ROVANIEMI UNIVERSITY OF APPLIED SCIENCE

SCHOOL OF TECHNOLOGY

Degree Programm of Informational Technology

EXERCISES OF ELECTRICITY AND MAGNETISMCourse 504D3B

Jouko Teeriaho

2007

Electrostatics1. The magnitude of the electric eld is 18 kV/m. There is a 15 nC charge in the eld. Calculate

the Coulomb's force on the charge in the eld. (Coulomb constant is κ = 9 ∗ 109Nm2/C2 )

2. Point charges 2.5 nC and -1.9 nC are at 0.50 m distance from each other. Calculate the forcebetween the charges.

3. Two charges 3 nC and -3 nC are at a distance of10 cm from each other. Calculate the magnitudeof the electric eld at point P.

4. Answer without using calculator. Four equal positive charges are at the edges of a square.a) What is the value of the electric eld at the center ofthe square?b) Assume, if only one of the charges would be present,the eld at the center of the square would be 15 kV/m.What is the value of the eld if any three of the chargesare present and one is missing?

5. How do you think, if there is somewhere on the straight line, which goes through 3 nC and-1 nC charges, a point where the electric eld is zero. If you think that such point exist, markits position on the picture. (Calculate it rst)

6. An electron (m = 9.11 ∗ 10−31 kg, q = 1.6 ∗ 10−19 C) is accelerated with 600 V potentialdierence. Calculate its end velocity using:a) classical;b) relativistic formula for kinetic energy;c) how many percents the result (a) diers from the correct result (b)?(Einsteins kinetic energy formula Ek= mc2√

1− v2

c2

−mc2, c=300 Mm/s)

7. During a thunderstorm the electric eld increases up to 3 MV/m. Electrons are removedfrom the atoms and lighting starts. The average free path of the electrons in air is 1.0 µmbefore colliding with other atoms in the air. Calculate the speed of electrons.

8. The electric potential V around a point charge Q is obtained from V = κQr, where κ = 9∗109.

Assume that Q=+1.0 µC.a) Calculate the voltage between points A and B, when A is at distance 0.5 m and B atdistance 2.5 m from the charge Q.b) Could this voltage light a 1000 V light tube (yes or no)?

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9. ABC is an equilateral triangle of side 4.0 cm in a vacuum. There are point charges of 8.0 µCat A and B. Find (a) the potential, (b) the electric eld intensity at C. State the direction ofthe eld.

10. A conducting sphere of radius 5.0 cm has a charge of 0.4 µC. Find the potential:a) 6.0 cm; b) 5.0 cm; c) 4.0 cm from the center of the sphere.(Assume 1

4πεo = 9.0 ∗ 109m F−1 )

11. Calculate the electric eld at a point 50 cm from a 1500 nC charge. Also calculate the energyon the air at that point.

12. A triangle has equal sides of 6 cm. The edges of thetriangle are 5 nC, 5 nC and -5 nC charges. Calculatethe electric eld in the middle of the triangle.

13. Calculate the speed of electron after it has been accelerated with 1000 V . (Use classicalSchuster's formula)

DC circuits

14. A capacitor is made of two 4.0 cm2 metal plates with 0.1 mm air between the plates. Calculate:a) surface charge density σ if the charge is 2 pC;b) electric eld between the plates if the charge is 2 pC;c) the capacitance C;d) maximum voltage between the plates;e) maximum energy of the capacitor.

15. Assume that the capacitor in upwards problem is charged with with 6 V battery and thebattery is still maintaining 6 V potential dierence between the poles of the capacitor. Athin plastic 0.1 mm foil is carefully put between the plates. (for plastic κ = 7) What happens?a) Does the eld E in the capacitor charge?b) Does the charge Q of the capacitor change?c) Does the energy of the capacitor increase, decrease or remain the same?d) Does the capacitance change?

16. Calculate the voltage between a cloud and the ground during a thunderstorm, when thelighting starts. The cloud is 1200 m above the ground. Calculate also the surface chargedensity σ of Earth under the cloud.

17. There are 12 bulbs in a candle series.a) Calculate the voltage and current over one candle, if the power of a candle is 3.0 W andwe use 220 V AC.b) If one of the candles is removed using a shortcut, what happens to the current and thepower output?

