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Natural Sciences I
lecture 11: Electricity & Magnetism
ELECTRONS as charged particles
In the late 1800s, cathode rays were understood to be electric currents.
They were known to impart a charge to objects in their path, but because they
had not (yet) been observed to be deflected by magnetic fields, their identity
as moving charges was not recognized. J.J. Thomson changed that with
come classic experiments, which showed unequivocally that cathode rays are
particles in motion. Thomson's device looked something like this:
evacuated glass tube
MAGNET
N
ANODE
wires to
high voltage
supply
( )
( )
m
ed bea
deflect
electron beam
(+)
(-)
CATHODE
(electron source)
S
fluorescing screen
Thomson observed deflection of cathode rays (i.e., the electron beam)
by the electric field, which established that electrons are particles. He was
also able to determine the charge-to-mass ratio (e/m) of these particles from
the amount of deflection caused by the electric field. Thomson proceeded
with attempts to measure the charge separately, but was not completely
successful. Knowledge of the charge (and therefore the mass) of the
electron would await R.A. Millikan's famous "oil drop" experiments in 1906.
voltage supply
(variable)
(+) (+) (+) (+)
oil drop with small
electric charge
(-)
(-) (-) (-) (-)
electric field force
gravity
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ELECTRONS as constituents of atoms
We saw on page 1 that electrons can be separated from atoms. Indeed,
physicists took advantage of this fact in order to measure the properties of
electrons. Before we go further in our discussion of electric forces, electricity
and magnetism, we should also recognize that electrons are key constituents
of atoms. We will examine atomic structure in detail in a couple of weeks, but
a quick summary is in order here.
proton
mass: 1.673E-27 kg
charge: (+) 1.602E-19 coulombs
neutron
mass: 1.6753E-27 kg
charge: none
electron
mass: 9.109E-31 kg
charge: (-) 1.602E-19 coulombs
The above cartoon of the atom has some major shortcomings.
Neutrons and protons are roughly 1840 times as massive as electrons, but
the nucleus is nowhere near as big as depicted (in proportion to the overall
atom). The nucleus is actually thousands of times smaller than the atom, so
there's lots of "empty" space in the volume occupied by the atom. It is also
inaccurate to think of electrons orbiting the nucleus in circular paths. [We'll
discuss these issues in more detail later in the course.]
Under most circumstances, atoms have the same number of protons
and electrons, so the overall charge on the atom is neutral. However,
electrons can be removed by a variety of processes – some of them
"chemical" in nature (as in the case of interactions with atoms of other
elements). Electrons can also be "stripped away" from atoms by friction.
For ease in discussing atomic charge and charge-transfer processes,
we arbitrarily assign a charge of -1 to the electron and +1 to the proton
(note the actual units in coulombs above; see also the discussion below
about the fundamental unit of charge – the coulomb).
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Consequently, an atom possessing one fewer electrons than protons has an
atomic (ionic) charge of +1. Similarly, an atom that has one excess electron
relative to the number of protons has an atomic (ionic) charge of -1:
-
+
+ +
-
3 protons
3 electrons
neutral atom
+1
-
-
-1
+
+ +
-
3 protons
2 electrons
atom charge = +1
-
+
+ +
-
3 protons
4 electrons
atom charge = -1
Your text describes in some detail the processes by which objects
acquire static charge:
1. Electrons can be stripped off by friction between two objects, leaving
one object with excess electrons (negative charge), the other with a deficit
of electrons (positive charge). The objects must be insulators themselves
or insulated from their surroundings (see below) – otherwise the charge
would be dissipated by flow of electrons through them.
2. Objects can acquire a static charge by contact with other charged
objects (electrons transferred from one object to the other).
3. A static charge may be induced on a surface by the close proximity of a
charged object:
comb with negative static charge
(excess electrons)
( )( ) ( )( ) ( )( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
sheet of paper
(thickness exaggerated)
these electrons have been repelled
to the side of the paper away from the comb
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A brief historical aside...
