Spring, 2000, Assignment

Last revised 2000/01/05

Go to assignment # 1 2 3 4 5 6 7 8 9 10 11 12
Due on Monday 02/14 02/21 02/28 03/06 03/13 03/20 04/03 04/10 04/17 04/24 05/01 05/08
Chapter 22 23 23, 24 25 26 27, 28 29 30 31 31 33 33

I will almost always have hints posted on the web beginning a week before the assignment is due. The day before the problem session I will try to post answers in the same location. I will frequently post solutions after the problem session is over.

In general, I have chosen problems like the more difficult ones in the chapter. It may be helpful to work a few of the simpler problems first, especially if you have trouble getting started on one of the assigned problems. If you have difficulty, see me in Sharp Lab 264, call x2013, or send EMail to mvb@udel.edu, preferably before turning the set in. It is as important to know why incorrect attempts do not work as it is to know what the correct solutions are. Please feel free to talk to me at any time either in person, by phone, or by E-MAIL.

The problems will be graded for the existence of diagrams, the use of units, and correct vector notation. Otherwise, a reasonable attempt will get full credit whether it is right or wrong. Since the problems are graded for technique and not for correctness, you can work with each other on the assignments. You should do so. You may also consult others outside the class, but I strongly advise against looking at their solutions before attempting the problems yourself. You must write up your own final copy, unless I have stated otherwise (no Xeroxes!).

Many of these problems are adapted from a book by Tipler, because I already have reliable answers and solutions for those problems. Similar problems appear in our text. I am also modifying some of the problems from the text so that they fit better with the other assigned problems. Whenever the assigned problem is very similar to one in the text, the assignment includes the statement "compare HRW # ...." If an assigned problem is nearly identical to one in the text, the assignment will say "see HRW # ...."

1. Due in recitation 02/14

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Chapter 22. Electric Charge:

1. (HRW p. 551 # 8) In the figure on the right (HRW figure 22-21), three charged particles lie on a straight line and are separated by a distance d. Charges q1 and q2 are held fixed [by external forces]. Charge q3 is free to move but happens to be in equilibrium (no net electrostatic force acts on it). Find q1 in terms of q2.

2. Four charges, each of magnitude q, are at the corners of a square of side L. The two charges at one pair of opposite corners are positive, and the other two charges are negative. Find the force exerted on either of the negative charges by the three remaining charges. See figure to the right.

3. (See HRW p. 551 #18 and p. 553 #41). A certain charge Q is divided into two parts alpha Q and (1 - alpha) Q, which are then separated by a certain distance. What must alpha be in order to maximize the electrostatic repulsion between the two charges? Draw a graph of the electrostatic force F as a function of alpha and verify the correctness of your answer.

4. (HRW p. 551 #20). In the figure on the right, two tiny conducting balls of identical mass m and identical charge q hang in equilibrium from nonconducting threads of length L. Assume that theta is so small that tan(theta) can be replaced by its approximate equal sin(theta) [both are in fact essentially equal to theta]. (a) Show that, for equilibrium,

x = [q2 L / ( 2 pi epsilon0 m g ]1/3

where x is the separation between the balls. (b) If L=120 cm , m=10 g, and x=5.0 cm, what is q?

2. Due in recitation 02/21

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Chapter 23. Electric Fields:

5. (Compare HRW p. 577 # 57) Two charged plates of length L are placed a distance d apart as shown in the figure to the right. An electron is injected between the plates, just clearing the lower plate as it enters. It feels a constant electric field E pointed straight upward due to the charges on the plates. If the electron's initial velocity is v0 at an angle theta with respect to the lower plate, for what values of theta will it not hit the upper plate? If it clears the upper plate, how far does it get before hitting the lower plate?

