2. Chapter 21 Current and Direct Current Circuits

3. 2 Fig. 21-CO, p. 683

4. 3 21.1 Electric Current Electric current is the rate of flow of charge through a surface The SI unit of current is the Ampere (A) 1 A = 1 C / s The symbol for electric current is I

5. 4 Average Electric Current Assume charges are moving perpendicular to a surface of area A If  Q is the amount of charge that passes through A in time  t, the average current is Fig 21.1

6. 5 Instantaneous Electric Current If the rate at which the charge flows varies with time, the instantaneous current, I, can be found

7. 6 Direction of Current The charges passing through the area could be positive or negative or both It is conventional to assign to the current the same direction as the flow of positive charges The direction of current flow is opposite the direction of the flow of electrons It is common to refer to any moving charge as a charge carrier

8. 7 Current and Drift Speed Charged particles move through a conductor of cross-sectional area A n is the number of charge carriers per unit volume n A Δx is the total number of charge carriers Fig 21.2

9. 8 Current and Drift Speed, cont The total charge is the number of carriers times the charge per carrier, q ΔQ = (n A Δ x) q The drift speed, v d, is the speed at which the carriers move v d = Δ x/ Δt Rewritten: ΔQ = (n A v d Δt) q Finally, current, I = ΔQ/Δt = nqv d A

10. 9 Charge Carrier Motion in a Conductor The zig-zag black line represents the motion of charge carrier in a conductor The net drift speed is small The sharp changes in direction are due to collisions The net motion of electrons is opposite the direction of the electric field Fig 21.4

11. 10 If you can't see the image above, please install Shockwave Flash Player.Shockwave Flash Player. If this active figure can’t auto-play, please click right button, then click play. NEXT Active Figure 21.4

12. 11 Motion of Charge Carriers, cont When the potential difference is applied, an electric field is established in the conductor The electric field exerts a force on the electrons The force accelerates the electrons and produces a current

13. 12 Motion of Charge Carriers, final The changes in the electric field that drives the free electrons travel through the conductor with a speed near that of light This is why the effect of flipping a switch is effectively instantaneous Electrons do not have to travel from the light switch to the light bulb in order for the light to operate The electrons are already in the light filament They respond to the electric field set up by the battery

14. 13 Drift Velocity, Example Assume a copper wire, with one free electron per atom contributed to the current The drift velocity for a 12 gauge copper wire carrying a current of 10 A is 2.22 x 10 -4 m/s This is a typical order of magnitude for drift velocities

15. 14 Current Density J is the current density of a conductor It is defined as the current per unit area J = I / A = n q v d This expression is valid only if the current density is uniform and A is perpendicular to the direction of the current J has SI units of A / m 2 The current density is in the direction of the positive charge carriers

16. 15 Conductivity A current density J and an electric field E are established in a conductor whenever a potential difference is maintained across the conductor J =  E The constant of proportionality, , is called the conductivity of the conductor

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19. 18 21.2 Resistance In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor The constant of proportionality is the resistance of the conductor

20. 19 Fig. 21-5, p. 687

21. 20 Resistance, cont SI units of resistance are ohms (Ω) 1 Ω = 1 V / A Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor

22. 21 Ohm’s Law Ohm’s Law states that for many materials, the resistance is constant over a wide range of applied voltages Most metals obey Ohm’s Law Materials that obey Ohm’s Law are said to be ohmic

23. 22 Ohm’s Law, cont Not all materials follow Ohm’s Law Materials that do not obey Ohm’s Law are said to be nonohmic Ohm’s Law is not a fundamental law of nature Ohm’s Law is an empirical relationship valid only for certain materials

24. 23 Ohmic Material, Graph An ohmic device The resistance is constant over a wide range of voltages The relationship between current and voltage is linear The slope is related to the resistance Fig 21.6(a)

25. 24 Nonohmic Material, Graph Non-ohmic materials are those whose resistance changes with voltage or current The current-voltage relationship is nonlinear A diode is a common example of a non-ohmic device Fig 21.6(b)

26. 25 Resistivity Resistance is related to the geometry of the device:  is called the resistivity of the material The inverse of the resistivity is the conductivity:  = 1 /  and R = l /  A Resistivity has SI units of ohm-meters ( . m)

27. 26 p. 688

28. 27 Table 21-1, p. 689

29. 28 Resistance and Resistivity, Summary Resistivity is a property of a substance Resistance is a property of an object The resistance of a material depends on its geometry and its resistivity An ideal (perfect) conductor would have zero resistivity An ideal insulator would have infinite resistivity

30. 29 Resistors Most circuits use elements called resistors Resistors are used to control the current level in parts of the circuit Resistors can be composite or wire- wound