An important discovery about the nature of electrical charges was
made by French scientist Charles Dufay in the early 1700s. Dufay
rubbed rods made of amber or glass using a silk cloth or a piece of fur.
Results:
Two charged glass rods repel one another
Two charged amber rods repel one another
Charged rods of amber and glass attract one another
On the basis of these experiments, Dufay concluded that there
are two types of electric charge; he called one "resinous electricity"
(produced by rubbing amber), the other "vitreous electricity" (produced
by rubbing glass). Although fundamentally correct, this conclusion led
to confusion: some scientists in Europe who followed Dufay decided
that there exist "resinous fluids" and "vitreous fluids" that carry these
different charges. What Dufay had in fact discovered is what we now
recognize as positive and negative charges. Implicit in his discovery is
that like charges repel each other and opposite charges attract.
CONDUCTORS and INSULATORS
These are probably familiar to you in a general way.
Conductors allow charged particles (usually electrons) to move through
them readily. Two factors are important in determining whether a given
material is a good electrical conductor:
(next page)
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conductors (cont'd)
1. availability of mobile electrons
(metals have lots)
2. "regularity" of the atomic
structure
In insulators, the availability of mobile electrons is limited – the
electrons are tightly bound to the atoms. Glasses are particularly good
insulators: not only are the electrons tightly bound in typical oxide glasses,
the atomic structure is also "chaotic" (in contrast to the structure shown
above). Electrons can still be added or removed from the surfaces of
insulators, so they can readily acquire a static charge.
Semiconductors, as their name suggests, fall in between conductors
and insulators. Their properties can be changed by "doping" with extra
electrons that may be very mobile, or by creating electron "holes" that
promote mobility. Silicon and germanium are the semiconductors
commonly used in technological applications.
COULOMB'S LAW: the force exerted by charges
More than a century before J.J. Thomson showed that electrons are
particles, French physicist Charles Coulomb had worked out the
relationship between the magnitude of two charges and the forces between
them. He used a torsion balance...
q = magnitude of charge
long fiber
F
( )
fixed rod
q2
( )
( )
d
q1
) (
) (
) (
insulating rod
F
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Coulomb knew the force required to twist the fiber by a given
amount, so he could use the simple device shown on page 5 to measure
the force pulling opposite charges together and pushing like charges
apart.
He learned two things about electric force, Fe:
when the charges on the two spheres are maintained at fixed
values,
Fe
1/d
2
(Another inverse square law!)
and when the distance, R, between the centers of the two spheres
is maintained at a fixed value,
Fe
q1 q2
Putting these two results together, Coulomb obtained the law that bears
his name...
Fe =
kcq1 q2
d
2
Coulomb's Law has the
same form as the law of
universal gravitation
The constant of proportionality, kc, is analogous to G in the equation for
universal gravitation.
9
2
kc = 9 10 N m / C
2
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One coulomb (C) of charge is the charge on 6.24 x 10 electrons
The coulomb is the SI unit of charge; it is a fundamental property.
We can turn this around and observe that the charge on the electron,
e = 1.6022 x 10
-19
coulombs
q=ne
number of electrons
charge on the electron
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FORCE FIELDS
On the previous page, the similarity between the equations governing
gravitational forces and electrical forces was noted. This is satisfying in a
way, because both gravity and electric charges are capable of exerting forces
at a distance – it's tempting to consider them as having something in
common.
To pre-19th century physicists trying to come to terms with the behavior
of natural systems, the notion of forces acting at a distance was difficult to
rationalize (as it still is for today for students of science). The equations had
been worked out by Newton (in the case of gravity) and by Coulomb (in the
case of electric charge), but there existed no intuitively satisfying model to
explain the phenomena of "non-contact" forces.
