Transcript Electric Charge

Electrostatics
 Electric charge
 Van de Graff generators
 Conservation of charge
 Equilibrium problems
 Insulators & conductors
 Grounding
 Charging objects
 Static electricity
 Electroscopes
 Coulomb’s law
 Lightning
 Systems of charges
Electric Charge
• Just as most particles have an attribute known as mass, many
possess another attribute called charge. Charge and mass are intrinsic
properties, defining properties that particles possess by their very
nature.
• Unlike mass, there are two different kinds of charge: positive and
negative.
• Particles with a unlike charges attract, while those with like charges
repel.
• Most everyday objects are comprised of billions of charged, but
usually there are about the same number of positive charges as
negative, leaving the object as a whole neutral.
• A charged object is an object that has an excess of one type of
charge, e.g., more positive than negative. The amount of excess
charge is the charge we assign to that object.
Conservation of Charge
Charged particles can be transferred from one object to another, but the total
amount of charge is conserved. Experiments have shown that whenever
subatomic particles are transferred between objects or interact to produce
other subatomic particles, the total charge before and after is the same (along
with the total energy and momentum). Example: An object with 5 excess
units of positive charge and another with 2 units of excess negative charge
are released from rest and attract each other. (By Newton’s 3rd law, the forces
are equal strength, opposite directions, but their accelerations depend on
their masses too.) Since there is no net force on the system, their center of
mass does not accelerate, and they collide there. As they “fall” toward each
other, electric potential energy is converted to kinetic energy. When contact
is made charge may be exchanged but they total amount before and after
must be the same. After the collision the total momentum must still be zero.
Before
+5
After
-2
Total charge: +3
+1.5
+1.5
Total charge: +3
Conservation of Charge: β-decay
• The stability of the nucleus of an atom depends on its size and its
proton-neutron ratio. This instability sometimes results in a
radioactive process known as β-decay.
• A neutron can turn into a proton, but in the process an electron
(beta particle) is ejected at high speed from the nucleus to conserve
charge.
• A proton can turn into a neutron. In this case the beta particle is an
positron (an antielectron: same mass as an electron but a positive
charge) to make up for the loss of positive charge of the proton.
• In either case, charge, momentum, and energy are conserved.
SI unit of Charge: the Coulomb
• Just as we have an SI unit for mass, the kilogram, we have one
for charge as well. It’s called the coulomb, and its symbol is C.
• It’s named after a French physicist, Charles Coulomb, who did
research on charges in the mid and late 1700’s.
• A coulomb is a fairly large amount of charge, so sometimes we
measure small amounts of charge in μC (mircocoloumbs).
• An electron has a charge of -1.6  10-19 C.
• A proton has a charge of +1.6  10-19 C.
• In a wire, if one coulomb of charge flows past a point in one
second, we say the current in the wire is one ampere.
Elementary Charge
• Charges come in small, discrete bundles. Another way to say this
is that charge is quantized. This means an object can possess charge
in incremental, rather than continuous, amounts.
• Imagine the graph of a linear function buy when you zoom in
very close you see that it really is a step function with very small
steps.
• The smallest amount of charge that can be added or removed from
an object is the elementary charge, e = 1.6  10-19 C.
• The charge of a proton is +e, an electron -e.
• The charge of an object, Q, is always a multiple of this
elementary charge: Q = Ne, where N is an integer.
• How many excess protons are required for an object to have 1 C
of charge?
Insulators vs. Conductors
• A conductor is a material in which excess charge freely flows.
Metals are typically excellent conductors because the valence (outer
shell) electrons in metal atoms are not confined to any one atom.
Rather, they roam freely about a metal object. Metal are excellent
conductors of electricity (and heat) for this reason.
• An insulator is a material in which excess charge, for the most
part, resides where it is deposited. That is, once placed, it does not
move. Most nonmetallic material are good insulators. Valence
electrons are much more tightly bound to the atoms and are not free
to roam about. Insulators are useful for studying electrostatics (the
study of charge that can be localized and contained).
• Semi-conductors, like silicon used in computer chips, have
electrical conductivity between that of conductors and insulators.
Details on Conductors, Semiconductors, and Insulators
Electrons and Chemical Bonds
All chemical bonding is due to forces between electrostatic charges.
Covalent bonding: A pair of electrons is shared between two nonmetal
atoms, allowing each atom to have access to enough electrons to fill
its outer shell. Except for hydrogen, this usually means 8 electrons in
the outer shell (octet rule).
Ionic bonding: One or more valence electrons of a metal atom are
“stolen” by a nonmetal atom, leaving a positive metal ion and a
negative nonmetal ion, which then attract one another.
