Current and Resistance

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 1

 Electric Current
Up to this point, our study of electricity has focused on
electrostatics, which deals with the properties of stationary
electric charges and fields.
Electric circuits were introduced in the discussion of
capacitors in Chapter 24, but it covered only situations
involving fully charged capacitors, where the charge is at
rest.
All electrical devices rely on some kind of current for their
operation.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 2

 Electric Current
Same
intensity

Dimmer Brighter Same

intensity

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 3

 Electric Current
We define the electric current, i, as the net charge
passing a given point in a given time.
Random motion of electrons in conductors, or the flow of
electrically neutral atoms, are not electric currents in spite
of the fact that large amounts of charge are moving past a
given point.
If net charge dq passes a point in time dt we define the
current i to be:

The amount of charge q passing a given point in time t is the

integral of the current with respect to time given by:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 4

 Electric Current
Total charge is conserved, which means that charge flowing
in a conductor is never lost.
The unit of current is coulombs per second, which has been
given the unit ampere (abbreviated A).
Some typical currents are:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 5

 Electric Current
A current that flows in only one direction, which does not
change with time, is called direct current.
Current that flows first in one direction and then in the
opposite direction is called alternating current.
The direction of the current flowing in a conductor is
indicated by an arrow.
Physically, the charge carriers in a conductor are electrons,
which are negatively charged.
However, by convention, positive current is defined as
flowing from the positive to the negative.
Current is defined as the flow of positive charge.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 6

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 7
 Iontophoresis
A painless method used to administer pain medication is
called iontophoresis, which uses (very weak) electrical
currents that are sent through the patients tissue.
The iontophoresis
device consists of a
battery and two
electrodes.
Dexamethasone is
applied to the
underside of the
negatively charged
electrode.
A current flows
through the patients skin and deposits the drug in the tissue.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 8
 Iontophoresis
PROBLEM:
A nurse wants to administer 80 g of dexamethasone to
the heel of an injured soccer player.
If she uses an iontophoresis device that applies a current
of 0.14 mA, how long does the administration of the
dose take?
Assume that the instrument has an application rate of
650 g/C and that the current flows at a constant rate.
SOLUTION:
If the drug application rate is 650 g/C, to apply 80 g
requires a total charge of:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 9

 Iontophoresis
The current flows at a constant rate so the charge is the
integral of the current:

Solving for the time and inserting the numbers, we find:

In 15 minutes the doctor can administer a strong, painless

dose of anti-inflammatory medication.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 10

 Current Density
Lets consider current flowing in a conductor.
Taking a plane through the conductor, the current per unit
area flowing through the conductor is the current density.

We take the direction of the current density as the direction

of the velocity of the charges crossing the plane.
If the cross sectional area is small, the magnitude of the
current density will be large.
If the cross section area is large, the magnitude of the
current density will be small.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 11
 Current Density
The current flowing through the surface is:

Here dA is the differential area element of the perpendicular

plane.
If the current is uniform and perpendicular to the surface,
then i = JA and we can write the current density as:

In a conductor that is not carrying current, the conduction

electrons move randomly.
When current flows through the conductor, electrons still
move randomly but also have an additional drift velocity,
vd , in the direction opposite to that of the electric field.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 12
 Current Density
The magnitude of the velocity of random motion is on the
order of 106 m/s, while the magnitude of the drift velocity is
on the order of 104 m/s or even less.
With such a slow drift velocity, you might wonder why a
light comes on almost immediately after you turn on a
switch.
The answer is that closing the switch establishes an electric
field almost immediately in the circuit (with a speed on the
order of 3108 m/s), causing the free electrons in the entire
circuit (including in the light bulb) to move almost instantly.
The current density is related to the drift velocity of the
moving electrons.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 13
 Electron Drift
Consider a conductor with cross sectional area A and
electric field E.
Suppose that there are n electrons per unit volume.
The negatively charged electrons will drift in a direction
opposite to the electric field.
We assume that all the electrons have the same drift velocity
vd and that the current density J is uniform.
In a time interval dt, each electron moves a distance vddt.
The volume that passes through area A is Avd dt.
The number of electrons is nAvd dt.
Each electron has charge e so that the charge dq that flows
through the area A in time dt is:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 14

 Drift Velocity
So the current is:

The current density is:

You can see that the drift velocity vector is antiparallel to

the current density vector, as stated before.