6. (Compare HRW p. 575 #22) Four charges of equal magnitude are arranged at the corners of a square of side L as shown in the figure to the right. Show that the electric field at the midpoint of one of the sides of the square is directed along that side, points toward the negative charge, and has a magnitude E given by

E = [1/4 pi epsilon0 ] [8 q / L2 ] [ 1 - 51/2 / 25]

7. A water molecule has its oxygen atom at the origin, one hydrogen nucleus at x = 0.077 nanometer, y = 0.058 nm, and the other hydrogen nucleus at x = - 0.077 nm, y = 0.058 nm. If the electrons from the hydrogen atoms are transferred completely to the oxygen atom so that it has a charge of -2e, what is the dipole moment of the water molecule? [This characterization of the chemical bonds of water as being totally ionic overestimates the dipole moment of a water molecule].

8. A positive point charge +Q is at the origin. A dipole of moment p is a distance r away from the charge and points in the radial direction, as shown in the diagram to the right.

(a) Show that the force exerted by the electric field of the point charge on the dipole is attractive and has a magnitude of approximately

F = 2 Q p / (4 pi epsilon0 r3 )

(b) Now assume that the dipole is centered at the origin and that the point charge Q is a distance r away along the line of the dipole. From your result in part (a) and Newton's Third Law, show that the magnitude of the electric field of the dipole along the line of the dipole a distance r away is approximately

E = 2 p / (4 pi epsilon0 r3 )

9. (HRW p. 575 # 26) Electric Quadrupole . The figure shows an electric quadrupole. It consists of two dipoles with dipole moments that are equal in magnitude but opposite in direction. Show that the value of the magnitude of the electric field E on the axis of the quadrupole for points a distance z from its center (assume z >> d) is given by

E = 3 Q / ( 4 pi epsilon0 z4 )

in which Q (=2 q d2) is known as the quadrupole moment of the charge distribution. [This is not the only way to get a quadrupole moment.]

10. (Compare HRW p. 576 #33) (a) A thin, nonconducting rod carries a charge of constant linear charge density lambda and lies on the x-axis from x = 0 to x = a. Show that the y-component of the electric field at any point on the y-axis is given by

Ey = [ lambda a / ( 4 pi epsilon0 y) ] ( y2 + a2) - 1/2

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(b) Show that if the rod extends from x = - b to x = a, the y-component of the electric field at the same point is given by

Ey = [ lambda a / ( 4 pi epsilon0 y) ] (y2 + a2) -1/2 + [ lambda b / ( 4 pi epsilon0 y) ] (y2 + b2) -1/2

(c) HRW asks for the special case a = b = L/2. Show that in this case you get

Ey = [ q / ( 2 pi epsilon0 y) ] ( 4 y2 + L2) - 1/2

(d) Show that if and only if a = b the x-component of the electric field vanishes.

11. A butterfly net is attached to a ring of radius R, and the ring is held in a plane perpendicular to a constant electric field of magnitude E. What is the electric flux

PHI=integral [E cos(theta) dA]

through the net? The angle theta is the angle between the normal to the net and the electric field.

3. Due in recitation 02/28

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Chapter 23. Electric Fields

12. A spherical shell of radius R carries a uniform surface charge density sigma which you may consider to be positive. The figure to the right shows a cross-section of the shell. A point P is in the plane of the cross-section but not at the center. Consider the two elements of the shell of areas s1 and s2 shown in the figure at distances r1 and r2 , respectively, from point P. (a) Show that the ratio of the charges on these elements is Q1 / Q2 = r1 2 / r2 2 (b) Which produces the greater field at point P? (c) What is the direction of the field at point P due to each element? What is the direction of the total electric field at point P? (d) Using the result in part (c), explain how it is possible to use Gauss's Law to calculate the electric field without taking into account any charges outside the Gaussian surface, provided that sufficient symmetry is present. What is the role of the symmetry requirement?