31. 30 Resistor Values Values of resistors are commonly marked by colored bands

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35. 34 Resistance and Temperature Over a limited temperature range, the resistivity of a conductor varies approximately linearly with the temperature ρ o is the resistivity at some reference temperature T o T o is usually taken to be 20° C  is the temperature coefficient of resistivity SI units of  are o C -1

36. 35 Temperature Variation of Resistance Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance

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39. 38 21.3 Resistivity and Temperature, Graphical View For metals, the resistivity is nearly proportional to the temperature A nonlinear region always exists at very low temperatures The resistivity usually reaches some finite value as the temperature approaches absolute zero Fig 21.8

40. 39 Residual Resistivity The residual resistivity near absolute zero is caused primarily by the collisions of electrons with impurities and imperfections in the metal High temperature resistivity is predominantly characterized by collisions between the electrons and the metal atoms This is the linear range on the graph

41. 40 Superconductors A class of metals and compounds whose resistances go to zero below a certain temperature, T C T C is called the critical temperature The graph is the same as a normal metal above T C, but suddenly drops to zero at T C Fig 21.9

42. 41 Superconductors, cont The value of T C is sensitive to Chemical composition Pressure Crystalline structure Once a current is set up in a superconductor, it persists without any applied voltage Since R = 0

43. 42 21.4 Superconductor Application An important application of superconductors is a superconducting magnet The magnitude of the magnetic field is about 10 times greater than a normal electromagnet

44. 43 Table 21-3, p. 693

45. 44 Electrical Conduction – A Model The free electrons in a conductor move with average speeds on the order of 10 6 m/s Not totally free since they are confined to the interior of the conductor The motion is random The electrons undergo many collisions The average velocity of the electrons is zero There is zero current in the conductor

46. 45 Conduction Model, 2 An electric field is applied The field modifies the motion of the charge carriers The electrons drift in the direction opposite of the field The average drift speed is on the order of 10 -4 m/s, much less than the average speed between collisions

47. 46 Conduction Model, 3 Assumptions: The excess energy acquired by the electrons in the field is lost to the atoms of the conductor during the collision The energy given up to the atoms increases their vibration and therefore the temperature of the conductor increases The motion of an electron after a collision is independent of its motion before the collision

48. 47 Conduction Model, 4 The force experienced by an electron is From Newton’s Second Law, the acceleration is Applying a motion equation Since the initial velocities are random, their average value is zero

49. 48 Conduction Model, 5 Let  be the average time interval between successive collisions The average value of the final velocity is the drift velocity This is also related to the current: I = n e v d A = (n e 2 E / m e )  A

50. 49 Conduction Model, final Using Ohm’s Law, an expression for the resistivity of a conductor can be found: Note, the resistivity does not depend on the strength of the field The average time is also related to the free mean path:  = l avg /v avg

51. 50 Conduction Model, Modifications A quantum mechanical model is needed to explain the incorrect predictions of the classical model developed so far The wave-like character of the electrons must be included The predictions of resistivity values then are in agreement with measured values

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55. 54 21.5 Electrical Power Assume a circuit as shown As a charge moves from a to b, the electric potential energy of the system increases by Q  V The chemical energy in the battery must decrease by this same amount Fig 21.10

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57. 56 Electrical Power, 2 As the charge moves through the resistor (c to d), the system loses this electric potential energy during collisions of the electrons with the atoms of the resistor This energy is transformed into internal energy in the resistor Corresponding to increased vibrational motion of the atoms in the resistor

58. 57 Electric Power, 3 The resistor is normally in contact with the air, so its increased temperature will result in a transfer of energy by heat into the air The resistor also emits thermal radiation After some time interval, the resistor reaches a constant temperature The input of energy from the battery is balanced by the output of energy by heat and radiation

59. 58 Electric Power, 4 The rate at which the system loses potential energy as the charge passes through the resistor is equal to the rate at which the system gains internal energy in the resistor The power is the rate at which the energy is delivered to the resistor

60. 59 Electric Power, final The power is given by the equation: Applying Ohm’s Law, alternative expressions can be found: Units: I is in A, R is in , V is in V, and P is in W

61. 60 Electric Power Transmission Real power lines have resistance Power companies transmit electricity at high voltages and low currents to minimize power losses

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72. 71 21.6 emf A source of emf (electromotive force) is an entity that maintains the constant voltage of a circuit The emf source supplies energy, it does not apply a force, to the circuit The battery will normally be the source of energy in the circuit Could also use generators

73. 72 Sample Circuit We consider the wires to have no resistance The positive terminal of the battery is at a higher potential than the negative terminal There is also an internal resistance in the battery Fig 21.12

