Chapter 21 Lecture Notes

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Nov 16, 2013 (3 years and 6 months ago)

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Lecture 1

Chapt. 21 Electric Charge and Electric Field

i.

This semester
we study electromagnetism

as an example of a “field.” In physics, a field is what
allows objects separated by some distance in space to exert forces on each other.

a.

There
were

four fields in physics: gravity, electromagnetism, the “strong” nuclear field,
and the “weak” nuclear field
.

b.

The latter three have over the past thirty years been shown to be just different aspects
of a single field, called the electroweak field. This uni
fication is considered to be one of
the most important advancements in twentieth century science.

c.

Currently researchers are trying to figure out if the electroweak field can be unified with
gravity into one single field that would explain the origin of all

forces in the universe.
There’s no guarantee that there is one single field that will explain all forces, and a lot of
progressive still needs to be made. But the discovery of the Higgs particle last summer
was a major advancement in this direction
.


ii.

Elec
tricity

and magnetism have
---
of course
---
been known about for centuries, but has only been
properly understood over the past two hundred or so years
.

a.

Electricity and magnetism were long thought to be unrelated
.

b.

They were shown to be two different aspects o
f the same field
---
the electromagnetic
field
---
by J. C. Ma
xwell around the time of the Am
erican civil war, even though a variety
of people had begun to realize that they were some
how related in the decades leading
up to Maxwell’s discovery
.

c.

Maxwell’s disco
very was the first example of unifying more than one field into a single
field; a magnetic field is essentially a changing electric field and
vice versa
. This is spelled
out in Maxwell’s equations, which are
partial
differential equations
.

21
-
1 Static Elec
tricity; Electric Charge and its Conservation

i.

There e
xists a property that objects may have called charge which enables them to push or pull
on each other
.

a.

This property is in addition to mass, which
only
enables objects to pull on each other
.

b.

Not al
l obje
cts have charge.


ii.

Charge comes in two kinds: positive and negative
.

a.

Opposite charges attract; like charges repel
.

b.

We will find out later that
positive charge probably should have been called negative
and
vice versa
. This is because the person who named the
m
---
Benjamin Franklin
---
got
them backwards
.

iii.

Charge is conserved; you can’t create or destroy it
.

a.

If it you start with something that
is
uncharged or “neutral” and end up with it being
charged, this isn’t because you created charge. It was made up of equal
amounts of
positive and negative charge that “cancelled” each other out. All you did was separate
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the positive from the negative charge
; you just moved the two types of charge around
to different places.

iv.

Presumably there is exactly the same amount of posit
ive charge in the universe as negative, but
this is something we don’t know for sure yet
.

v.

Charge is only a model, an abstraction about the way the universe around us works. No one has
ever seen a charge. We infer the existence of charge indirectly from the

results of experiments;
it works so well conceptually to explain experiments that we believe it exists. But there’s

always
the possibility that someday someone will make a discovery that is inconsistent with the idea of
charge and we have to come up with
a whole new theory to explain things. This is the way
things work in science.

You can’t prove theories right;

you can only prove them wrong. If a
theory works well for a long period of time under through many different experiments, then be
just end up believing in it.

21
-
2 Electric Charge in the Atom

i.

The origin of charge is a
t

the atomic level

ii.

Atoms are composed
of nuclei

made from positive protons and neutral neutrons; electrons
circle around the nucleus much like planets orbit the sun
---
sort of


a.

The way electrons circle around the nucleus in an atom is much more complicated than
the
way planets orbit the sun, an
d is explained by quantum mechanics
.

b.

Protons and neutrons are actually composed of even smaller charged particles called
quarks
.

c.

Electrons are currently believed to be “truly” elementary in nature; they are though
t

not

to be composed of even smaller partic
les
.

d.

Protons have the same charge as electrons; a neutral atom has the same no. of protons
and electrons; an atom with differing nos. of protons and electrons is called an ion
.

e.

Molecules are aggregates of atoms bound together by electrical attraction
.

i.

Like

atoms they can be neutral, or be ions
.

ii.

Frequently neutral molecules are polar; this means that some parts of the
molecule have an excess of positive charge, while other parts have an excess o
f
negative charge
---
the positive and negative charges aren’t eve
n
ly

distributed
.

1.

Water and ammonia are probably the most familiar examples of polar
molecules
.

2.

Water in liquid form is necessary for life
as we know it
. Many
biochemists believe this is partly due to the water molecule’s polar
nature. Since ammonia is
pola
r, biochemists have speculated about

the
possibility of life based on liquid ammonia on other planets where
ammonia might be present.

iii.

Molecular ions can be polar, too

21
-
3 Insulators and Conductors

i.

Electricity is the motion of charge
.

3


ii.