13. A disk of radius R carries a uniform charge density sigma. (a) Compare the approximation E = sigma / (2 epsilon0 ) with the exact expression for the electric field on the axis of the disk by computing the neglected term as a percentage of sigma / (2 epsilon0 ) for distances of x = R / 300 , x = R / 150 , and x = R / 10 . (b) At what distance is the neglected term 1% of sigma / (2 epsilon0 ) ?

14. (Hint: compare HRW p. 576 #34) In the figure to the right, two nonconducting rods of length L have charge q uniformly distributed along their lengths. (a) What is the linear charge density of each rod? (b) What is the electric field E due to the left-hand rod at a distance a beyond its right-hand end? (c) What is the force that one of the rods exerts on the other? (d) Show that when the distance d between the rods becomes large compared to the length of the rods, the force tends toward the expected result q2 / d2.

Chapter 24: Gauss's Law

15. (HRW p. 599 #51) A point charge +q is placed at the center of an electrically neutral, spherical conducting shell with inner radius a and outer radius b. What charge appears on (a) the inner surface of the shell and (b) the outer surface of the shell? Find expressions for the net electric field at a distance r from the center of the shell if (c) r < a , (d) b > r > a, and r > b. Sketch the field lines for those three regions. For r > b, what is the net electric field due to (f) the central point charge and inner surface charge and (g) the outer surface charge? A point charge -q is now placed outside the shell. Does this point charge change the charge distribution on (h) the outer surface and (i) the inner surface? Sketch the field lines now. (j) Is there an electrostatic force on the second point charge? (k) Is there a net electrostatic force on the first point charge? (l) Does this situation violate Newton's third law?

16. (HRW p. 599 #52). A solid nonconducting sphere of radius R has a nonuniform charge distribution of rho = rhos r / R, where rhos is a constant and r is the distance from the center of the sphere. Show that (a) the total charge on the sphere is Q = pi rhos R 3 and (b) the electric field inside the sphere has a magnitude given by

E = [1 / ( 4 pi epsilon0 ) ] [ Q r2 / R4 ]

17. (Compare HRW p. 598 #31) An infinitely long nonconducting cylindrical shell of inner radius a and outer radius b has a uniform volume charge density rho . Find the electric field everywhere.

18. An infinitely long nonconducting solid cylinder of radius R has a charge density of B / r , where r is the distance from the axis of the cylinder. Find the electric field everywhere.

4. Due in recitation 03/06

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Chapter 25. Electric Potential:

19. Two positive charges + q are on the x-axis at x = + a and x = - a . (a) Find the potential V(x) as a function of x for all points on the x-axis. (b) Sketch V(x) . (c) What is the significance of the minimum of your plot of V(x) ?

20. Three equal charges lie in the xy plane. Two are on the y-axis at y = - a and y = + a , and the third is on the x axis at x = a . (a) What is the potential V(x) due to these charge at any point on the x-axis? (b) Find Ex along the x-axis from the potential function V(x) . Evaluate your answers to (a) and (b) at the origin and at x = infinity to see if they yield the expected results. (c) If a fourth charge of + q is released at the origin, what is its velocity when it gets very far away from the other three?

21. (HRW p. 624 #41) The figure shows a plastic rod of length L and uniform positive charge Q lying on an x axis. With V = 0 at infinity, find the electric potential at point P1 on the axis, at distance d from one end of the rod.

22. Four equal charges Q are at the corners of a square of side L. The charges are released one at a time proceeding clockwise around the square. Each charge is allowed to reach its final speed a long distance from the square before the next charge is released. What is the final kinetic energy of (a) the first charge released, (b) the second charge released, (c) the third charge released, and (d) the fourth charge released?

23. (Compare HRW p. 624 # 46) The electric potential in a region of space is given by

V(x, y, z) = (2 Volt / m2) x2 + (1 Volt / m2)yz

Find the electric field at all points (x, y, z)

24. Consider two concentric spherical metal shells of radii a and b where b > a . The outer shell has a charge Q , but the inner shell is grounded [it is at the same potential as a point at infinity]. This means that the inner shell is at zero potential and implies that electric-field lines leave the outer shell and go to infinity but that other electric-field lines leave the outer shell and end on the inner shell. Find the charge on the inner shell.

