1. Current and Resistance
Chapter 17
Current and Resistance

2. Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte.
This is a simple electric cell.

3. A battery transforms chemical energy into electrical energy.
Chemical reactions within the cell create a potential difference between the terminals by slowly dissolving them. This potential difference can be maintained even if a current is kept flowing, until one or the other terminal is completely dissolved.

4. Several cells connected together make a battery, although now we refer to a single cell as a battery as well.

5. Electric Current
The current is the rate at which the charge flows through this surface
Look at the charges flowing perpendicularly to a surface of area A
The SI unit of current is Ampere (A)
1 A = 1 C/s

6. Active Figure

7. Electric Current, cont
The direction of the current is the direction positive charge would flow
This is known as conventional current direction
In a common conductor, such as copper, the current is due to the motion of the negatively charged electrons
It is common to refer to a moving charge as a mobile charge carrier
A charge carrier can be positive or negative

8. By convention, current is defined as flowing from + to -
By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.

9. Quick Quiz
Consider positive and negative charges moving horizontally through the four regions in Figure Rank the magnitudes of the currents in these four regions from lowest to highest. (Ia is the current in Figure 17.2a, Ib the current in Figure 17.2b, etc.) (a) Id , Ia , Ic , Ib (b) Ia , Ic , Ib , Id (c) Ic , Ia , Id , Ib (d) Id , Ib , Ic , Ia (e) Ia , Ib , Ic , Id (f) none of these

10. Answer
(d). Negative charges moving in one direction are equivalent to positive charges moving in the opposite direction. Thus, are equivalent to the movement of 5, 3, 4, and 2 charges respectively, giving .

11. 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

12. Current and Drift Speed, cont
The total charge is the number of carriers times the charge per carrier, q
ΔQ = (n A Δx) q
The drift speed, vd, is the speed at which the carriers move
vd = Δx/ Δt
Rewritten: ΔQ = (n A vd Δt) q
Finally, current, I = ΔQ/Δt = nqvdA

13. Current and Drift Speed, final
If the conductor is isolated, the electrons undergo random motion
When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current

14. Active Figure

15. Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity.

16. 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

17. Electrons in a Circuit
The drift speed is much smaller than the average speed between collisions
When a circuit is completed, the electric field travels with a speed close to the speed of light
Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s

18. Quick Quiz
Suppose a current-carrying wire has a cross-sectional area that gradually becomes smaller along the wire, so that the wire has the shape of a very long cone. How does the drift speed vary along the wire? (a) It slows down as the cross section becomes smaller. (b) It speeds up as the cross section becomes smaller. (c) It doesn’t change. (d) More information is needed.

19. Answer (b). Under steady-state conditions, the current is the same
in all parts of the wire. Thus, the drift velocity, given by
,is inversely proportional to the cross-sectional area.

20. Circuits
A circuit is a complete conducting path.

21. Meters in a Circuit – Ammeter
An ammeter is used to measure current
In line with the bulb, all the charge passing through the bulb also must pass through the meter

22. Meters in a Circuit – Voltmeter
A voltmeter is used to measure voltage (potential difference)
Connects to the two ends of the bulb

23. Quick Quiz
Look at the four “circuits” shown in Figure 17.6 and select those that will light the bulb.

24. Answer
(c), (d). Neither circuit (a) nor circuit (b) applies a difference in potential across the bulb. Circuit (a) has both lead wires connected to the same battery terminal. Circuit (b) has a low resistance path (a “short”) between the two battery terminals as well as between the bulb terminals.

25. Quick Quiz
Suppose an electrical wire is replaced with one having every linear dimension doubled (i.e. the length and radius have twice their original values). Does the wire now have (a) more resistance than before, (b) less resistance, or (c) the same resistance?

26. Answer (b). Consider the expression for resistance:
Doubling all linear dimensions increases the numerator of this expression by a factor of 2, but increases the denominator by a factor of 4. Thus, the net result is that the resistance will be reduced to one-half of its original value.

27. Example 1
A certain conductor has 7.50 × 1028 free electrons per cubic meter, a cross-sectional area of 4.00 × 10−6 m2, and carries a current of 2.50 A. Find the drift speed of the electrons in the conductor.

28. Example 2
In a particular television picture tube, the measured beam current is 60.0 μA. How many electrons strike the screen every second?

29. Example 3
An aluminum wire with a cross-sectional area of 4.0 × 10−6 m2 carries a current of 5.0 A. Find the drift speed of the electrons in the wire. The density of aluminum is 2.7 g/cm3. (Assume that one electron is supplied by each atom.)

30. Practice 1
A 200-km-long high-voltage transmission line 2.0 cm in diameter carries a steady current of A. If the conductor is copper with a free charge density of 8.5 × 1028 electrons per cubic meter, how many years does it take one electron to travel the full length of the cable?