In the early 19th century, Michael Faraday and James Maxwell
developed a rather different way of looking at forces associated with charged
objects (actually, it was Faraday who provided most of the fundamental
observations; Maxwell formalized the mathematics). The conceptual
breakthrough was that an electric charge came to be viewed as affecting the
space around it in a particular way. Just as a large mass carries with it a
gravitational field, an electric charge is surrounded by an electric field (and
a magnet is surrounded by a magnetic field). All three are characterized as
force fields. The concept of a force field may seem like a subtle change in
thinking, but it freed scientists from the idea that gravitational and electric
forces are like invisible ropes connecting objects together.
Of the three types listed above, the magnetic force field is the easiest to
"see"...
iron filings
bar magnet
N
S
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Evaluating Electric Fields
Electric fields are more complex to represent than are gravitational
fields, in part because there exist both positive and negative charges.
Electric fields are represented as circles or spheres with lines of force
radiating toward or away from them:
Remember that
forces are vector
quantities. By
convention, the
lines of force are
directed away
from the charge
if it is positive,
and toward the
charge if it is
negative
-q
+q
The manner in which a charge is evaluated is by bringing a test
charge near it (the test charge is positive by convention). Consider the
electric force Fe acting on a test charge q that is brought near another
charge. The electric field at the location of the test charge is defined as
E
Fe / q
Since Fe is a vector, E is also a vector – it has direction as well as
magnitude.
We can rearrange this equation to answer the question What is the
electric force on a charged body (of charge q) interacting with the electric
field due to other charged bodies?:
Fe
qE
This equation can be combined with Coulomb's law to allow calculation of
the electric field E at a point distance d from charge q:
E =
kc q
d
2
= 9
10
9
q
d
2
9
Electric fields (cont'd)
Let's look at the interaction between two charges, first considering
opposite charges of equal magnitude...
At any point in the "neighborhood"
of the two charges, the total force
field is represented by the vector
sum of contributions from the two
charges (this diagram has bilateral
symmetry because the two charges
are equal, but this does not have to
be the case)
-q2
+q1
vector sum
vector sum
q2
q1
contri
bution
contribution
q1 contribution
q2
contribution
Now consider two charges that are equal in magnitude and charge...
Because the force-field vectors
are pointing outward from both
charges, their sums in the
region around (and especially
between) the two charges have
a very different orientation.
vector sum
q1
contribution
q2 contribution
+q1
+q2
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Electric fields
In the examples on page 9, the electric field E at any point in the vicinity
of the two charges q1 and q2 can be calculated from the relation:
E = n1
kc q1
d1
2
+
n2
kc q2
d2
2
You don't need to remember
this equation if the diagrams on
page 9 make sense to you
where d1 and d2 are the distances to charges q1 and q2, respectively, and n1
and n2 are vectors indicating the direction to the two charges. This is the
vector sum shown schematically for each case on page 9.
Electric Potential
Two or three weeks ago, we talked about the creation of gravitational
potential energy by raising an object in a gravitational field (PE = mgh).
Given the mathematical similarities between gravitational and electric forces,
it probably won't come as a surprise to you that we also speak in terms of
electric potential energy.
A charge (or charged particle) is surrounded by an electric field that is
conceptually similar to the Earth's gravitational field. Just as moving an object
in the Earth's gravity field changes the potential energy of that object, so too
does moving a charge in the electric field of another charge. Bringing
charged particle into the electric field of another particle of the same charge
involves the performance of work: likecharged particles repel each other, so
work must be done against the repulsive
force to bring the particles together.
Squeezing two tennis balls together is
not a bad analogy.
Electric potential energy is usually measured in terms of a potential
difference (PD), which is defined as the work required to create the potential
divided by the charge moved:
PD = W/q
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Consider two spheres, one anchored
by an insulating rod, the other
moveable. Both are positively
charged; the moveable one at 1
coulomb (C).
At the point where 1 joule of work
has been done in bringing the 1-C
sphere closer to the stationary one,
we have created a potential
difference of 1 volt.