Note that the electric field, current density, and current are
to the right, while the drift velocity is to the left.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 15
 PROBLEM:
You are playing Galactic Destroyer on your video game
console.
Your game controller operates at 12 V and is connected to
the main box with an 18-gauge copper wire of length 1.5 m.
One mole of copper has a mass of 63.5 g and contains
6.021023 atoms.

As you fly your spaceship into battle, you hold the joystick
in the forward position for 5.3 s, sending a current of 0.78
mA to the console.
How far have the electrons in the wire moved during those
few seconds, while on the screen your spaceship crossed
half of a star system?
16
 Drift Velocity of Electrons in a Copper Wire
SOLUTION:
Think
To find out how far electrons in a wire move during a given
time interval, we need to calculate their drift velocity.
To determine the drift velocity for electrons in a copper
wire carrying a current, we need to find the density of
charge-carrying electrons in copper.
Then, we can apply the definition of the charge density to
calculate the drift velocity.
Sketch

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 17

 Drift Velocity of Electrons in a Copper Wire
Research
The distance x traveled by the electrons during time t is:

The drift velocity is related to the current density via:

The density of electrons is defined as:

One mole of copper has a mass of 63.5 g and contains

6.021023 atoms.
The density of electrons is:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 18

 Drift Velocity of Electrons in a Copper Wire
Simplify
We solve for the magnitude of the drift velocity:

The distance traveled is:

Calculate
Putting in the numerical values, we get:

x=
( 0.78 10-3 A (5.3 s ) ) = 3.69318 10-7 m
( )(
8.49 1028 m -3 1.602 10-19 C 0.823 mm 2 )( )

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 19

 Drift Velocity of Electrons in a Copper Wire
Round
We report our result with two significant figures:

Double-check
Our result for the the drift velocity is stunningly small.
Typical drift velocities are on the order of 104 m/s.
Since the current is proportional to the drift velocity, a
small current implies a small drift velocity.
An 18-gauge wire can carry a current of several amperes,
so the current specified in the problem statement is less
than 1% of the maximum current.
The fact that our calculated drift velocity is less than 1% of
104 m/s, a typical drift velocity for high currents, is
reasonable.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 20
 Resistivity and Resistance
Some materials conduct electricity better than others.
If we apply a given voltage across a conductor, we get a
large current.
If we apply the same voltage across an insulator, we get
very little current.
The property of a material that describes its ability to
conduct electric currents is called the resistivity, .
The property of a particular device or object that describes
its ability to conduct electric currents is called the
resistance, R.
Resistivity is a property of the material.
Resistance is a property of a particular object made from
that material.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 21
 Resistance
If we apply an electric potential difference V across a
conductor and measure the resulting current i in the
conductor, we define the resistance R of that conductor as:

The unit of resistance is volt per ampere.

In honor of George Simon Ohm (1789-1854), resistance has
been given the unit ohm, .

We can rearrange our equation to get Ohms Law:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 22

 Resistivity
We will assume that the resistance of the device is uniform
for all directions of the current; e.g., uniform metals.
The resistance R of a device depends on the material from
which the device is constructed as well as the geometry of
the device.
The conducting properties of a material are characterized in
terms of its resistivity.
We define the resistivity of a material by the ratio:

The units of resistivity are:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 23

 Typical Resistivities
The resistivities of some representative conductors at 20C
are listed in the table below (more in Table 25.1).

As you can see, typical values for the resistivity of metals

used in wires are on the order of 10-8 m.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 24

 Concept Check

If the diameter of a wire is doubled, its resistance will

A. increase by a factor of 4.
B. increase by a factor of 2.
C. stay the same.
D. decrease by a factor of 2.
E. decrease by a factor of 4.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 25

 Resistors
In many electronics applications one needs
a range of resistances in various parts of the
circuit.
For this purpose one can use commercially
available resistors.
Resistors are commonly made from carbon,
inside a plastic cover with two wires sticking
out at the two ends for electrical connection.
The value of the resistance is indicated by four color-bands
on the plastic capsule.
The first two bands are numbers for the mantissa, the third
is a power of ten, and the fourth is a tolerance for the range
of values.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 26
 Resistor Color Codes
Color Digit
Black 0
Brown 1
Red 2
Orange 3
Yellow 4
Green 5
Blue 6
Violet 7
Grey 8
White 9

Color Tolerance
Brown 1%
Red 2%
Gold 5%
Silver 10%
None 20%
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 27
 Resistor Color Codes

Band 1 Mantissa Digit 1 - Brown 1

Band 2 Mantissa Digit 2 - Green 5
Band 3 Power of 10 - Brown 1
Band 4 Tolerance - Gold
15101 = 150
5% tolerance