18. The length of a conducting wire is 2.87 m and diameter is 0.2 mm. Using 1.5 V we get33.4 mA current. Calculate the resistance and the resistivity of the wire.

19. A capacitor of 0.012 mF is charged with 300 V . Calculate the amount of work needed.

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20. Two bulbs with texts (40 W , 220 V ) and (100 W , 220 V ) are connected (a) in series (b) inparallel to 220 V power supply. Calculate the total current and power taken from the powersupply.

21. You have three 10 Ω resistors in your positions. What values of resistance can you build fromthese resistors using dierent combinations. Draw pictures.

22. You have lots of 10 Ω resistors. Combine them in a way, which gives you exactly 3 Ω resistance.Draw picture.

23. Find the (steady) current in a circuit when a charge of 40 C passes in 5 s.

24. What is the resistance of a copper cylinder of length 12 cm and cross-sectional area 0.4 cm2?(Resistivity of copper = 1.7 ∗ 10−8 Ωm)

25. Find the value of the current, I, in each of the two circuits below.

26. A millimeter with a full-scale deection of 5.0 mA and a coil resistance of 50 Ω is to be usedas a voltmeter with a full-scale deection of 2.0 V . What size resistance needs to be placedin series with the meter?

27. A millimeter with a full-scale deection of 20 mA and a coil resistance of 40 Ω is to be usedas a ammeter with a full-scale deection of 500 mA. What size shunt is required?

28. When a 12 V battery (i.e. a battery of EMF 12 V ) is connected across a lamp with a resistanceof 6.8 Ω the PD across the lamp is 10.2 V . Find:a) the current through the lamp;b) the inertial resistance of the battery.

29. A battery of EMF 6.0 V and internal resistance 2.0 Ω is connected across a torch bulb witha resistance of 10 Ω. Calculate:a) the current supplied by the battery;b) the power consumed by the bulb;c) the power consumed in the internal resistance of the battery.Verify that the total power supplied by the battery is equal to the product of its EMF andthe current it supplies.

30. A 3.0 µF capacitor and a 5.0 µF capacitor are each in parallel with a 12 V supply. Calculatethe charge on each capacitor.

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31. Refer to Fig. 3. Calculate:a) the charge on each capacitor;b) the PD across each capacitorin the circuit shown below.

32. A 4.0 µF capacitor and a 6.0 µF capacitorare connected in series with a 20 V supply.Calculate:a) the charge on each capacitor;b) the PD across each capacitor. (Fig.3 )

33. A 500 µF capacitor with a charge of 3000 µC is discharged through a 200 κΩ resistor.What isa) the initial discharge current;b) the current after 20 s?

34. Calculate the charge on the capacitor in the below circuit.

35. The following circuit is set up with both switchesopen. Assume that the internal resistance of the6.0 V cell is negligible. Calculate the charges storeon the capacitors M and N:a) with S1 closed but S2 left open;b) with both switches closed.

Magnetic elds

36. Calculate the radius of the trajectory of a proton which comes with the velocity of 260 Mm/sat 90o angle to the magnetic eld of the Earth (55µT ). The mass and the charge of protonsare 1.7 ∗ 10−27 kg and 1.6 ∗ 10−19 C.

37. In radiation treatment of cancer electrons are accelerated with a small cyclotron with radius10 cm and magnetic eld of 0.005 T.a) What is the speed of the electrons when they come out from the cyclotron?b) Calculate also the period of the electrons in the circular motion and the frequency of theaccelerating electric eld.The mass of an electrons is 9.11 ∗ 10−31 kg.

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38. A 20 m long wire with 12 A current is going from East to West in the Earth's magnetic eld55µT . Calculate the force acting on the wire due to the magnetic eld.

39. In a DC motor a 10 A current is going through a coil with 300 windings and cross sectionalarea 0.6 dm2. Magnetic eld of the motor is 0.5 T. The motor is running at 600 RPM. Graphthe torque during 0-5 seconds. What is the maximum value of the torque?

40. The coil of a DC motor has a resistance of 2.0 Ω. The motor is running by a battery of 12 Vand an inertial resistance of 0.5 Ω.Calculate:a) the current at the moment the motor is started;b) the current when the motor is running at 600 RPM, when there is a counted EMF of 10V in the coil.

