accelerator (sometimes called an “atom smasher” in the popular press) is a device for accelerating charged particles, such as electrons and protons, to speeds close to the speed of light. The basic design is quite simple. The particle is bent in a circular path by a uniform magnetic field. An electric field is pulsed periodically to increase the speed of the particle. The charged particle (or ion) of mass m and charge q is introduced into the cyclotron so that it is moving perpendicular to a uniform magnetic field B. (a) Starting with the radius of the circular path of a charge moving in a uniform magnetic field, show that the time T for this particle to make one complete circle is T = 2πm/(|q|B). (b) Which would take longer to complete one circle, an ion moving in a large circle or one moving in a small circle? Explain.

90. Medical uses of cyclotrons. The largest cyclotron (see the previous problem) in the United States is the Tevatron at Fermilab, near Chicago, Illinois. It is called a Tevatron because it can accelerate particles to energies in the TeV range (1 tera-eV = 1012 eV). Its circumference is 6.4 km, and it currently can produce a maximum energy of 2.0 TeV. In a certain medical experiment, protons will be accelerated to energies of 1.25 MeV and aimed at a tumor to destroy its cells. (a) How fast are these protons moving when they hit the tumor? (b) How strong must the magnetic field be to bend the protons in the circle indicated?

91. A plastic circular loop has radius R, and a positive charge q is distributed uniformly around the circumference of the loop. The loop is now rotated around its central axis, perpendicular to the plane of the loop, with angular speed ω. If the loop is in a region where there is a uniform magnetic field B directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

92. A long wire carrying 6.50 A of current makes two bends, as shown in Figure 20.80. The bent part of the wire passes through a uniform 0.280 T magnetic

field directed as shown in the figure and confined to a limited region of space. Find the magnitude and direction of the force that the magnetic field exerts on the wire.

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