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 28

 Temperature Dependence of Resistivity
Demo
The resistivity and resistance vary with temperature.
For metals, this dependence on temperature is linear over a
broad range of temperatures.
An empirical relationship for the temperature dependence of
the resistivity of metals is given by:

where
is the resistivity at temperature T.
0 is the resistivity at temperature T0.
is the temperature coefficient of electric resistivity.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 29

 Temperature Dependence of Resistance
In everyday applications we are interested in the
temperature dependence of the resistance of various devices.
The resistance of a device depends on the length and the
cross sectional area.
These quantities depend on temperature.
However, the temperature dependence of linear expansion is
much smaller than the temperature dependence of resistivity
of a particular conductor.
So the temperature dependence of the resistance of a
conductor is, to a good approximation:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 30

 Temperature Dependence
Our equations for temperature dependence deal with relative
temperatures so that one can use C as well as K.
Values of for representative metals are shown below
(more can be found in Table 25.1).

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 31

 Concept Check
If the temperature of a copper wire ( = 3.910-3 K-1) with a
resistance of 100 is increased by 25 K, the resistance will
A. increase by approximately 10 .
B. increase by approximately 4 m.
C. decrease by approximately 4 m.
D. decrease by approximately 10 .
E. stay the same.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 32

 Other Temperature Dependence
Most materials have a resistivity that varies linearly with
the temperature under ordinary circumstances.
However, some materials do not follow this rule at low
temperatures.
At very low temperatures the resistivity of some materials
goes to exactly zero.
These materials are called superconductors.
There are many applications, including MRIs.
The resistance of some semiconducting materials actually
decreases with increasing temperature.
These materials are often found in high-resolution detection
devices for optical measurements or particle detectors.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 33

 Thermistor
A thermistor is a semiconductor whose resistance depends
strongly on temperature.
Thermistors are used to measure temperature.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 34

 Microscopic Basis of Conduction
Conduction of current results from the motion of electrons.
In a metal, the atoms of the metal form a crystal lattice.
The outermost electrons of each atom are essentially free.
When an electric field is applied, the electrons drift in the
direction opposite to that of the electric field.
Resistance to drift occurs when electrons interact with the
metal atoms in the lattice.
When the temperature of the metal is increased, the motion
of the atoms in the lattice increases.
This, in turn, increases the probability that electrons will
interact with the atoms, effectively increasing the resistance
of the metal.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 35
 Microscopic Basis of Conduction
The atoms of a semiconductor are in a crystal lattice also.
However, the outermost electrons of the atoms of the
semiconductor are not free to move.
To move about, the electrons must be given enough energy
to attain an energy state where they can move freely.
Thus, a typical semiconductor has a higher resistance than
a metal conductor because it has many fewer conduction
electrons.
In addition, when a semiconductor is heated, many more
electrons gain enough energy to move freely.
The resistance of the semiconductor decreases as its temperature
increases.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 36
 Electromotive Force and Ohms Law
To make current flow through a resistor one must
establish a potential difference across the resistor.
This potential difference is termed an electromotive force
or emf.
A device that maintains a potential difference is called an
emf device and does work on the charge carriers.
The emf device not only produces a potential difference but
supplies current.
The potential difference created by the emf device is termed
Vemf.
We will assume that emf devices have terminals that we can
connect and the emf device is assumed to maintain Vemf
between these terminals.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 37
 Circuit
Examples of emf devices are:
Batteries that produce emf through chemical reactions.
Electric generators that create emf from electromagnetic induction.
Solar cells that convert energy from the Sun to electric energy.
We will often use DC (direct current) power supplies, which
supply emf just like a battery.
A circuit is an arrangement of electrical components
connected together with ideal conducting wires (i.e., having
no resistance).
Electrical components can be sources of emf, capacitors,
resistors, or other electrical devices.
We will begin with simple circuits that consist of resistors
and sources of emf.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 38
 Batteries
We use batteries as devices that provide direct current
in circuits.
If you examine a battery, you will find its voltage.
This voltage is the potential difference that it can provide
to a circuit.
You will also find its rating in units of mAh.
This rating provides information on the total charge that a single
battery can deliver over its lifetime.
The quantity mAh is another unit of charge:

iPhone battery: 1150 mAh = 4140 C

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 39

 Ohms Law
Here a source of emf provides a potential
difference Vemf across a resistor with
resistance R.
The relationship between the potential
difference and the resistance is given by
Ohms Law:

The current in the circuit flows through the resistor, the

source of emf, and the wires.
The change in potential of the current in the circuit must
occur in the resistor.
The change is called the potential drop across the resistor.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 40

 Ohms Law
We can represent the previous circuit in a different way,
making it clearer where the potential drop happens and
showing which parts of the circuit are at which potential.
On the next slide, the top part of the figure shows the
previous circuit while the bottom part shows the same
circuit but with the vertical dimension representing the
value of the electric potential at different points around
the circuit.
The potential difference is supplied by the source of emf,
and the entire potential drop occurs across the single
resistor.
Ohms Law applies for the potential drop across the resistor.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 41
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 42
 Resistance of the Human Body
This short introduction of resistance and Ohms Law leads
to a point about electrical safety.
Currents above 100 mA can be deadly if they flow through
human heart muscle.
The resistance of the human body determines whether a
given potential differencesay, from a car batterycan be
dangerous.
The most relevant measure for the human bodys resistance,
Rbody, is the resistance along a path from the fingertips of
one hand to the fingertips of the other hand.
For most people this resistance is in the range
500 k < Rbody < 2M.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 43
 Resistance of the Human Body
Most of this resistance comes from the skin, particularly the
dead layers on the outside.
If the skin is wet, the resistance is drastically lowered and
the bodys resistance is drastically lowered.
Handling electrical devices in wet environments is a bad
idea.
If a sharp wire penetrates the skin, the resistance can also be
be drastically lowered.
In these situations, even low potential differences can
produce dangerous currents.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 44

 Resistors in Series
Circuits can contain more than one resistor and/or more
than one source of emf.
The analysis of circuits with multiple resistors requires
different techniques.
Resistors connected such that all the current in a
circuit must flow through each of the resistors are
connected in series.
For example, two resistors R1 and R2
in series with one source of emf with
potential difference Vemf are shown
in this circuit.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 45

 Resistors in Series
The potential drop across resistor R1 is V1.
The potential drop across resistor R2 is V2.
The potential drops must sum to the potential difference:

The current flowing through each resistor is the same:

The two resistors can be replaced with an equivalent

resistance equal to the sum of the two resistances.
We can generalize this for n resistors in series:

We can visualize two resistors in series.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 46
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 47
 Concept Check
What are the relative values of the two resistances in the
figure?
A. R1 < R2
B. R1 = R2
C. R1 > R2
D. Not enough
information
is given in the
figure to
compare the
resistances.
Copyright The McGraw-Hill Companies. Permission required for reproduction of display.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 48

 Internal Resistance of a Battery
When a battery is not connected in a circuit, the voltage
across its terminals is Vt.
When the battery is connected in series with a resistor with
resistance R, current i flows through the circuit.
When current is flowing, the potential difference, Vemf,
across the terminals of the battery is less than Vt.
This drop occurs because the
battery has an internal resistance
Ri that can be thought of as being
in series with the external resistor:

The battery terminals are points

A and B.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 49
 Internal Resistance of a Battery
PROBLEM:
Consider a battery that has Vt = 12.0 V when it is not
connected to a circuit.
When a 10.0 resistor is connected with the battery, the
potential difference across the batterys terminals drops to
10.9 V.
What is the internal resistance of the battery?
SOLUTION:
The current flowing through the external resistor is:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 50

 Internal Resistance of a Battery
The current flowing through the circuit must be the same as
the current flowing through the external resistor:

The internal resistance of the battery is 1.00 .

You cannot determine if a battery is still functional by
simply measuring the potential difference across the
terminals.
You must connect a resistance across the terminals and
then measure the potential difference.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 51
 Concept Check
Three identical resistors, R1, R2, and R3, are wired together:

An electric current is flowing through the three resistors

The current through R2
A. is the same as the current through R1 and R3.
B. is a third of the current through R1 and R3.
C. is twice the sum of the current through R1 and R3.
D. is three times the current through R1 and R3.
E. cannot be determined.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 52
 Resistors in Parallel
Instead of being connected in series so that all the current
must pass through both resistors, two resistors can be
connected in parallel, which divides the current between
them.
The potential drop across
each resistor is equal to the
potential difference provided
by the source of emf.
We can better visualize the potential drops, we can look at
the same circuit in 3D:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 53

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 54
 Resistors in Parallel
The voltage drop across each resistor is equal to the
potential difference provided by the source of emf.
Using Ohms Law we can write the current in each resistor:

The total current in the circuit must equal the sum of these
currents:

Which we can rewrite as:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 55

 Resistors in Parallel
We can then rewrite Ohms Law for the complete circuit as:

Here we have:

We can generalize this result for two parallel resistors to

n parallel resistors:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 56

 Concept Check
Three identical resistors,
R1, R2, and R3, are wired together.
An electric current is flowing
from point A to point B.
The current flowing through R2
A. is the same as the current through R1 and R3.
B. is a third of the current through R1 and R3.
C. is twice the sum of the current through R1 and R3.
D. is three times the current through R1 and R3.
E. cannot be determined.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 57

 Concept Check
Which combination of
resistors has the highest
equivalent resistance?
A. (a)
B. (b)
C. (c)
D. (d)
E. The equivalent
resistance is the same
for all four.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 58

 Equivalent Resistance
Copyright The McGraw-Hill Companies.
PROBLEM: Permission required for reproduction of display.

The figure shows a circuit with six

resistors.
What is the current flowing through
resistors R2 and R3 in terms of Vemf
and R1 through R6?
SOLUTION:
We begin by identifying parts of the circuit that are clearly
wired in parallel or series.
The current flowing through R2 is the current flowing from
the source of emf.
We can see that R3 and R4 are in series so:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 59

 Equivalent Resistance
Copyright The McGraw-Hill Companies.
Permission required for reproduction of display.

We make the substitution and draw a new

circuit diagram.
Now we see that R34 and R1 are in parallel:

We make this substitution and draw a new

circuit diagram.
In this circuit, we see that R2, R5, R6, and
R134 are in series:

Copyright The McGraw-Hill Companies.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 Permission required for reproduction of display. 60
 Equivalent Resistance
We make the substitution and draw a new
circuit diagram:
Now we substitute in R3 and R134:

Copyright The McGraw-Hill Companies.

Permission required for reproduction of display.

So the current flowing through R2 is: Copyright The McGraw-Hill Companies.

Permission required for reproduction of display.

To get the current flowing through R3, we

note that current i2 is flowing through R134
that contains R3.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 61
 Equivalent Resistance
Copyright The McGraw-Hill Companies.

So we can write:
Permission required for reproduction of display.

The resistor R1 and the equivalent resistor

R34 are in parallel.
Therefore the potential drop V34 across R34
is the same as the potential drop V134 across
R134.
The resistors R3 and R4 are in series
and i3, the current flowing through R3,
is the same as i34, the current flowing
through R34, so we can write:

Copyright The McGraw-Hill Companies.

Permission required for reproduction of display.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 62
 Equivalent Resistance
So we can write i3 in terms of Vemf and R1 through R6

Or

Yikes!
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 63
 Concept Check
As more identical resistors R are added to the parallel circuit
shown here, the total resistance between points A and B

A) increases.
B) remains the same.
C) decreases.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 64

 Concept Check
Three light bulbs are connected in series with a battery that
delivers a constant potential difference, Vemf.
When a wire is connected across light bulb 2 as shown in
the figure, light bulbs 1 and 3

A. burn just as brightly as they did

before the wire was connected.
B. burn more brightly than they did
before the wire was connected.
C. burn less brightly than they did
before the wire was connected.
D. go out.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 65

 Energy and Power in Electric Circuits
Consider a circuit in which a source of emf with potential
difference V causes a current i to flow in a circuit.
The work required to move a differential amount of charge
dq around the circuit is equal to the differential electric
potential energy dU given by:

The definition of current is i = dq/dt so we can rewrite the

differential electric potential energy as:

Using the definition of power P = dU /dt we get:

So the power is the product of the current and the potential

difference.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 66
 Energy and Power
Using Ohms Law we can write equivalent formulations of
the power:

The unit of power is the watt (W).

Electrical devices are rated by the amount of power they
consume in watts.
An electricity bill is based on how many kilowatt-hours of
electrical energy you consume:
kW h = power times time.
1 kW h = 1000 W X 3600 s = 3.6106 joules.
The energy is converted to heat, motion, light, etc.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 67

 Temperature Dependence of a Light Bulbs Resistance

A 100 W light bulb is connected to a source of emf with

Vemf = 100 V.
When the light bulb is operating, the temperature of its
tungsten filament is 2520 C.
PROBLEM:
What is the resistance of the light bulb at room temperature
(20 C)?