25. A nonconducting sphere of radius R has a volume charge density rho = rho0 r / R , where rho0 is a constant. (a) Show that the total charge is Q = pi R3 rho0 . (b) Use Gauss's Law to find the electric field Er everywhere. (c) From the electric field, find the electric potential everywhere, assuming that V = 0 at infinity. (Remember that V is continuous at r = R .

5. Due in recitation 03/13

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Chapter 26. Capacitance:

26. (Compare HRW p. 649 # 53, 57) You are asked to construct a parallel-plate, air-gap capacitor that will store 100 kJ of energy. (a) What minimum volume is required between the plates of the capacitor? (b) Suppose you have developed a dielectric that can withstand a maximum electric field Emax and has a dielectric constant of kappa. What volume of this dielectric between the plates of the capacitor is required for it to be able to store 100 kJ of energy?

27. Consider two parallel-plate capacitors, C1 and C2, that are connected in parallel. The capacitors are identical except that C2 has a dielectric inserted between its plates. A voltage source of V volts is connected across the capacitors to charge them and is then disconnected. (a) What is the charge on each capacitor? (b) What is the total stored energy of the capacitors? (c) The dielectric is removed from C2. What is the final stored energy of the capacitors? (d) What is the final voltage across the two capacitors?

28. (Compare HRW p. 650 # 63) A parallel-plate capacitor of area A and separation d is charged to a potential difference V and is then removed from the charging source. A dielectric slab of constant kappa, thickness d, and area A/2 is inserted along one edge of the capacitor as shown in the figure. Let sigma1 be the free charge density at the conductor-dielectric surface and sigma2 be the free charge density at the conductor-air surface. (a) Why must the electric field have the same value inside the dielectric as in the free space between the plates? (b) Show that sigma1 = kappa sigma2. (c) Show that the new capacitance is

C = ( kappa + 1 ) epsilon0 A / ( 2 d )

and that the new potential difference is 2 V / ( kappa + 1 )

29. HRW p. 650 # 68: The space between two concentric conducting spherical shells of radii b and a (where b > a) is filled with a substance of dielectric constant kappa. A potential difference V exists between the inner and outer shells. Determine (a) the capacitance of the device, (b) the free charge q on the inner shell, and (c) the charge q' induced along the surface of the inner shell.

6. Due in recitation 03/20

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Chapter 27. Current and Resistance:

30. The wires in a house must be large enough in diameter so that they do not get hot enough to start a fire. Suppose a certain wire is to carry a current of 20 A, and it is determined that the joule heating of the wire should not exceed 2 W/m. What diameter must a copper wire have to be "safe" for this current?

31. HRW p. 672 #59: A linear accelerator produces a pulsed beam of electrons. The pulse current is 0.50 A, and the pulse duration is 0.10 microsecond. (a) How many electrons are accelerated per pulse? (b) What is the average current for a machine operating at 500 pulses/s? (c) If the electrons are accelerated to an energy of 50 MeV, what are the average and peak powers of the accelerator?

32. HRW p. 671 #41: A resistor has the shape of a truncated right-circular cone (see figure 27-25 in text). The end radii are a and b, and the altitude is L. If the taper is small, we may assume that the current density is uniform across any cross section. (a) Calculate the resistance of this object. (b) Show that your answer reduces to rho L / A for the special case of zero taper (that is, for a = b).

Chapter 28. Circuits:

33. A resistor of size R1 is placed in series with a parallel combination of resistors R2 and R3. A battery of voltage V is connected across the entire system, as shown in the diagram. Find the effective resistance of the system, the voltage across each resistor, the current through each resistor, and the current through the battery. [When you get the correct answers, keep them for use in the assignment on AC circuits.]