31. 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

32. Active Figure

33. Resistance, cont 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

34. Georg Simon Ohm 1787 – 1854 Formulated the concept of resistance
Discovered the proportionality between current and voltages

35. Ohm’s Law
Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents
This statement has become known as Ohm’s Law
ΔV = I R
Ohm’s Law is an empirical relationship that is valid only for certain materials
Materials that obey Ohm’s Law are said to be ohmic

36. Ohm’s Law, cont 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

37. Ohm’s Law, final
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

38. Resistivity
The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A
ρ is the constant of proportionality and is called the resistivity of the material
See table 17.1

39. Temperature Variation of Resistivity
For most metals, resistivity increases with increasing temperature
With a higher temperature, the metal’s constituent atoms vibrate with increasing amplitude
The electrons find it more difficult to pass through the atoms

40. Temperature Variation of Resistivity, cont
For most metals, resistivity increases approximately linearly with temperature over a limited temperature range
ρ is the resistivity at some temperature T
ρo is the resistivity at some reference temperature To
To is usually taken to be 20° C
 is the temperature coefficient of resistivity

41. 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

42. Quick Quiz
Two wires, A and B, are made of the same metal and have equal length, but the resistance of wire A is four times the resistance of wire B. How do their diameters compare?
1) dA = 4 dB
2) dA = 2 dB
3) dA = dB
4) dA = 1/2 dB
5) dA = 1/4 dB

43. Answer
1) dA = 4 dB
2) dA = 2 dB
3) dA = dB
4) dA = 1/2 dB
5) dA = 1/4 dB
Two wires, A and B, are made of the same metal and have equal length, but the resistance of wire A is four times the resistance of wire B. How do their diameters compare?
The resistance of wire A is greater because its area is less than wire B. Since area is related to radius (or diameter) squared, the diameter of A must be two times less than B.

44. Quick Quiz
1) it decreases by a factor 4
2) it decreases by a factor 2
3) it stays the same
4) it increases by a factor 2
5) it increases by a factor 4
A wire of resistance R is stretched uniformly (keeping its volume constant) until it is twice its original length. What happens to the resistance?

45. Answer
1) it decreases by a factor 4
2) it decreases by a factor 2
3) it stays the same
4) it increases by a factor 2
5) it increases by a factor 4
A wire of resistance R is stretched uniformly (keeping its volume constant) until it is twice its original length. What happens to the resistance?
Keeping the volume (= area x length) constant means that if the length is doubled, the area is halved.
Since R=L/A , this increases the resistance by four.

46. Example 4
A lightbulb has a resistance of 240 Ω when operating at a voltage of 120 V. What is the current in the bulb?

47. Example 5
Eighteen-gauge wire has a diameter of mm. Calculate the resistance of 15 m of 18-gauge copper wire at 20°C.

48. Example 6
A rectangular block of copper has sides of length 10 cm, 20 cm, and 40 cm. If the block is connected to a 6.0-V source across two of its opposite faces, what are (a) the maximum current and (b) the minimum current that the block can carry?

49. Practice 2
Suppose that you wish to fabricate a uniform wire out of 1.00 g of copper. If the wire is to have a resistance R = Ω, and if all of the copper is to be used, what will be (a) the length and (b) the diameter of the wire?

50. Example 7
A wire 3.00 m long and mm2 in cross-sectional area has a resistance of 41.0 Ω at 20°C. If its resistance increases to 41.4 Ω at 29.0°C, what is the temperature coefficient of resistivity?

51. Example 8
A 100-cm-long copper wire of radius 0.50 cm has a potential difference across it sufficient to produce a current of 3.0 A at 20°C. (a) What is the potential difference? (b) If the temperature of the wire is increased to 200°C, what potential difference is now required to produce a current of 3.0 A?

52. Practice 3
The copper wire used in a house has a cross-sectional area of 3.00 mm2. If 10.0 m of this wire is used to wire a circuit in the house at 20.0°C, find the resistance of the wire at temperatures of (a) 30.0°C and (b) 10.0°C.

53. Superconductors
A class of materials and compounds whose resistances fall to virtually 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

54. 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

55. Superconductor Timeline
1911
Superconductivity discovered by H. Kamerlingh Onnes
1986
High temperature superconductivity discovered by Bednorz and Müller
Superconductivity near 30 K
1987
Superconductivity at 96 K and 105 K
Current
More materials and more applications

56. Superconductor, final
Good conductors do not necessarily exhibit superconductivity
One application is superconducting magnets

57. Electrical Energy and Power
In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQΔV
The chemical potential energy of the battery decreases by the same amount
As the charge moves through a resistor, it loses this potential energy during collisions with atoms in the resistor
The temperature of the resistor will increase

58. Energy Transfer in the Circuit
Consider the circuit shown
Imagine a quantity of positive charge, DQ, moving around the circuit from point A back to point A

59. Energy Transfer in the Circuit, cont
Point A is the reference point
It is grounded and its potential is taken to be zero
As the charge moves through the battery from A to B, the potential energy of the system increases by DQDV
The chemical energy of the battery decreases by the same amount