41. Calculate the magnetic eld at 10 cm distance of a wire with DC of 100 A.

42. Calculate the magnetic eld in the center of a current loop with radius of 20 cm. The currentis 0.6 A.

43. Calculate the magnetic eld inside a 10 cm long coil with 300 windings. The coil has an ironcore. Relative permeability of iron is 1200. Current through the col is 3.0 A.

44. A 2000 turn solenoid of length 40 cm and resistance 16 Ω is connected to a 20 V supply.What is the ux density at the mid point of the axis of the solenoid?

45. A long wire (X) carrying a current of 30 A is placed parallel to, and 3.0 cm away from, asimilar wire (Y) carrying a current of 6.0 A. What is the ux density midway between thewires:a) when the currents are in the same direction;b)when they are in opposite direction;c) when the currents are in the same direction there is a point somewhere between X and Yat which the ux density is zero. How far from X is this point?

Magnetic induction, AC generators

46. A conductor rod (length 80 cm) is moving perpendicular to a magnetic eld of 5.0 T with aspeed of 20 m/s. Calculate the induced EMF between the ends of the road.

47. The magnetic elds inside a coil (length 10 cm, cross section area 8 cm * 8 cm, 500 windings)increases uniformly from 0 to 0.5 T in 2 s. Calculate the induced EMF in the coil.

48. A coil with 1200 windings rotates with a frequency of 100 Hz in a magnetic eld of 2.0 T.Calculate the induced EMF if the cross section area of the coil is 1.2 dm2.

49. In a hydroelectric power plant a cylinder (height 1.8 m, radius 2.0 m) with columns ofelectromagnets rotates slowly in a way, that the north poles (followed by south poles) passby a 1.8 high iron rod 50 times per second. The magnetic eld of the poles is 10 T. Thereare 25 rows of North-South pairs on the surface of the cylinder. Calculate the EMF createdin the rod.

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Self inductors and transformers

50. Calculate the inductance of a coil with 1200 windings, length of 10 cm and cross sectionalarea 1.2 dm2. Calculate the value also when there is an iron coil inside (for iron κ = 1000).

51. Calculate:a) the resistance of the copper wires between Helsinki and Rovaniemi (2*800 km). Thediameter of the wire is 2.5 cm.b) the power loss and voltage loss in the wires if 250 MW is transferred from Rovaniemi toHelsinki using 400 kV .

52. An engine needs 220 V , 4.0 kV electric power. The power is brought to the engine from anearby power station with 1.5 Ω wires. Calculate:a) the power loss in the wires and the EMF needed in the generator if transformers are notbeen used;b) the power loss and EMF in the case that there is two similar ideal transformers in theboth ends. The transformers have 33 windings in the primary coil and 660 windings in thesecondary coil.

53. A 12 V battery, a switch, a 4 Ω resistors and a 2 H coil are in series forming a closed circuit.Calculate:a) the current I at the moment the circuit is closed (switch is turned on);b) the nal current in the circuit;c) the time constant of the circuit;d) the value of the time derivative I'(t) when the current is 1.5 A.

54. A coil with an inductance of 20 H and a resistance of 10 Ω is connected in series with abattery of EMF 12 V and a switch. What is:a) the rate of change of current immediately after the switch is closed;b) the nal current;c) the current after 3.0 S;d) how long after the switch is closed will the current be 0.4 A.

55. A coil with a resistance of 6.0 Ω and an inductance of 30 mH is connected to a 12 V supply.What is the energy stored in the coil when the current has reached its equilibrium value?

56. A farmer installs a private hydroelectric generator to provide power for equipment rated at120 kW 240 V AC. The generator is connected to the equipment by two conductors whichhave a total resistance of 0.2 Ω. The system is shown schematically in the gure below.

a) The equipment is operating at its rate power. Calculate:(i) the power loss in the cables;(ii) the voltage which must be developed by the generator;(iii) the eciency of the transmission system.

b) An engineer suggest that the farmer uses a transformer to convert the generator outputto give a PD of 2400 V at the end of the transmission line, as shown in the next gure. Asecond transformer is to be used to step down this PD to 240 V .

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(i) Explain briey how a transformer makes use of electromagnetic induction to produce anoutput voltage several times bigger than the input voltage.(ii) The transformer are 100 % ecient. Calculate the power loss in the new transmissionsystem.