1 volt = 1 joule / 1 C
(+)
1.0 C
(+)
1 joule of work done
to bring the moveable
sphere into position
In the set-up at the bottom of page 10, we created a potential difference of
1 volt by moving the charged spheres together. The electric potential energy
we created is now capable of performing 1 joule of work. In other words,
if we let go of the moveable
sphere (allowing the repulsive
force to push it away) that force
could be used to do 1 joule of
work – just as the energy stored
in the compressed tennis balls
could also do work.
ELECTRIC CURRENT
Up to this point we have been considering charges that are static in
nature. As we saw last week, these static charges can be produced by
physical processes that involve motion, but apart from the experiments in which
we deliberately moved charges together to assess their character, the charges
themselves did not move.
The sustained flow or movement of charge is referred to as electric current,
and it is given the symbol I.
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Electric circuits
If you acquire a static charge on your clothing by rubbing on a carpet or
chair seat, you may become aware of the fact when your hand approaches a
conductor such as a metal doorknob. Suddenly a pathway is available for
electrons to move, so the static charge is quickly dissipated by flow of
electrons – often with an uncomfortable spark. The resulting shock you feel is
very brief, however, because the electric potential is eliminated by flow of
electrons. In order for current to flow continuously, you need two things: a
circuit and a sustained voltage source, such as a battery or generator.
In many respects, electric current is like the flow of water:
PUMP
Water will keep flowing
in this system as long
as the pump is on. The
pump maintains the
potential energy that
sustains flow.
Most car batteries have a
potential of 12 volts. This
means that each coulomb of
charge that moves through
the circuit can do 12 joules of
work.
Voltage describes the
potential difference between
any two points in the circuit.
It has aspects in common
conducting wire
with water pressure or water
voltage source:
voltage drop "force", and so is sometimes
(maintains potential;
(work is done here) called electromotive force
does work)
(emf).
A typical electric circuit
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The flow rate, I, of the electric current is the amount of charge that
passes through a given point in the circuit in a given amount of time:
electric current =
I =
quantity of charge
time
q
t
The units of charge are therefore coulombs per second; a coulomb per second
is called an ampere (amp for short).
1 amp (A) =
1 coulomb (C)
1 second (s)
The Nature of Electric Current
Because of the manner in which our understanding of electricity
progressed, we are left today with two different conventions for describing the
flow of current in a circuit:
1. Conventional Current refers to the flow of positive charges from the
positive to the negative terminal (pre-dates discovery of the electron –
but still used in circuit diagrams!)
2. Electron Current refers to flow of negative charges.
conventional current
)+(
( )
(+)
)+(
electron current
)+(
(+)
( )
(+)
( )
( )
(+)
( )
( )
( )
(+)
( )
The electron current is actually the "correct" description of flow in the
circuit, because it refers to the drift of electrons from the negative to the
positive terminal in a battery. However, the two "models" are mathematically
equivalent, and there's no difference between them when it comes to
calculating current in circuits.
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Physical details
Our everyday experience is that electric current appears to travel very fast
in wires and other conductors. After an electrical outage, for example, when the
power is turned back on, the lights come on over an entire city instantaneously.
What we actually observe in such a case is not incredibly fast flow of electrons,
but the near-instantaneoous establishment of the electric field.
SECTION OF METAL CONDUCTOR
no electric field
electron
ion
electric field present
Unattached electrons move about
randomly as in a gas. Motion is
chaotic and collisions are frequent
– no net movement of electrons in
any direction
Motion of electrons retains its
random, chaotic aspect, but there
is also some net motion (flow) due
to the presence of the field – the
electrons make some progress...
It is the net motion of electrons that constitutes the electric current; the drift
velocity is ~ 0.01 cm/s. The electric field that causes the net motion
propagates much, much faster – close to the speed of light.