60. Energy Transfer in the Circuit, final
As the charge moves through the resistor, from C to D, it loses energy in collisions with the atoms of the resistor
The energy is transferred to internal energy
When the charge returns to A, the net result is that some chemical energy of the battery has been delivered to the resistor and caused its temperature to rise

61. Electrical Energy and Power, cont
The rate at which the energy is lost is the power
From Ohm’s Law, alternate forms of power are

62. Electrical Energy and Power, final
The SI unit of power is Watt (W)
I must be in Amperes, R in ohms and DV in Volts
The unit of energy used by electric companies is the kilowatt-hour
This is defined in terms of the unit of power and the amount of time it is supplied
1 kWh = 3.60 x 106 J

63. Active Figure

64. Quick Quiz
When you rotate the knob of a light dimmer, what is being changed in the electric circuit?
1) the power
2) the current
3) the voltage
4) both (1) and (2)
5) both (2) and (3)

65. Answer
1) the power
2) the current
3) the voltage
4) both (1) and (2)
5) both (2) and (3)
When you rotate the knob of a light dimmer, what is being changed in the electric circuit?
The voltage is provided at 120 V from the outside. The light dimmer increases the resistance and therefore decreases the current that flows through the lightbulb.

66. Quick Quiz
1) the 25 W bulb
2) the 100 W bulb
3) both have the same
4) this has nothing to do with resistance
Two lightbulbs operate at 120 V, but one has a power rating of 25 W while the other has a power rating of 100 W. Which one has the greater resistance?

67. Answer
1) the 25 W bulb
2) the 100 W bulb
3) both have the same
4) this has nothing to do with resistance
Two lightbulbs operate at 120 V, but one has a power rating of 25 W while the other has a power rating of 100 W. Which one has the greater resistance?
Since P = V2 / R the bulb with the lower power rating has to have the higher resistance.

68. Quick Quiz
1) heater 1
2) heater 2
3) both equally
Two space heaters in your living room are operated at 120 V. Heater 1 has twice the resistance of heater 2. Which one will give off more heat?

69. Answer
1) heater 1
2) heater 2
3) both equally
Two space heaters in your living room are operated at 120 V. Heater 1 has twice the resistance of heater 2. Which one will give off more heat?
Using P = V2 / R, the heater with the smaller resistance will have the larger power output. Thus, heater 2 will give off more heat.

70. Quick Quiz
A voltage ΔV is applied across the ends of a nichrome heater wire having a cross-sectional area A and length L. The same voltage is applied across the ends of a second heater wire having a cross-sectional area A and length 2L. Which wire gets hotter? (a) the shorter wire, (b) the longer wire, or (c) more information is needed.

71. Answer
(a). The resistance of the shorter wire is half that of the longer wire. The power dissipated, , (and hence the rate of heating) will be greater for the shorter wire. Consideration of the expression might initially lead one to think that the reverse would be true. However, one must realize that the currents will not be the same in the two wires.

72. Quick Quiz
For the two resistors shown in Figure 17.12, rank the currents at points a through f from largest to smallest.
(a) Ia = Ib > Ie = If > Ic = Id
(b) Ia = Ib > Ic = Id > Ie = If
(c) Ie = If > Ic = Id > Ia = Ib

73. Answer (b). . Charges constituting the current
leave the positive terminal of the battery and then split to flow through the two bulbs; thus, Because the potential difference is the same across the two bulbs and because the power delivered to a device is , the 60–W bulb with the higher power rating must carry the greater current, meaning that Because charge does not accumulate in the bulbs, all the charge flowing into a bulb from the left has to flow out on the right; consequently and The two currents leaving the bulbs recombine to form the current back into the battery,

74. Active Figure

75. Example 9
A high-voltage transmission line with a resistance of 0.31 Ω/km carries a current of A. The line is at a potential of 700 kV at the power station and carries the current to a city located 160 km from the station. (a) What is the power loss due to resistance in the line? (b) What fraction of the transmitted power does this loss represent?

76. Example 10
What is the required resistance of an immersion heater that will increase the temperature of 1.50 kg of water from 10.0°C to 50.0°C in 10.0 min while operating at 120 V?

77. Example 11
How much does it cost to watch a complete 21-hour-long World Series on a 180-W television set? Assume that electricity costs $0.070/kWh.

78. Example 12
A house is heated by a 24.0-kW electric furnace that uses resistance heating. The rate for electrical energy is $0.080/kWh. If the heating bill for January is $200, how long must the furnace have been running on an average January day?

79. Practice 4
It has been estimated that there are 270 million plug-in electric clocks in the United States, approximately one clock for each person. The clocks convert energy at the average rate of 2.50 W. To supply this energy, how many metric tons of coal are burned per hour in coal-fired electric generating plants that are, on average, 25.0% efficient? The heat of combustion for coal is 33.0 MJ/kg.