34. For the circuit shown in the figure, find (a) the current in each resistor, (b) the power supplied by each EMF, and (c) the power dissipated in each resistor. (d) Is energy conservation obeyed?

35. Nine resistors of resistance R are connected as shown in the figure, and a potential difference V is applied between points a and b. What is the equivalent resistance of this network? (b) Find the current in each of the nine resistors.

36. Find the current through each of the elements in the diagram to the right. The use of a symbolic-algebra computer program is recommended. [When you have correct answers, keep them for use in the chapter on AC circuits.]

7. Due in recitation 04/03

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Chapter 29. Magnetic Fields:

37. HRW p. 725 #55: Figure 29-46 in the text shows a rectangular, 20 turn coil of wire, 10 cm by 5.0 cm. It carries a current of 0.10 A and is hinged along one side. It is mounted in the xy plane, at an angle of 30o to the direction of a uniform magnetic field of 0.50 T. Find the magnitude and direction of the torque acting on the coil about the hinge line.

38. A particle of charge q and mass M moves in a circle of radius r and with angular velocity omega. (a) Show that the average current is I = q omega / (2 pi ) and that the magnetic moment has the magnitude mu = (1/2) q omega r2. (b) Show that the angular momentum of this particle has the magnitude L = M r2 omega and that the magnetic moment and angular momentum vectors are related by mu = [q/(2M)]L.

39. A current-carrying wire is bent into a semicircular loop of radius R, which lies in the xy plane. There is a uniform magnetic field B = B 1z perpendicular to the plane of the loop. Show that the force acting on the loop is F = 2 I R B 1y.

40. A nonconducting rod of mass M and length l has a uniform charge per unit length lambda and rotates with angular velocity omega about an axis through one end and perpendicular to the rod. (a) Consider a small segment of the rod of length dx and charge dq = lambda dx at a distance x from the pivot. Show that the magnetic moment of this segment is (1/2) lambda omega x2 dx. (b) Integrate your result to show that the total magnetic moment of the rod is mu = (1/6) lambda omega l3  (c) Show that the magnetic moment mu and angular momentum L are related by mu = [(Q/(2M)] L, where Q is the total charge on the rod.

8. Due in recitation 04/10

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Chapter 30: Magnetic Fields Due to Currents

41: HRW p. 748 # 45 [reworded without change of content]. Figure 30-61 shows an approximation to the magnetic field in a region, and an arrow representing a path through that region. Show that the uniform magnetic field B cannot drop abruptly to zero as one moves along a path perpendicular to B, as shown at point a on the figure. (Hint: Apply Ampere's law to the rectangular path shown by the dashed lines.) In actual magnets "fringing" of the magnetic field lines always occurs, which means that B approaches zero in a gradual manner. Modify the field lines in the figure to indicate a more realistic set of lines.

42: HRW p. 747 # 38: In the figure to the right (HRW figure 30-55), the long straight wire carries a current of 30 A and the rectangular loop carries a current of 20 A. Calculate the resultant force acting on the loop. Assume that the distance a = 1.0 cm, b = 8.0 cm, and L = 30 cm.

43: HRW p. 749 #50. Figure 30-64 of the text shows a cross section of a long cylindrical conductor of radius a containing a long cylindrical hole of radius b. The axes of the cylinder and hole are parallel and are a distance d apart; a current i is uniformly distributed in the conductor outside the hole. (a) Use superposition to show that the magnetic field at the center of the hole is

                      mu0 i d
          B = ------------------------
                  2 pi (a2 - b2)

(b) Discuss the two special cases b = 0 and d = 0. (c) Use Ampere's Law to show that the magnetic field in the hole is uniform.
(Hint: Regard the cylindrical hole as filled with two equal currents moving in opposite directions, thus canceling each other. Assume that each of these currents has the same current density as that in the actual conductor. Then superimpose the fields due to two complete cylinders of current, of radii a and b, each cylinder having the same current density.)