Properties of electric current – Summary
a potential difference establishes an electric field throughout a circuit;
this occurs very rapidly (near light-speed)
net motion of electrons results from the presence of the field – this is
the electric current
the net electron motion is relatively slow (~0.01 cm/s on average)
(collisions occur among electrons and between electrons and positivelycharged atom nuclei – this is the cause of resistance to electron flow)
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RESISTANCE
Now that you are familiar with the physical nature of electric current,
we are almost ready to discuss the properties of circuits and do some
example calculations. Let's first look at resistance a little more closely...
When a current flows in a conductor,
random collisions of mobile electrons with
positive ions result in loss of energy from
the electrons. Kinetic energy of the electrons is partially transferred to the ions,
whose increased vibrational amplitude
causes the temperature of the conductor
to rise.
This can be a sort of "runaway" effect,
because the resistance of a conductor
increases with increasing temperature:
the "thermal agitation" of the positive ions
increases the likelihood of collisions with
electrons.
As noted last week, electrical resistance varies greatly from one material to
another, due to differences in the electronic structure of the atoms making up
the material and their physical arrangement. For a given potential difference,
the resistance of a circuit affects the current passing through it, as we'll see
momentarily...
Let's return briefly to the analogy between current in an electric circuit
and water flow in a plumbing "circuit". If the potential difference between two
points in an electric circuit is analogous to a water pressure difference, it
follows that
I
PD
I
1/R
It is reasonable, also, that
where R is the resistance of the circuit. The "water-flow" analogy is valid
here, too (see next page)...
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A constriction in a pipe
restricts the flow of water
A high-resistance circuit reduces
the current (at a given PD)
I
I
(-)
(+) (-)
low R
A large diameter pipe can carry
more water than a small one
(at a fixed pressure difference)
(+)
high R
A large diameter wire can carry
more current than a small one
of the same material (for a given PD)
I
(-)
(+)
I
(-)
(+)
Combining the two proportionalities from p. 15, we obtain
I = PD / R
R = PD / I
or
which many of you will recognize as Ohm's law. It is usually written
V = IR
The unit of resistance is the ohm:
1 ohm (W) =
1 volt (V)
1 amp (A)
This relationship indicates that the resistance of a conductor is 1 ohm if a
potential difference of 1 volt will maintain a current of 1 amp through that
conductor.
Summarizing (and extending) conclusions from the previous page... the
resistance depends on four factors: material identity; length; crosssectional area; and temperature
length
area
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The resistances of materials vary over many orders of magnitude. Most
metals are reasonably good conductors; most oxides and organic materials
are reasonably good insulators (i.e., they have high resistances).
The best conductors (rank
order)
incandescent
light filaments
conductors
insulators
silver
rubber
copper
glass
gold
ceramics
aluminum
diamond
graphite
plastics
tungsten
wood
iron
most rocks
Nichrome
dry air
nickel/chromium alloy used in
toaster windings and the like
the length factor is a
direct multiplier – i.e.,
twice the length means
twice the resistance (all
else being equal)
The resistance of all conductors
decreases at low temperatures;
in the case of superconductors, it
goes to zero.
DIRECT CURRENT (DC) vs. ALTERNATING CURRENT (AC)
Up to this point we have spoken only about direct current (DC), which
involves the actual flow of charges (electrons) through conductors, always in
the same direction.
In the case of alternating current (AC), the potential difference between
the two terminals of the voltage source switches back and forth between
positive (+) and negative (-) –-- hence the term "alternating" current. In the
U.S., the frequency of the alternation is 60 changes from (+) to (-) and 60
changes from (-) to (+) per second, for 60 complete cycles (60 Hz) each
second (in the U.K., the frequency is 50 Hz; which is why some electrical
devices made for the American market require an adaptor).
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1/60 sec
+120
potential
difference
(volts)
-120
time
Alternating current was adopted for long-distance power transmission,
regional grids, and household current because it is more efficient than direct
current (resistance of the wires does not take as big a toll). The switching of
the potential in AC means that there is no net electron drift. The electric field
moves back and forth through the circuit at near-light speed.