SOLUTION:
The power when lighted is:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 68

 Temperature Dependence of a Light Bulbs Resistance
We can rearrange our equation for power to get the
resistance of the filament:

The temperature dependence of the filaments resistance is:

We solve for the resistance at room temperature:

Taking the temperature coefficient from Table 25.1 we get:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 69

 Concept Check
A current of 2.00 A is maintained in a circuit with a total
resistance of 5.00 . How much heat is generated in 4.00 s?
A. 55.2 J
B. 80.0 J
C. 116 J
D. 168 J
E. 244 J

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 70

 Total Energy in a Flashlight Battery
A standard flashlight battery can deliver about 2.0 W h of
energy before it runs down.
If a battery costs $0.80, what is the cost of operating a
100 W lamp for 8.0 hours using standard batteries compared
with power from the grid?
Energy provided by one battery: 2.0 W h.
Energy needed: (100 W)(8 h) = 800 W h.
Number of batteries needed: (800 W h)/(2.0 W h) = 400.
400 batteries cost (400)($0.80) = $320.
On the grid, power costs $0.10 per kW h.
Cost on grid = (800 W h)($0.10)/(1000 W h) = $0.08.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 71

 Diodes
Many resistors obey Ohms Law.
A diode is a device that does not obey Ohms Law.
A diode is designed to conduct current in one direction and
not in the other direction.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 72

 Diodes
Diodes are useful for converting alternating current to
direct current.
One particular kind of diode is the light-emitting diode
(LED), which not only regulates current in a circuit but
emits light of a single wavelength.
LEDs can save up to 90% of the power used for lighting.
LEDs can be used in large display screens such as the one at
Cowboy Stadium in Arlington, Texas:
10,584,064 LEDs.
4 LEDs for each pixel.
2,176 x 4,864 pixel display.
Largest HD screen in the world.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 73
 High-Voltage Direct Current Power Transmission
Often power plants are located in remote areas.
Power must be transmitted over long distances.
The power transmitted to users is the product of the current
and the potential difference, P = iV.
The current required for a given power is i = P/V:
Higher V means lower current in the power line.
The power lost during transmission is:

Reducing the losses means reducing the current.

You might argue that we could also write:

The V in this equation is potential drop across the power

line, not the potential difference at which the power is sent.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 74
 High-Voltage Direct Current Power Transmission

The expression for the dissipated power is:

This means that the losses decrease with the square of the
potential difference used to transmit the power.
Normally power transmission is done using alternating
current, but alternating currents have the inherent
disadvantage of high power losses.
High-voltage direct current (HVDC) transmission lines do
not have this problem.
The largest HVDC power line carries power from the Itaipu
Dam 800 km to Sao Paulo, Brazil:
6300 MW power at V = 1200 kV .

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 75

 Size of Wire for a Power Line
PROBLEM:
Imagine you are designing the HVDC power line from the
Itaipu to the city of Sao Paulo in Brazil.
The power line is 800 km long and transmits 6300 MW of
power at a potential difference of 1.20 MV.
The electric company requires that no more than 25% of the
power be lost in transmission.
If the line consists of one wire made out of copper and
having a circular cross section, what is the minimum
diameter of the wire?

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 Size of Wire for a Power Line
SOLUTION:
Think
Knowing the power transmitted and the potential difference
with which it is transmitted, we can calculate the current
carried in the line.
We can then express the power lost in terms of the
resistance of the transmission line.
With the current and the resistance of the wire, we can write
an expression for the power lost during transmission.
The resistance of the wire is a function of the diameter of
the wire, the length of the wire, and the resistivity of copper.
We can then solve for the diameter of the wire that will
keep the power loss within the specified limit.
October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 77
 Size of Wire for a Power Line
Sketch

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 78

 Size of Wire for a Power Line
Research
The power carried in the line is given by:
P = iDV
The power lost in transmission is given by:

The resistance of the wire is given by:

The cross-sectional area of the wire is:

So the resistance of the wire is:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 79

 Size of Wire for a Power Line
Simplify
We solve for the current in the wire:

The power lost is then:

The fraction of lost power relative to the total power is:

Solving for the diameter of the wire gives us:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 80

 Size of Wire for a Power Line
Calculate
Putting in the numerical values, we get:

Round

Double-check
To double-check our result, lets calculate the resistance of
this transmission line:
Using our calculated value for the diameter, we can find the
cross-sectional area and the resistance:

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 81

 Size of Wire for a Power Line
The current transmitted is:

The power lost is:

This is close to 25% of the total power of 6300 MW.

We are confident in our result.

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 82

 Assignments
25.28
25.31
25.40
25.43
25.48
25.57

October 17, 2017 Physics for Scientists & Engineers 2, Chapter 25 83