Chapter 17
Current and Resistance
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
Q
I 
t
The SI unit of current is Ampere (A)

1 A = 1 C/s
Active Figure
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 Quiz
Consider 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 crosssectional 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


The drift speed, vd, is the speed at
which the carriers move



ΔQ = (n A Δx) q
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
Active Figure
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
vd  I nqA ,is inversely proportional to the cross-sectional
area.
Circuits
A circuit is a complete conducting
path.
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:
R  
l
A
 
l
r
2
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.
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
R 
V
I
Active Figure
Resistance, cont

Units of resistance are ohms (Ω)


1Ω=1V/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 nonohmic device
Resistivity

The resistance of an ohmic conductor is
proportional to its length, L, and
inversely proportional to its crosssectional area, A
L
R  
A


ρ is the constant of proportionality and is
called the resistivity of the material
See table 17.1
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
  o [1  (T  To )]


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
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
R  Ro [1  (T  To )]
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
1) dA = 4 dB
Answer
2) dA = 2 dB
3) dA = 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.
4) dA = 1/2 dB
5) dA = 1/4 dB
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.
Quick Quiz
1) it decreases by a factor 4
2) it decreases by a factor 2
A wire of resistance R is
3) it stays the same
stretched uniformly
4) it increases by a factor 2
(keeping its volume
5) it increases by a factor 4
constant) until it is twice its
original length. What
happens to the resistance?
Answer
1) it decreases by a factor 4
2) it decreases by a factor 2
3) it stays the same
A wire of resistance R is
4) it increases by a factor 2
stretched uniformly (keeping
5) it increases by a factor 4
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.
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


1986



High temperature superconductivity
discovered by Bednorz and Müller
Superconductivity near 30 K
1987


Superconductivity discovered by H.
Kamerlingh Onnes
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
QV

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


Q
t
V  I V
From Ohm’s Law, alternate forms
of power are
2
 I R 
V
R
2
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
Active Figure
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)
Answer
1) the power
When you rotate the
3) the voltage
knob of a light
2) the current
4) both (1) and (2)
5) both (2) and (3)
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.
Quick Quiz
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?
1) the 25 W bulb
2) the 100 W bulb
3) both have the same
4) this has nothing to do
with resistance
Answer
1) the 25 W bulb
2) the 100 W bulb
3) both have the same
Two lightbulbs operate at 120 V,
but one has a power rating of 25 W
4) this has nothing to do
with resistance
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.
Quick Quiz
1) heater 1
Two space heaters
2) heater 2
in your living room
3) both equally
are operated at 120
V. Heater 1 has
twice the resistance
of heater 2. Which
one will give off
more heat?
Answer
1) heater 1
2) heater 2
Two space heaters in your
3) both equally
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.
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.
Answer
(a). The resistance of the shorter wire is half that of the
longer wire. The power dissipated, P    V  R , (and hence
the rate of heating) will be greater for the shorter wire.
Consideration of the expression P  I2 R 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.
2
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).
Ia  Ib  Ic  Id  Ie  If
. Charges constituting the current
leave the positive terminal of the battery and then
split to flow through the two bulbs; thus, Ia  Ic  Ie .
Because the potential difference is the same across the
two bulbs and because the power delivered to a device
is P  I  V  , the 60–W bulb with the higher power rating
must carry the greater current, meaning that Ic  Ie .
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 Ic  Id and I  I . The
two currents leaving the bulbs recombine to form the
current back into the battery, If  Id  Ib .
Ia
e
f
Active Figure
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.