Conductors are materi
als which conduct charge relatively easily; insulators are the opposite
.

iii.

When we think about conductors and insulators, we usually

think about them in solid form, but
they don’t have to be in solid form.

iv.

Metals tend to be good conductors
.

a.

Silver is the best, copper the second best. If silver had a conduction

value


of 100, then
copper would have a conduction value of 93
.

v.

Most non
-
metals are good insulators
.

vi.

There are a few materials which are “in
-
between” conductors and insulators; they ar
e called
semiconductors
.

a.

Examples are silicon and germanium
.

b.

Semiconductors are very important in electronics; they’re used to make transistors
.

vii.

Metals conduct because the outermost electrons
in the atoms
that make them up are free to
jump from one atom to

the next to produce a flow of electricity
.

In insulators these electrons
can’t move freely. Why is this so?

To understand the reason things work out this way in detail
would take us into the study of quantum mechanics and condensed
-
matter physics, which
isn’t
the subject of Physics 2049C. But a simple explanation is this:
the atoms

in metals arrange
themselves in very regular, symmetric patterns call lattices. The atoms
in insulators don’t; the
insulators’ atoms are arranged in a very disorderly fashion.
Certain quantum
-
mechanical
effects
arise only in the case where a lot of symmetry is present that allow the free motion of electrons,
and thus allow them to conduct electricity well. These effects come about large
ly

from the fact
that when atoms are arrang
ed in a regular fashion, the atoms effectively become
indistinguishable from each other. In this case an electron can’t really “tell” which electron it

belongs


to.

So with just a tiny

push, it can move from one atom to the next.

21
-
4 Induced Charge; the
Electroscope

i.

Bring a
positively
charge
d conductor next to a neutral

one
, here’s what happens
:

a.

The neutral conductor becomes polarized; the negative charges in it move to the end
closest to the charge
d

conductor, leaving an excess
positive charge at the oth
er end
.

b.

But the total charge on the neutral conductor is still zero
.

c.

If you cut the neutral conductor in half, one of the halves is positively charge
d
, the other
has an equal negative charge
.

d.

When you do this you are charging by

induction
.”

ii.

An electroscop
e is used to determine if an object is charged
.

a.

It’s made of two movable metal vanes that are connected
.

b.

Touch an object to the end of the vanes where they connect. If the object is charged,
then some of its charge flows into the conducting vanes. Since th
is charge all has the
same sign, the vanes repel each other
, and move apart.

c.

This doesn’t enable you to tell the sign of charge, just if an object has some charge

instead of being neutral.

21
-
5 Coulomb’s Law

4


i.

How to express the forces charges exert on each
other quantitatively? Through Coulomb’s law…

a.

Named after guy who discovered it

b.

Was discovered a little before the year 1800

ii.

Coulomb’s law holds for charges that are points
;

it says
:








, where


is the strength of one charge,


is
the strength of the other,


is the distance
between the two charges, and


is a constant whose value account for whether you are using
English unit
s
, metric units, or some other system of units
.

a.

Has same form as law of gravitation

b.

Force is directed
along the line joining the charges

c.

Values of charge are measured in units called “Coulombs
;


an electron, for example, has
a charge of about 1.602 x 10
-
19

C
, where “C” is the symbol for Coulomb.

d.

Metric value of
k

is about 8.99 x 10
9

Nm
2
/C
2

e.

Often write Coul
omb

s law in terms of


,
where







; have










.


is called
the permittivity of free space

iii.

Reme
m
ber
, Coulomb’s law only works for charges which are points
.

a.

If you have a charged object that isn’t a point, then think of it as a bunch
of points, and
add up the contribution each point makes to the total force by using Coulomb’s law
.

b.

This involves doing a sum or an integral; we’ll see how this works in the problems
.

c.

Adding things up like this to get a net result is called
linear superposi
tion
; linear
superposition only holds for certain special situations, and is the subject of linear
algebra
.

iv.

Coulomb’s law can be

written in vector form as







̂
, where

̂

is the unit vector pointing
from one charge to the other
.

21
-
6 The Electric
Field

i.

Action
-
at
-
a
-
distance is
a
bothersome

idea
; how can two objects that don’t actually touch each
other exert forces on each other?

a.

Use idea of field to help understand action
-
at
-
a
-
distance

b.

Any charge


c
hanges the region of space around it by producing

an electric field

.

c.

Strength of
the
field at any point determines force on another charge


p
laced at that
point

d.



is the ratio

F
/
q
, in the limit that
q

becomes very small (goes to zero)
. Electric field is a
vector just like force is.

e.


Must take limit so
q

has no effect on
Q

ii.

For point charges this gives







̂

21
-
7 Electric Field Calculations for Continuous Charge Distributions

i.