44. A solenoid has n turns per unit length and radius R and carries a current I. Its axis is along the x axis with one end at x = -L/2 and the other end at x = +L/2, where L is the total length of the solenoid. Show that the magnetic field B at a point on the axis outside the solenoid is given by

          B = (1/2) mu0 n l (cos theta1 - cos theta2)


                            x + (1/2) L
          cos theta1 = -------------------------
                        [ R2 + (x + L/2)2 ]1/2


                            x - (1/2) L
          cos theta2 = -------------------------
                        [ R2 + (x - L/2)2 ]1/2

45. In Problem 44, a formula for the magnetic field along the axis of a solenoid is given. For x >> L and L > R, the angles theta1 and theta2 in that problem are very small, so that the small angle approximation cos theta = 1 - theta2 / 2 is valid. (a) Draw a diagram and show that

     tan (theta1) = ------ 
                    x + L/2


     tan (theta2) = ------- 
                    x - L/2

(b) Show that the magnetic field at a point far from either end of the solenoid can be written

              mu0    qm    qm
          B = ---- [ -- - -- ]
              2 pi   r12   r22

where r1 = x - L/2 is the distance to the near end of the solenoid, r2 = x + L/2 is the distance to the far end, and qm = n I pi R2 is the magnetic moment of the solenoid divided by its length.

9. Due in recitation 04/17

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Chapter 31. Induction and Inductance:

46. HRW p. 779 # 25: A rectangular coil of N turns and of length a and width b is rotated at frequency f = omega/(2 pi) in a uniform magnetic field B, as indicated in Figure 31-50 in the text. The coil is connected to co-rotating cylinders, against which metal brushes slide to make contact. (a) Show that the emf induced in the coil is given (as a function of time t) by

EMF = 2 pi f N a b B sin(2 pi f t) = EMF0 sin (2 pi f t).

This is the priciple of the commercial alternating-current generator. (b) Design a loop that will produce an emf of 150 V when rotated at 60.0 rev/s in a magnetic field of 0.500 T.

47. HRW p. 780 # 29: A rectangular loop of wire with length a, width b, and resistance R is placed near an infinitely long wire carrying current i, as shown in Fig. 31-53 of the text. The distance from the long wire to the center of the loop is r. Find (a) the magnitude of the magnetic flux through the loop and (b) the current in the loop as it moves away from the long wire with speed v. [Neglect the magnetic field generated by the current in the loop itself. - MVB]

48. HRW p. 781 # 38: In Fig. 31-57 of the text a conducting rod of mass m, resistance R, and length L slides without friction on two long, horizontal, resistanceless rails. A uniform vertical magnetic field B fills the region in which the rod is free to move. The generator G supplies a constant current i directed as shown in the figure. (a) Find the velocity of the rod as a function of time, assuming it to be at rest at t = 0. The generator is now replaced by a battery that supplies a constant EMF. (b) Show that the velocity of the rod now approaches a constant terminal value v and give its magnitude and direction. (c) What is the current in the rod when this terminal velocity is reached? (d) Analyze this situation and that with the generator from the point of view of energy transfers.

49. A long solenoid has n turns per unit length and carries a current given by I = I0 sin (omega t). The solenoid has a circular cross section of radius R. Find the induced electric field at a radius r from the axis of the solenoid for (a) r < R and (b) r > R.

10. Due in recitation 04/24

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Chapter 31. Induction and Inductance:

50. HRW p. 782 # 57: Two inductors L1 and L2 and connected in parallel and separated by a large distance. (a) Show that the equivalent inductance is given by 1/Leq = 1/L1 + 1/L2. (Hint: Review the derivations for resistors in parallel and capacitors in parallel. Which is similar here?) (b) Why must their separation be large for this relationship to hold? (c) What is the generalization of (a) for N inductors in parallel?