An interesting historical note...
Thomas Edison is one of the huge names in electricity. We probably owe
more to him than to any other single individual for the range and breadth
of his electrical inventions. The one big battle he lost, however, was that
he was a proponent of DC power. He set up a small DC grid in northern
New Jersey that was never a financial success. George Westinghouse
carried the day in his advocacy of AC power, mainly because the voltage
can easily be stepped up and down for different applications.
We'll talk more about AC circuits next lecture, in connection with transformers, induction and related topics...
19
ELECTRICAL POWER and ELECTRICAL WORK
(equations and example applications)
As summarized on page 3, electrical circuits consist of three types of
components:
a voltage source
conducting wires
an electrical device (can be more than one)
In a DC circuit, the electric field moves from the voltage source through
one length of wire to the electrical device (e.g., a light bulb) and back along
another length of wire to the other terminal of the voltage source,
maintaining the potential difference across the device:
conducting wire
voltage source
electrical device
In an AC circuit, one wire (the black or "hot" one) supplies the
alternating electric field to the device. A second wire from the device is
"grounded" – that is, connected to a metal rod hammered into the ground.
As noted on page 3, it is the voltage source that does work in an electrical
circuit. This voltage source could be a battery (DC) or an electric power
generator (AC). The total work done by the voltage source is...
work done in the device by the electric field
+
energy lost to resistance in the wires
(this is kept small)
Resistance is analogous to friction in a mechanical device – we'll ignore it as
we often do in the case of friction.
20
We already know (page 10) that
potential difference = work / charge
PD = W/q
so we can write
W = (PD) q
Note the units:
joules =
joules
coulomb
coulomb
As in mechanical processes, a joule is a unit of work, However, as we
learned yesterday, an "electrical" joule is related to moving a charge to a
higher potential rather than to a force acting through a distance (these two
"kinds" of joules are really the same thing, of course, as you could discover
with the appropriate electromechanical device.
Remember from a few weeks ago that power is work per unit time
P = W/t
so electrical power must be
P =
power = J/s = watts
(PD) q
q
=
PD
t
t
current (C/s) = I
voltage (V = J/C)
We end up with...
power = current x potential difference = amps x volts
P = IV
If W = P x t, then
W = IVt
units:
C
s
J
C
s = J
21
Power is what the utility company bills you for: Power is work per
unit time, so what the utility company wants to know is the amount of
power you used times the amount of time you used it:
power x time = kilowatt x hour = kWhr
But remember that
P = IV
so kWhr can be calculated by multiplying
amps x volts x time = watts x hr
If you were to use 1000 watts (1 kilowatt) of power for 1 hour, you would
be billed for 1 kilowatt-hour (kWhr). As a point of calibration, a typical
hair dryer is 1000-1500 watts (let's say 1200 on average). Note that if
the hair dryer is operating at 120 volts, this means it draws a current of
10 amps (P = I V).
EXAMPLE CALCULATIONS
Following on the hair dryer example above, let's do another hair
dryer calculation in which we keep track of all the units. If you use a
1,100-watt hair dryer designed to operate at 120 V, how much current
does this hair dryer draw?
given
P = 1,100 W
V = 120 V
I = ?A
solution
P=IV
joule
P = second
joule
V = coulomb
I = P/V
joule
1,100 second
=
joule
120
coulomb
= 9.2 C/s = 9.2 A
22
Another example...
What is the cost of operating a 75-watt light bulb for 24 hours if the cost
of electricity is $0.12 per kWhr?
given
P = 75 watts
t = 24 hr
rate = $0.12 / kWhr
cost = ?$
solution
kWhr = P t
= (watts) (time)
= (75 W) (24 hr)
= 1,800 Whr
= 1.8 kWhr
1.8 kWhr x
$0.12
kWhr
= $0.22