Three phase current

57. Calculate the current in the neutral line, if:a) all the phased are loaded equally;b) current taken from phases A,B and C are 10 A, 12 A and 15 A.

58. The peak value of the voltage in all three phases is 155.6 V . Calculate:a) the eective phase voltage (voltage between a phase and neutral line);b) the eective voltage that can be taken between two phase lines.

AC circuits

59. An 24 V AC power supply with adjustable frequency and a 100 Ω resistors are in series.Calculate the current when:a) 50 Hz frequency is used;b) 500 Hz frequency is used.

60. An 24 V AC power supply and a 0.2 H coil with negligible resistance are in series. Calculatethe current when:a) 50 Hz frequency is used;b) 500 Hz frequency is used;c) 5.0 kHz frequency is used.

61. An 24 V AC power supply and a 4 µF capacitor are in series. Calculate the current when:a) 50 Hz frequency is used;b) 500 Hz frequency is used;c) 5.0 kHz frequency is used.

62. An 24 V AC power supply, a 100 Ω resistors, 2.0 H coil and 4 µF capacitor are in series.Calculate:a) the frequency that gives the largest value for the current. What is the value of the currentin that case?b) the current and the phase angle when 500 Hz frequency is used.

63. Calculate the value of the capacitor needed when 96.7 MHz radio program is received. Thecoil used in the receiving circuit has 2.0 µH inductance.

64. A 47.0 µF capacitor and a resistor of resistance 30.0 Ω are connected in series. The pair areconnected across a 50 Hz supply with a peak value of 100 V . Find:a) the reactance of the capacitor;b) the impedance of the circuit;

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c) the peak value of the current in the circuit;d) the peak value of the PD across the capacitor;e) the peak value of the PD across the resistor.

65. A resistor of resistance 120 Ω, a capacitor of capacitance 22 µF and a coil with a resistance of10 Ω and an inductance of 300 mH are connected in series with a 100 Hz alternating supply.Find:a) the impedance of the circuit;b) the angle by which the applied PD leads the current.

66. A 20 µF capacitor and a 100 Ω resistor are connected in series with a 240 V RMS 50 Hzsupply. Calculate:a) the reactance of the capacitor;b) the impedance of the circuit;c) the RMS current;d) the average power.

67. A 47 µF capacitor and a 2 mH inductor with a resistance of 100 Ω are connected in serieswith a 50 V RMS supply. What is:a) the resonant frequency of the circuit;b) the average power consumed at this frequency (Note: you should be able to do part (b) inyour head!).

68. A 60 W light bulb, designed for use with a 120 V supply, may be operated at the correctrating from a 240 V , 50 Hz supply by connecting it in series with a resistor, as shown ingure below.

a) Calculate, for normal working conditions,(i) the current owing in the lamp(ii) the resistance R of the lamp(iii) a value for a suitable series resistor.

b) The same lamp may also be operated at the correct rating from the 240 V , 50 Hz supplyby connecting it in series with a capacitor as shown in the next gure.

(i) Write down an expression for the impedance of the circuit containing the lamp and thecapacitor of reactance Xc

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(ii) Calculate the value for Xc

(iii) Calculate a value for the capacitance of the capacitor.

(c) By considering the power dissipate in each circuit explain which circuit provides the bettersolution.

69. A is a coil having a low resistance and a high inductance.A resistor B has the same resistance as A, but negligibleinductance. C and D are identical lament lamps and thebattery can be assumed to have negligible resistance.a) Describe and explain how the appearance of each lampchanges in the period after the switch S is closed.b) The DC battery is replaced by a sinusoidal alternatingsource of RMS potential dierence 9 V olts and negligibleimpedance. Describe and explain the appearance of eachlamp after S is closed.

70. In the circuit,the source has negligible internal impedance. Find:a) the RMS current in the circuit;b) the mean rate of production of heat.

71. A tuning capacitor is to be used with a 10 mH inductor in an LC circuit of a radio to providetuning of all stations broadcasting in the band from 500 kHz to 1.50 MHz.a) The largest value of capacitance required is 10.1 pF . What is the smallest value used?b) The capacitor is to be made as shown in the diagram using two semi-circular plates whichare separated by an air gap of 1.00 mm.

