Transcript Chapter 17

Chapter 17
Current and Resistance
Electric Current
• Let us look at the charges flowing perpendicularly to a
surface of area A
• The electric current is the rate at which the charge
flows through this surface
Q
I 
t
• The SI unit of current is Ampere (A): 1 A = 1 C/s
André-Marie Ampère
1775 – 1836
Electric Current
• The conventional direction of the current is the
direction positive charge would flow
• In a common conductor (e.g., 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
Current and Drift Speed
• Charged particles move through a conductor of crosssectional area A and a charge carrier density n
• The total number of charge carriers: n A Δx
• 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
ΔQ = (n A vd Δt) q
I = ΔQ / Δt = n q vd A
Current and Drift Speed
• If the conductor is isolated, the electrons undergo
random motion (due to collisions with the atoms)
• When an electric field is set up in the conductor, it
creates an electric force on the electrons and hence a
current
• The zigzag line represents the motion of charge
carrier in a conductor
Current and Drift Speed
• 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
• Therefore, 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
Meters in a Circuit – Ammeter, Voltmeter
• 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
• A voltmeter is used to measure voltage (potential
difference) – connects to the two ends of the bulb
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 – it arises due to collisions between the
electrons carrying the current with the fixed atoms
inside the conductor
V
R
I
• SI unit of resistance is ohm (Ω): 1 Ω = 1 V / A
Georg Simon Ohm
1787 – 1854
Ohm’s Law
• For certain 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
• Materials that obey Ohm’s Law are said to be ohmic
(the relationship between current and voltage is linear)
Non-ohmic
Resistivity
• The resistance of an ohmic conductor is
proportional to its length, L, and inversely
proportional to its cross-sectional area, A
L
R=r
A
• ρ is the constant of proportionality and is called the
resistivity of the material (See table 17.1)
Chapter 17
Problem 12
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
= 0.500 Ω, and if all of the copper is to be used, what
will be (a) the length and (b) the diameter of the wire?
Temperature Variation of Resistivity
• For most metals, resistivity increases with increasing
temperature – the atoms vibrate with increasing
amplitude so the electrons find it more difficult to pass
through the atoms
• For most metals, resistivity increases approximately
linearly with temperature over a limited temperature
range
r = r o [1   ( T  To )]
• ρ0 – resistivity at some reference temperature T0
(usually taken to be 20° C); α – is the temperature
coefficient of resistivity
Temperature Variation of Resistance
• Since the resistance of a conductor with uniform
cross sectional area is proportional to the resistivity,
the effect of temperature on resistance is similar
R = R o [1   ( T  To )]
Chapter 17
Problem 52
A given copper wire has a resistance of 5.00 Ω at 20.0°C
while a tungsten wire of the same diameter has a
resistance of 4.75 Ω at 20.0°C. At what temperature will the
two wires have the same resistance?
Superconductors
• Superconductors – a class of materials
whose resistances fall virtually to zero
below a certain temperature, TC (critical
temperature)
• The value of TC is sensitive to chemical
composition, pressure, and crystalline
structure
• Once a current is set up in a
superconductor, it persists without any
applied voltage (since R = 0)
• One application is superconducting
magnets
Electrical Energy and Power
• In a circuit, as a charge moves through the battery, the
electrical potential energy of the system is increased
by ΔQ ΔV (the chemical potential energy of the battery
decreases by the same amount)
• The charge moving through a resistor loses this
potential energy during collisions with atoms in the
resistor (the temperature of the resistor increases)
• 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
Electrical Energy and Power
• The rate at which the energy is lost is the power
Q
=
V = I V
t
• From Ohm’s Law, alternate forms of power are
V
= I R =
R
2
2
• The SI unit of power is Watt (W) (I must be in Amperes,
R in ohms and ΔV in Volts)
• The unit of energy used by electric companies is the
kilowatt-hour (defined in terms of the unit of power and
the amount of time it is supplied): 1 kWh = 3.60 x 106 J
Chapter 17
Problem 62
In a certain stereo system, each speaker has a resistance
of 4.00 Ω. The system is rated at 60.0 W in each channel.
Each speaker circuit includes a fuse rated at a maximum
current of 4.00 A. Is this system adequately protected
against overload?
Chapter 17
Problem 40
A certain toaster has a heating element made of Nichrome resistance
wire. When the toaster is first connected to a 120-V source of potential
difference (and the wire is at a temperature of 20.0°C), the initial
current is 1.80 A but the current begins to decrease as the resistive
element warms up. When the toaster reaches its final operating
temperature, the current has dropped to 1.53 A. (a) Find the power the
toaster converts when it is at its operating temperature. (b) What is
the final temperature of the heating element?
Answers to Even Numbered Problems
Chapter 17:
Problem 2
(a) 5.57 × 10−5 m / s
(b) the drift speed is smaller
Answers to Even Numbered Problems
Chapter 17:
Problem 4
3.4 × 1021 electrons
Answers to Even Numbered Problems
Chapter 17:
Problem 10
8.89 Ω
Answers to Even Numbered Problems
Chapter 17:
Problem 18
(a) 2.8 × 108 A
(b) 1.8 × 107 A
Answers to Even Numbered Problems
Chapter 17:
Problem 22
6.3 Ω
Answers to Even Numbered Problems
Chapter 17:
Problem 34
(a) $0.29
(b) $2.6