chapter 17 current and resistance. volta discovered that electricity could be created if dissimilar...
TRANSCRIPT
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.
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.
Several cells connected together make a battery, although now we refer to a single cell as a battery as well.
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
QI
t
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
By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.
Quick QuizConsider positive and negative charges moving horizontally through the four regions in Figure 17.2. 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
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 .
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
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
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
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.
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
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
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.
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.
dv I nqA
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
Meters in a Circuit – Voltmeter
A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb
Quick Quiz
Look at the four “circuits” shown in Figure 17.6 and select those that will light the bulb.
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.
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?
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.
2RA r
l l
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.
Example 2
In a particular television picture tube, the measured beam current is 60.0 μA. How many electrons strike the screen every second?
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.)
Practice 1
A 200-km-long high-voltage transmission line 2.0 cm in diameter carries a steady current of 1 000 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?
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
VR
I
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
Georg Simon Ohm 1787 – 1854 Formulated the
concept of resistance
Discovered the proportionality between current and voltages
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
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
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
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
LR
A
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
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
)]TT(1[ oo
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
)]TT(1[RR oo
Two wires, Two wires, AA and and BB, are made of , are made of
the same metal and have equal the same metal and have equal
length, but the resistance of wire length, but the resistance of wire AA
is four times the resistance of wire is four times the resistance of wire
BB. How do their diameters . How do their diameters
compare?compare?
1) ddAA = 4 = 4 ddBB
2) ddAA = 2 = 2 ddBB
3) ddAA = = ddBB
4) 4) ddAA = 1/2 = 1/2 ddBB
5) 5) ddAA = 1/4 = 1/4 ddBB
Quick Quiz
The resistance of wire A is greater because its area is lessarea is less than
wire B. Since areaarea is related to radiusradius (or diameter) squaredsquared,
the diameter of diameter of AA must be two times less than must be two times less than BB.
Two wires, Two wires, AA and and BB, are made of the , are made of the
same metalsame metal and have and have equal lengthequal length, but , but
the resistance of wire the resistance of wire AA is is four timesfour times
the resistance of wire the resistance of wire BB. How do their . How do their
diameters compare?diameters compare?
1) ddAA = 4 = 4 ddBB
2) ddAA = 2 = 2 ddBB
3) ddAA = = ddBB
4) 4) ddAA = 1/2 = 1/2 ddBB
5) 5) ddAA = 1/4 = 1/4 ddBB
Answer
A wire of resistance A wire of resistance RR is is
stretched uniformly (keeping stretched uniformly (keeping
its volume constant) until it is its volume constant) until it is
twice its original length. twice its original length.
What happens to the What happens to the
resistance?resistance?
1) it decreasesit decreases by a factor 4by a factor 4
2) it decreasesit decreases by a factor 2by a factor 2
3) it stays the sameit stays the same
4) it increases4) it increases by a factor 2by a factor 2
5) it increases5) it increases by a factor 4by a factor 4
Quick Quiz
Keeping the volume (= area x length) constant means that if the
length is doubleddoubled, the area is halvedhalved.
Since R=L/A , this increases the resistance by fourfour.
A wire of resistance A wire of resistance RR is is
stretched uniformly (keeping stretched uniformly (keeping
its volume constant) until it is its volume constant) until it is
twice its original length. What twice its original length. What
happens to the resistance?happens to the resistance?
1) it decreasesit decreases by a factor 4by a factor 4
2) it decreasesit decreases by a factor 2by a factor 2
3) it stays the sameit stays the same
4) it increases4) it increases by a factor 2by a factor 2
5) it increases5) it increases by a factor 4by a factor 4
Answer
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?
Example 5
Eighteen-gauge wire has a diameter of 1.024 mm. Calculate the resistance of 15 m of 18-gauge copper wire at 20°C.
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?
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 = 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?
Example 7
A wire 3.00 m long and 0.450 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?
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?
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.
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
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
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
Superconductor, final Good conductors
do not necessarily exhibit superconductivity
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 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
Energy Transfer in the Circuit Consider the
circuit shown Imagine a
quantity of positive charge, Q, moving around the circuit from point A back to point A
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 QV The chemical energy of the battery
decreases by the same amount
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
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
QV I V
t
22 V
I RR
Electrical Energy and Power, final The SI unit of power is Watt (W)
I must be in Amperes, R in ohms and V 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
When you rotate the knob of a When you rotate the knob of a
light dimmer, what is being light dimmer, what is being
changed in the electric circuit?changed in the electric circuit?
1) the power
2) the current
3) the voltage
4) both (1) and (2)
5) both (2) and (3)
Quick Quiz
The voltage is provided at 120 V from the
outside. The light dimmer increases the increases the
resistanceresistance and therefore decreases the decreases the
currentcurrent that flows through the lightbulb.
When you rotate the knob of a When you rotate the knob of a
light dimmer, what is being light dimmer, what is being
changed in the electric circuit?changed in the electric circuit?
1) the power
2) the current
3) the voltage
4) both (1) and (2)
5) both (2) and (3)
Answer
Two lightbulbs operate at 120 V, Two lightbulbs operate at 120 V,
but one has a power rating of but one has a power rating of 25 W25 W
while the other has a power rating while the other has a power rating
of of 100 W100 W. Which one has the . Which one has the
greater resistance? greater resistance?
1) the 25 W bulb
2) the 100 W bulb
3) both have the same
4) this has nothing to do with resistance
Quick Quiz
Since P = VP = V22 / R / R the bulb with the lower lower
power ratingpower rating has to have the higher higher
resistanceresistance.
Two lightbulbs operate at 120 V, but Two lightbulbs operate at 120 V, but
one has a power rating of one has a power rating of 25 W25 W while while
the other has a power rating of the other has a power rating of 100 W100 W. .
Which one has the greater resistance? Which one has the greater resistance?
1) the 25 W bulb
2) the 100 W bulb
3) both have the same
4) this has nothing to do with resistance
Answer
Two space heaters in your Two space heaters in your
living room are operated at living room are operated at
120 V. Heater 1 has 120 V. Heater 1 has twicetwice
the resistance of heater 2. the resistance of heater 2.
Which one will give off more Which one will give off more
heat?heat?
1) heater 1
2) heater 2
3) both equally
Quick Quiz
Using P = VP = V22 / R, / R, the heater with the smaller resistancesmaller resistance
will have the larger powerlarger power output. Thus, heater 2 will
give off more heat.
Two space heaters in your Two space heaters in your
living room are operated at 120 living room are operated at 120
V. Heater 1 has V. Heater 1 has twicetwice the the
resistance of heater 2. Which resistance of heater 2. Which
one will give off more heat?one will give off more heat?
1) heater 1
2) heater 2
3) both equally
Answer
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.
Quick Quiz
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.
2V R P
2I RP
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
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, .
a b c d e fI I I I I I
aI
a c eI I I
I V P
c eI I
c dI I e fI I
f d bI I I
Example 9
A high-voltage transmission line with a resistance of 0.31 Ω/km carries a current of 1 000 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?
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?
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.
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?
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.