Calculate the diameter of the plates required. Permittivity of air = 8.9 ∗ 10−12F m−1

c) State two ways of designing the capacitor so that the diameter can be reduced whilst givingthe same maximum capacitance.d) To change the range of tuning capacitance a 22 pH capacitor is connected in series withthe variable capacitor. What is the new maximum capacitance?

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72. A series LCR circuit is set up as shown below. The frequency of the source S (of zeroimpedance) is 1000

2πHz.

Calculate for the circuit:a) the impedance;b) the RMS current;c) the power dissipate.

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Electrostatics1. 0.27 mN ; 2. 1.7 ∗ 10−7 N ; 3. 7.6 kV/m; 6. a)14.18 Mm/s b)14.50 Mm/s c)n 2.3%; 7.

1.0 Mm/s; 8. a)14.4 kV b)yes; 9. a)3.6 ∗ 106 V b)7.8 ∗ 107 NC−1 at 90o to AB, directed away from AB;10. a)6.0 ∗ 105 V b)7.2 ∗ 105 V c)7.2 ∗ 105 V ; 11. 54 kV/m 13 mJ/m3; 12. 75 kV/m; 13.18.7 Mm/s.

DC circuits14. a)5 nC/m2 b)n 565 V/m c)35 pF d)300 V e)1.6 µJ ; 16. 3.6 GV 27 µC/m2; 17. a)18.3 V ,

0.16 A b)0.179 A, 39.4 W ; 18. resistance 45 Ω resistivity 49 ∗ 10−8 Ωm; 19. 0.54 J ; 20. a)current 0.13 A, power outputs 20.3 W and 8.2 W= together 28.5 W ; 23. 8.0 A; 24. 5.1 ∗ 10−5 Ω;25. a)2.0 A b)2 A; 26. 3.5 ∗ 102 Ω; 27. 1.7 Ω; 28. a)1.5 A b)1.2 Ω; 29. a)0.50 Ab)2.5 Wc)0.50 W ; 30. 36 µC (3.0 µF ), 60 µC (5.0 µF ); 31. a) 36 µC(X), 36 µC(Y), 24 µC(Z) b) 9 V (X),3 V (Y), 12 V (Z); 32. a)48 µCon each b)12 V (4.0 µF ), 8.0 V (6.0 µF ); 33. a)30.0 µA b)24.6 µA;34. 13.3 µC; 35. a)4 µCon each b)2 µC(M), 8 µC(N).

Magnetic elds36. 50227 m; 37. a)88 Mm/s b)7 ns; 38. 13 mN ; 39. max torque 9 Nm; 40. a)24 A

b)4 A; 41. 0.2 mT ; 42. 18.8 µT ; 43. 13.6 T ; 44. 7.9 ∗ 10−3 T ; 45. a)3.2 ∗ 10−4 T b)4.8 ∗ 10−4 Tc)2.5 cm.

Magnetic induction, AC generators46. 80 V ; 47. 0.8 V ; 48. peak value 18 kV ; 49. peak value 453 V .

Self inductors and transformers50. 217 H; 51. a)55 Ωb)22 MW ; 52. a) 497 W , 247 V b) 1.24 W , 220 V ; 53. a)0 A b)3 A

c)0.5 s d)3 A/s; 54. a)0.60 As−1 b)1.2 A c)0.93 A d)0.81 s; 55. 60 mJ ; 56. a) (i)50 kW (ii) 340 V(iii) 70.6 % b) (ii)500 W .

Three phase current57. a)0 A b)4.4 A; 58. a)110 V b)191 V .

AC circuits59. 0.24 A; 60. a)380 mA b)38 mA c)3.8 mA; 61. a)30 mA b)300 mA c)3 A; 62. a)

56 Hz, 0.24 A b) 3.9 A, 89o; 63. 1.4 pF ; 64. a)67.7 Ω b)74.1 Ω c)1.35 A d)91.4 V e)40.5 V ;65. a)174 Ω b)41.8o; 66. a)159 Ω b)188 Ω c)1.28 A d)163 W ; 67. a)519 Hz b)25 W ; 68.a) (i)0.50 A (ii) 240 Ω (iii) 240 Ω b) (ii)416 Ω (iii) 7.7 µF ; 70. a)47.2 A b)8.91 ∗ 103 W ; 71.a)1.13 pF b)5.4 cm d)6.9 pF ; 72. a)141 Ω b)0.20 A c)4.0 W .

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