51. Show that the inductance of a toroid of rectangular cross section, half of which is shown in the figure, is given by

L = mu0 N2 h ln(b/a) / (2 pi)

where N is the total number of turns, a is the inside radius of the toroid, b is its outside radius, and h is its height.

52. HRW p. 783 # 69. In figure 31-65 of the text, E = 100 V, R1 = 10.0 Ohms, R2 = 20 Ohms, R3 = 30 Ohms, and L = 2.00 H. Find the values of i1 and i2 (a) immediately after the closing of switch S, (b) a long time later, (c) immediately after the reopening of switch S, and (d) a long time after the reopening.

53. For the circuit in the figure, (a) find the rate of change of current in each inductor and in the resistor just after the switch is closed. (b) What is the current through the resistor after a long time has passed?

54. In the circuit in the figure, the switch is closed at t = 0. From time t to time L/R, find (a) the total energy that has been supplied by the battery, (b) the total energy that has been dissipated in the resistor, and (c) the energy that has been stored in the inductor. Hint: Find the rates as functions of time and integrate from t = 0 to t = L/R

11. Due in recitation 05/01

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Chapter 33. AC Circuits

55. Sketch as a function of frequency the magnitude of the total impedance Z for (a) a series LR circuit, (b) a series RC circuit, and (c) and a series LRC circuit.

56. A resistance R and a 1.4-H inductance are in series across a 60-Hz AC voltage. The voltage across the resistor is 30 V and the voltage across the inductor is 40 V. (a) What is the value of the resistance? (b) What is the AC input voltage?

57. A coil draws 15 A when connected to a 220-V AC 60-Hz line. When it is in series with a 4-ohm resistor and the combination is connected to a 100-V battery, the battery current after a long time is observed to be 10 A. (a) What is the resistance in the coil? (b) What is the inductance of the coil?

12. Due in recitation 05/08

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Chapter 33. AC Circuits:

58. HRW p. 839 # 85 (Compare figure 33-35): A series LRC circuit has R = 15.0 ohms, C = 470 microfarads, and L = 25.0 mH. The generator driving the circuit provides a sinusoidal voltage of 75.0 V (rms) and frequency f = 550 Hz. (a) Calculate the rms current. (b) Find the rms voltages across the resistor, the capacitor, the inductor, the capacitor and inductor together, and across the resistor, inductor, and capacitor together. (c) At what average rate is energy dissipated by each of the three circuit elements?

59. The circuit shown in the figure is called an RC high-pass filter because high input frequencies are transmitted with greater amplitude than low input frequencies. (a) If the input voltage is

       Vin = V0 cos(omega t),

show that the output voltage [the potential difference between the two black dots in the figure] is

         V0 cos (omega t + phi)
   Vout = -------------------------
         [ 1 + (omega R C)-2 ]1/2
and determine the value of phi. (b) At what angular frequency is the output voltage half the input voltage? (c) Sketch a graph of the amplitude of Vout/V0 as a function of omega.

60. A resistor and an ideal inductor are connected in parallel across an emf V = V0 cos(omega t). Show that
(a) the current in the resistor is IR = (V0/R) cos(omega t),
(b) the current in the inductor is IL = (V0/XL)cos(omega t - pi / 2),
(c) I = IR + IL = Imaxcos(omega t - delta) where tan(delta) = R/XL and Imax = V0/Z with Z -2 = R -2 + XL -2.

61. Solve for the amplitudes of all the currents in a circuit which has a capacitor in series with a parallel combination of a resistor and an ideal inductor. Use the values R=10 ohms, |XL|=30 ohms, |XC|=10 ohms, and |V0|=20 volts.

62. Find the amplitudes of all the currents in the 3-loop circuit in the figure. Use R1 = 10 ohms, R2 = 40 ohms, R3 = 20 ohms, XL = 30 ohms, XC = 10 ohms, and V = 15 Volts. Do the calculation numerically or use a computer-algebra program to work the problem analytically.