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- 1. ROVANIEMI UNIVERSITY OF APPLIED SCIENCE SCHOOL OF TECHNOLOGY Degree Programm of Informational Technology EXERCISES OF ELECTRICITY AND MAGNETISM Course 504D3B Jouko Teeriaho 2007
- 2. Electrostatics 1. 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 ∗ 109 Nm2 /C 2 ) 2. Point charges 2.5 nC and -1.9 nC are at 0.50 m distance from each other. Calculate the force between the charges. 3. Two charges 3 nC and -3 nC are at a distance of 10 cm from each other. Calculate the magnitude of 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 of the 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 charges are 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, mark its 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 potential dierence. 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)? 2 (Einsteins kinetic energy formula Ek = mc 2 − mc2 , c=300 M m/s) 1− v2 c 7. During a thunderstorm the electric eld increases up to 3 M V /m. Electrons are removed from the atoms and lighting starts. The average free path of the electrons in air is 1.0 µm before 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 = κ Q , where κ = 9 ∗ 109 . r 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 at distance 2.5 m from the charge Q. b) Could this voltage light a 1000 V light tube (yes or no)? 2
- 3. 9. ABC is an equilateral triangle of side 4.0 cm in a vacuum. There are point charges of 8.0 µC at A and B. Find (a) the potential, (b) the electric eld intensity at C. State the direction of the 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 πεo = 9.0 ∗ 109 m F −1 ) 4 11. Calculate the electric eld at a point 50 cm from a 1500 nC charge. Also calculate the energy on the air at that point. 12. A triangle has equal sides of 6 cm. The edges of the triangle are 5 nC , 5 nC and -5 nC charges. Calculate the electric eld in the middle of the triangle. 13. Calculate the speed of electron after it has been accelerated with 1000 V . (Use classical Schuster'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 the battery is still maintaining 6 V potential dierence between the poles of the capacitor. A thin 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 the lighting starts. The cloud is 1200 m above the ground. Calculate also the surface charge density 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 and we use 220 V AC. b) If one of the candles is removed using a shortcut, what happens to the current and the power output? 18. The length of a conducting wire is 2.87 m and diameter is 0.2 mm. Using 1.5 V we get 33.4 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. 3
- 4. 20. Two bulbs with texts (40 W , 220 V ) and (100 W , 220 V ) are connected (a) in series (b) in parallel to 220 V power supply. Calculate the total current and power taken from the power supply. 21. You have three 10 Ω resistors in your positions. What values of resistance can you build from these 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 used as a voltmeter with a full-scale deection of 2.0 V . What size resistance needs to be placed in 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 used as 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 resistance of 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 with a 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 and the current it supplies. 30. A 3.0 µF capacitor and a 5.0 µF capacitor are each in parallel with a 12 V supply. Calculate the charge on each capacitor. 4
- 5. 31. Refer to Fig. 3. Calculate: a) the charge on each capacitor; b) the PD across each capacitor in the circuit shown below. 32. A 4.0 µF capacitor and a 6.0 µF capacitor are 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 is a) 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 switches open. Assume that the internal resistance of the 6.0 V cell is negligible. Calculate the charges store on 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/s at 90o angle to the magnetic eld of the Earth (55µT ). The mass and the charge of protons are 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 radius 10 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 the accelerating electric eld. The mass of an electrons is 9.11 ∗ 10−31 kg . 5
- 6. 38. A 20 m long wire with 12 A current is going from East to West in the Earth's magnetic eld 55µT . 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 sectional area 0.6 dm2 . Magnetic eld of the motor is 0.5 T. The motor is running at 600 RPM. Graph the 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 V and 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 10 V 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 current is 0.6 A. 43. Calculate the magnetic eld inside a 10 cm long coil with 300 windings. The coil has an iron core. 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, a similar wire (Y) carrying a current of 6.0 A. What is the ux density midway between the wires: 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 Y at 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 a speed 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 of electromagnets rotates slowly in a way, that the north poles (followed by south poles) pass by a 1.8 high iron rod 50 times per second. The magnetic eld of the poles is 10 T. There are 25 rows of North-South pairs on the surface of the cylinder. Calculate the EMF created in the rod. 6