Fields can be linearly superposed just like forces; we calculate the field produced by a
continuous distr
ibution of charge by thinking of it as made up of lots of point charges, then we
just add up the contributions of each point charge by doing a sum or i
ntegral.

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ii.

We’ll see how this works when we get to the problems
.

21
-
8 Field Lines

i.

Recall that e
lectric field is a vector. Could represent an electric field by drawing an arrow at
every point in space.
But there
would be too many arrows
.

ii.

Easier to draw curves along the direction the field is in
:

a.

For a positive point

charge it would look like Fig. 1.

b.

For a positive and an equal negative charge
---
which is called an electric
dipole
---
it would
look like Fig. 2.

(We get this adding the field of the positive and negative charges
together
---
linear superposition. But here we don’t actually do the calculation;

we only
make a visual guess at its result
.)

c.

The curves are called field lines
.

iii.

Important n
ote:

a.

The arrows point in the direction a positive test charge would be pushed by the field
;
this force is tangential to the line
.

b.

The stronger the field, the closer
adjacent lines are
.

c.

Lines always start and end on charges
.

d.

Two field lines can never cross. (Why? Because then the force would be in two
directions at the point in space where they cross; this makes no sense…
the forces in the
two different directions would

just linearly superpose to create a new net force
described by a single field line in this case anyway…
)

iv.

Just like electric fields, we can think about gravitational fields. Everything works out in a
completely analogous way, just like you would expect
.

21
-
9 Electric Fields and Conductors

i.

The electric field inside a conductor is zero
.

a.

The electrons inside the conductor are free to move
.

b.

If you subject them to a field, then they move under its influence redistributing
themselves
.

c.

This new distribution of cha
rge creates its own electric field; the electrons keep moving
‘till the
ir

electric field exactly balances the field you originally subjected them to. With
the two fields in balance, the net field is zero, and there is no longer any force
accelerating the e
lectrons
.

d.

Because of this, conductors are used for shielding against unwanted electric fields
.

ii.

Implications of zero field inside a conductor:

a.

Any net charge must reside on the surface of a conductor. If there were charge inside,
then it would set up a fiel
d. This would push the charge along until it ended up on the
surface. (Once the charge gets to the surface, it can’t magically jump off, so it stops
there.) Another way to think of this is that electrons are negatively
-
charged, so they
push on each other.
They try to push each other as far apart as they can
---
which is to
the conductor’s surface
.

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b.

The field at the surface of the conductor is perpendicular to the surface. If it weren’t
then it would have some component tangent to the surface. This would cause charge to
move along the surface ‘till it redistributed itself in such a way that the tangen
tial
component were zero. Then the force would be zero and the redi
stribution of charge
would stop

sort of; we will revisit these ideas and their consistency with Newton’s
laws of motion later when we talk about “superconductors.”

21
-
10 Motion of a Charged

Particle in an Electric Field

i.

By previous eqns., know





, on a charge


ii.

Also know from last semester that





iii.

Can combine these and use
more eqns. from last semester to figure out how the charge moves
under the influence of the electric field

21
-
11 Electric Dipoles

i.

Already saw that an electric dipole is a combination of two equal and opposite charges


and


. Let them be separated by distance


ii.

Quantity




is called dipole moment; can think of the dipole moment as a vector


that
point from the negative to the positive charge
.

’s

magnitude is

.

iii.

If a dipole is put in an external field

, then there is no
net

force (because there’s no
net

charge).
But there is a torque, because there’s a force on each
individual

charge.

Can show the torque is






iv.

Also is a potential energy associated with this torque of the form







v.

The dipole additionally produces its own electric field. Can use linear superposition to show that
at a distance



from the center of the dipole it’s








(







)




, for the case where




,
this becomes











vi.

All this is very important when understanding how polar molecules interact

21
-
12 Electric Forces in Molecular Biology; DNA

i.

Read this sectio
n; it’s very interesting and explains
how DNA replication comes about through
electrical forces, and why even cloned animals aren’t
exactly

identical

ii.

But we won’t talk about these issues; this isn’t biology class
. You still really should read this, it’s
qu
ite interesting and involves some very important applications of physics that you might be
interested in should you decide to study biomedical engineering.

21
-
13 Photocopy Machines and Computer Printers use Electrostatics

i.

Read this section, too. It’s also
interesting

ii.

It tells how electrical forces are used in printers; again we won’t talk about these issues
. But it’s
also interesting and illustrates the kind of work professional electrical engineers do in industry.

Summary of Equations:

7


i.

Force one point char
ge exerts on another:








̂

ii.

Electric field due to a point charge
:







̂

iii.

Relationship between


and


:








iv.

Relationship between force and electric field
:





v.

Relationships for electric dipoles:

a.

Dipole Moment:




b.

Electric field
“relatively” far away from an electric dipole:













8





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