1. Current and Direct Current Circuits
Chapter 21
Current
and
Direct Current Circuits

2. 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

3. Average Electric Current
Assume charges are moving perpendicular to a surface of area A
If DQ is the amount of charge that passes through A in time Dt, the average current is

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

5. 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

6. 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

7. 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, vd, is the speed at which the carriers move
vd = Δ x/ Δt
Rewritten: ΔQ = (n A vd Δt) q
Finally, current, I = ΔQ/Δt = nqvdA

8. 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

9. 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

10. 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

11. 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

12. Current Density J is the current density of a conductor
It is defined as the current per unit area
J = I / A = n q vd
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 / m2
The current density is in the direction of the positive charge carriers

13. 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 = s E
The constant of proportionality, s, is called the conductivity of the conductor

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. Resistivity Resistance is related to the geometry of the device:
r is called the resistivity of the material
The inverse of the resistivity is the conductivity:
s = 1 / r and R = l / sA
Resistivity has SI units of ohm-meters (W . m)

21. Some Resistivity Values

22. 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

23. 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

24. Resistor Values
Values of resistors are commonly marked by colored bands

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

26. 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

27. 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

28. 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

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

30. Superconductors, cont The value of TC 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

31. 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

32. Electrical Conduction – A Model
The free electrons in a conductor move with average speeds on the order of 106 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

33. 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

34. 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

35. 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

36. Conduction Model, 5
Let t 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 vd A = (n e2 E / me) t A

37. 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: t = l avg/vavg

38. 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

39. Electrical Power Assume a circuit as shown
As a charge moves from a to b, the electric potential energy of the system increases by QDV
The chemical energy in the battery must decrease by this same amount

40. 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

41. 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

42. 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

43. 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 W, V is in V, and P is in W

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

45. 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

46. 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

47. 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, DV = e - Ir

48. 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

49. 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

50. Power The total power output of the battery is P = IDV =Ie
This power is delivered to the external resistor (I2 R) and to the internal resistor (I2r)
P = Ie = I2 R + I2 r
The current depends on the internal and external resistances

51. 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

52. Resistors in Series, cont
Potentials add
ΔV = IR1 + IR2
= I (R1+R2)
Consequence of Conservation of Energy
The equivalent resistance has the same effect on the circuit as the original combination of resistors

53. Equivalent Resistance – Series
Req = R1 + R2 + R3 + …
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

54. Equivalent Resistance – Series – An Example
Two resistors are replaced with their equivalent resistance

55. 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 = I1 + I2
The currents are generally not the same
Consequence of Conservation of Charge

56. Equivalent Resistance – Parallel
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

57. 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

58. 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

59. 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

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

61. 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

62. 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

63. Mathematical Statement of Kirchhoff’s Rules
Junction Rule:
SIin = SIout
Loop Rule:

64. More About the Junction Rule
I1 - I2 - I3 = 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

65. 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

66. 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

67. 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

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

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

76. Time Constant, Charging
The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum
t has units of time
The energy stored in the charged capacitor is ½ Qe = ½ Ce2

77. 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

78. Discharging Capacitor
At t =  = RC, the charge decreases to Qmax
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 t = RC

79. 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

80. 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

81. 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

82. 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

83. 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 104 A
Power is in the billions of watts range

84. 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 A / m2
During the lightning stoke, J ~ 105 A/m2
Fair-weather current is in the opposite direction from the lightning current