74. 73 Internal Battery Resistance If the internal resistance is zero, the terminal voltage equals the emf In a real battery, there is internal resistance, r The terminal voltage,  V =  - Ir Fig 21.13

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76. 75 emf, cont The emf is equivalent to the open-circuit voltage This is the terminal voltage when no current is in the circuit This is the voltage labeled on the battery The actual potential difference between the terminals of the battery depends on the current in the circuit

77. 76 21.7 Load Resistance The terminal voltage also equals the voltage across the external resistance This external resistor is called the load resistance In the previous circuit, the load resistance is the external resistor In general, the load resistance could be any electrical device

78. 77 Power The total power output of the battery is P = I  V =I  This power is delivered to the external resistor (I 2 R) and to the internal resistor (I 2 r) P = I  I 2 R + I 2 r The current depends on the internal and external resistances

79. 78 Resistors in Series When two or more resistors are connected end- to-end, they are said to be in series For a series combination of resistors, the currents are the same in all the resistors because the amount of charge that passes through one resistor must also pass through the other resistors in the same time interval The potential difference will divide among the resistors such that the sum of the potential differences across the resistors is equal to the total potential difference across the combination

80. 79 Resistors in Series, cont Potentials add ΔV = IR 1 + IR 2 = I (R 1 +R 2 ) Consequence of Conservation of Energy The equivalent resistance has the same effect on the circuit as the original combination of resistors Fig 21.14

81. 80 Equivalent Resistance – Series R eq = R 1 + R 2 + R 3 + … The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistances If one device in the series circuit creates an open circuit, all devices are inoperative

82. 81 Equivalent Resistance – Series – An Example Two resistors are replaced with their equivalent resistance Fig 21.14

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84. 83 Resistors in Parallel The potential difference across each resistor is the same because each is connected directly across the battery terminals The current, I, that enters a point must be equal to the total current leaving that point I = I 1 + I 2 The currents are generally not the same Consequence of Conservation of Charge

85. 84 Equivalent Resistance – Parallel Equivalent Resistance The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance The equivalent is always less than the smallest resistor in the group Fig 21.16

86. 85 Equivalent Resistance – Parallel, Examples Equivalent resistance replaces the two original resistances Household circuits are wired so the electrical devices are connected in parallel Circuit breakers may be used in series with other circuit elements for safety purposes

87. 86 If you can't see the image above, please install Shockwave Flash Player.Shockwave Flash Player. If this active figure can’t auto-play, please click right button, then click play. NEXT Active Figure 21.16

88. 87 Resistors in Parallel, Final In parallel, each device operates independently of the others so that if one is switched off, the others remain on In parallel, all of the devices operate on the same voltage The current takes all the paths The lower resistance will have higher currents Even very high resistances will have some current

89. 88 Fig. 21-18, p. 703

90. 89 Circuit Reduction A circuit consisting of resistors can often be reduced to a simple circuit containing only one resistor Examine the original circuit and replace any resistors in series with their equivalents and any resistors in parallel with their equivalents Sketch the new circuit Examine it and replace any new series or parallel combinations with their equivalents Continue until a single equivalent resistance is found

91. 90 Fig. 21-19, p. 704

92. 91 Circuit Reduction – Example The 8.0  and 4.0  resistors are in series and can be replaced with their equivalent, 12.0  The 6.0  and 3.0  resistors are in parallel and can be replaced with their equivalent, 2.0  These equivalent resistances are in series and can be replaced with their equivalent resistance, 14.0  Fig 20.20

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101. 100 21.8 Kirchhoff’s Rules There are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor Two rules, called Kirchhoff’s Rules, can be used instead

102. 101 Fig. 21-22, p. 705

103. 102 Statement of Kirchhoff’s Rules Junction Rule At any junction, the sum of the currents must equal zero A statement of Conservation of Charge Loop Rule The sum of the potential differences across all the elements around any closed circuit loop must be zero A statement of Conservation of Energy

104. 103 Mathematical Statement of Kirchhoff’s Rules Junction Rule:  I in =  I out Loop Rule:

105. 104 More About the Junction Rule I 1 - I 2 - I 3 = 0 Use +I for currents entering a junction Use –I for currents leaving a junction From Conservation of Charge Diagram b shows a mechanical analog Fig 21.23

106. 105 More About the Loop Rule Traveling around the loop from a to b In a, the resistor is transversed in the direction of the current, the potential across the resistor is –IR In b, the resistor is transversed in the direction opposite of the current, the potential across the resistor is is +IR Fig 21.24

107. 106 Loop Rule, final In c, the source of emf is transversed in the direction of the emf (from – to +), the change in the electric potential is +ε In d, the source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is -ε Fig 21.24

108. 107 Junction Equations from Kirchhoff’s Rules Use the junction rule as often as needed, so long as, each time you write an equation, you include in it a current that has not been used in a previous junction rule equation In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit

109. 108 Loop Equations from Kirchhoff’s Rules The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation You need as many independent equations as you have unknowns

110. 109 Kirchhoff’s Rules’ Equations, final In order to solve a particular circuit problem, the number of independent equations you need to obtain from the two rules equals the number of unknown currents

111. 110 Problem-Solving Hints – Kirchhoff’s Rules Conceptualize Study the circuit diagram Identify all the elements in the circuit Identify the polarity of all the batteries and imagine the directions in which the current would exist through the batteries Categorize Determine if the circuit can be reduced by combining series and parallel resistors If not, continue with the application of Kirchhoff’s rules

112. 111 Problem-Solving Hints – Kirchhoff’s Rules, cont Analyze Draw the circuit diagram and assign labels and symbols to all known and unknown quantities. Assign directions to the currents The direction is arbitrary, but you must adhere to the assigned directions when applying Kirchhoff’s Rules Apply the junction rule to any junction in the circuit that provides new relationships among the various currents

113. 112 Problem-Solving Hints, cont Analyze, cont Apply the loop rule to as many loops as are needed to solve for the unknowns To apply the loop rule, you must choose a direction to travel around the loop and correctly identify the potential difference as you cross various elements Solve the equations simultaneously for the unknown quantities If a current turns out to be negative, the magnitude will be correct and the direction is opposite to that which you assigned Finalize Check your answers for consistency

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119. 118 21.9 RC Circuits A direct current circuit may contain capacitors and resistors, the current will vary with time When the circuit is completed, the capacitor starts to charge The capacitor continues to charge until it reaches its maximum charge (Q = Cε) Once the capacitor is fully charged, the current in the circuit is zero

120. 119 If you can't see the image above, please install Shockwave Flash Player.Shockwave Flash Player. If this active figure can’t auto-play, please click right button, then click play. NEXT Active Figure 21.25

121. 120 Charging an RC Circuit As the plates are being charged, the potential difference across the capacitor increases At the instant the switch is closed, the charge on the capacitor is zero Once the maximum charge is reached, the current in the circuit is zero The potential difference across the capacitor matches that supplied by the battery

122. 121 Charging Capacitor in an RC Circuit The charge on the capacitor varies with time q = C  (1 – e -t/RC ) = Q(1 – e -t/RC )  is the time constant  =RC The current can be found Fig 21.26

123. 122 Time Constant, Charging The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum  has units of time The energy stored in the charged capacitor is ½ Q  = ½ C  2

124. 123 Discharging Capacitor in an RC Circuit When a charged capacitor is placed in the circuit, it can be discharged q = Qe -t/RC The charge decreases exponentially Fig 21.26

125. 124 Discharging Discharging Capacitor At t =  = RC, the charge decreases to 0.368 Q max In other words, in one time constant, the capacitor loses 63.2% of its initial charge The current can be found Both charge and current decay exponentially at a rate characterized by  = RC

126. 125 Fig. 21-27, p. 711

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128. 127 Fig. 21-28, p. 711

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136. 135 21.10 Atmosphere as a Conductor Lightning and sparks are examples of currents existing in air Earlier examples of the air as an insulator were a simplification model Whenever a strong electric field exists in air, it is possible for the air to undergo electrical breakdown in which the effective resistivity of the air drops and the air becomes a conductor

137. 136 Creating a Spark (a) A molecule is ionized as a result of a random event Cosmic rays and other events produce the ionized molecules (b) The ion accelerates slowly and the electron accelerates rapidly due to the force from the electric field This is if there is a strong electric field In a weak field, they both accelerate slowly and eventually neutralize as they recombine Fig 21.29

138. 137 Creating a Spark, cont (c) The accelerated electron approaches another molecule at a high speed (d) If the field is strong enough, the electron may have enough energy to ionize the molecule during the collision Fig 21.29

139. 138 Creating a Spark, final (e) There are now two electrons to be accelerated by the field Each of these electrons can strike another molecule and repeat the process The result is a very rapid increase in the number of charge carriers available in the air and a corresponding decrease in the resistance of the air Fig 21.29

140. 139 Lightning Lightning occurs when a large current in the air neutralizes the charges that established the initial potential difference Typical currents during lightning can be very high Stepped leader current is in the range of 200 – 300 A Peak currents are about 5 x 10 4 A Power is in the billions of watts range

141. 140 Fair Weather Currents The average fair-weather current in the atmosphere is about 1000 A This is spread out over the entire globe The average fair-weather charge density is 2 x 10 -12 A / m 2 During the lightning stoke, J ~ 10 5 A/m 2 Fair-weather current is in the opposite direction from the lightning current