- 7. Self inductors and transformers 50. Calculate the inductance of a coil with 1200 windings, length of 10 cm and cross sectional area 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). The diameter of the wire is 2.5 cm. b) the power loss and voltage loss in the wires if 250 M W is transferred from Rovaniemi to Helsinki using 400 kV . 52. An engine needs 220 V , 4.0 kV electric power. The power is brought to the engine from a nearby 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 not been used; b) the power loss and EMF in the case that there is two similar ideal transformers in the both ends. The transformers have 33 windings in the primary coil and 660 windings in the secondary 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 a battery 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 at 120 kW 240 V AC. The generator is connected to the equipment by two conductors which have 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 output to give a PD of 2400 V at the end of the transmission line, as shown in the next gure. A second transformer is to be used to step down this PD to 240 V . 7
- 8. (i) Explain briey how a transformer makes use of electromagnetic induction to produce an output voltage several times bigger than the input voltage. (ii) The transformer are 100 % ecient. Calculate the power loss in the new transmission system. 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. Calculate the 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 current in 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 M Hz radio program is received. The coil 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 are connected 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; 8
- 9. 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 of 10 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 Hz supply. 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 series with 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) in your head!). 68. A 60 W light bulb, designed for use with a 120 V supply, may be operated at the correct rating from a 240 V , 50 Hz supply by connecting it in series with a resistor, as shown in gure 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 supply by 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 the capacitor of reactance Xc 9
- 10. (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 better solution. 69. A is a coil having a low resistance and a high inductance. A resistor B has the same resistance as A, but negligible inductance. C and D are identical lament lamps and the battery can be assumed to have negligible resistance. a) Describe and explain how the appearance of each lamp changes in the period after the switch S is closed. b) The DC battery is replaced by a sinusoidal alternating source of RMS potential dierence 9 V olts and negligible impedance. Describe and explain the appearance of each lamp 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 provide tuning of all stations broadcasting in the band from 500 kHz to 1.50 M Hz . 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 which are separated by an air gap of 1.00 mm. Calculate the diameter of the plates required. Permittivity of air = 8.9 ∗ 10−12 F m−1 c) State two ways of designing the capacitor so that the diameter can be reduced whilst giving the same maximum capacitance. d) To change the range of tuning capacitance a 22 pH capacitor is connected in series with the variable capacitor. What is the new maximum capacitance? 10
- 11. 72. A series LCR circuit is set up as shown below. The frequency of the source S (of zero impedance) is 1000 Hz . 2π Calculate for the circuit: a) the impedance; b) the RMS current; c) the power dissipate. 11
- 12. Electrostatics 1. 0.27 mN ; 2. 1.7 ∗ 10−7 N ; 3. 7.6 kV /m; 6. a)14.18 M m/s b)14.50 M m/s c)n 2.3%; 7. 1.0 M m/s; 8. a)14.4 kV b)yes; 9. a)3.6 ∗ 106 V b)7.8 ∗ 107 N C −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 M m/s. DC circuits 14. 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 W c)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 µC on 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 µC on each b)2 µC (M), 8 µC (N). Magnetic elds 36. 50227 m; 37. a)88 M m/s b)7 ns; 38. 13 mN ; 39. max torque 9 N m; 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 T c)2.5 cm. Magnetic induction, AC generators 46. 80 V ; 47. 0.8 V ; 48. peak value 18 kV ; 49. peak value 453 V . Self inductors and transformers 50. 217 H ; 51. a)55 Ωb)22 M W ; 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 current 57. a)0 A b)4.4 A; 58. a)110 V b)191 V . AC circuits 59. 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 . 12