Metallic bonding: Valence electrons of metals flow freely throughout
a metal object. These delocalized electrons are attracted to the nuclei
of the atoms through which they are moving about. This produces a
strong binding force that holds the atoms together. In an iron bar, for
example, there is no covalent or ionic bonding. Metallic bonding hold
the metal together.
Charging up Objects
Charging up an object does not mean creating new charges. Charging
implies either adding electrons to an object, removing electrons from
an object, or separating out positive and negative charges within an
object. This can be accomplish in 3 different ways:
• Friction: Rubbing two materials together can rub electrons off of
one and onto the other.
• Conduction: Touching an object to a charged object could lead to a
flow of charge between them.
• Induction: If a charged object is brought near (but not touching) a
second object, the charged object could attract or repel electrons
(depending on its charge) in the second object. This yields a
separation charge in the second object, an induced charge separation.
Electroscopes
An electroscope is an apparatus comprised of a metal sphere and
very light metal leaves. A metal rod connects the leaves to the
sphere. The leaves are enclosed in an insulating, transparent
container. When the electroscope is uncharged the leaves hang
vertically. The scope is charged by placing a charged rod near the
sphere. The rod is charged by friction. If a rubber rod is rubbed
with fur, electrons will be rubbed off the fur and
onto the rubber rod, leaving the rod negatively
charged. If a glass rod is rubbed with silk,
electrons will be rubbed off the rod onto the silk,
leaving the glass rod positively charged. Either
rod, if brought near, will charge the scope by
induction. Also, either rod, if contact is made with
the sphere, will charge the scope by conduction.
Electroscopes
uncharged
continued…
Electroscopes
(cont.)
When a positively charged rod is placed near but not touching the
metal sphere, some of the valence electrons in the metal leaves are
drawn up into the sphere, leaving the sphere negatively charged and
the leaves positively charged. Thus, the rod has induced a charge
separation in the scope. The light,
++++++++++++++
positive leaves repel each other and
separate. The electroscope as a whole
-- - -is still electrically neutral, but it has
-undergone a charge separation. As
soon as the rod is removed from the
vicinity, the charge separation will
+
+
cease to exist and the leaves the drop.
Note: Only the electron are mobile;
+
+
the positives on the leaves represent
+
+
missing electrons.
continued…
Electroscopes
(cont.)
When a negatively charged rod is placed near but not touching the
metal sphere, some of the valence electrons in the sphere are repelled
down into the metal leaves, leaving the sphere positively charged and
the leaves negatively charged. The rod has again induced a charge
separation in the scope. The light,
------------------negative leaves repel each other as
before. Again, the electroscope as a
++
whole is electrically neutral, but the
+ +
++
charge separation will remain so long
as the rod remains nearby. Note that
this situation is indistinguishable from
the situation with the positive rod.
Since the effects are the same, how do
we know that the rods really do have
different charges?
continued…
Electroscopes
(cont.)
Now let’s touch the negative rod to the sphere. Some of the electrons
can actually hop onto the sphere and spread throughout the scope.
This is charging by conduction since, instead of rearranging charges
in the scope, new charges have been added; the scope is no longer
neutral. The extra electrons force the leaves apart, even when the rod
is removed. If the negative rod returns, it charges the leaves further,
but this time by induction (by driving some of
--------------electrons on the sphere down
- - - to
the
leaves).
This
causes
an
- - increased separation of the
leaves. When the rod is
removed, the scope will return
to the state on the left.
Continued…
extra e- ’s added
leaf spread increases
Electroscopes
(cont.)
The pic on the left shows a scope that has acquired extra electrons
from a negative rod that has since been removed. Now we bring a
positive rod nearby. This has the opposite effect of bringing the
negative rod near. This time some of the extra electrons in the leaves
head to the sphere and the spread of the leaves diminishes. Note: the
scope is still negatively charged overall, but the presence of the
positive rod means more of the
+++++++++++
excess negative charge will
- - - reside in the sphere and less in
- - the leaves. When the rod is
removed, the scope return to
the state on the left.
Continued…
extra e- ’s added
leaf spread decreases
Grounding an Electroscope
Whether a scope has charged by conduction, either positively or
negatively, the quickest way to “uncharge” it is by grounding it. To do
this we simply touch the sphere. When a negatively charged scope is
grounded by your hand, the excess electrons from the scope travel
into your body and, from there, into your surroundings. When a
positively charged scope is
grounded, electrons from your
body flow into the scope until
- ++
it is neutral. Your surroundings
- + +
- ++
will replace the electrons
you’ve donated to the scope.
As always, it’s only the
+
+
electrons that move around.
+
+
+
+
Electroscope Practice Problem
For the following scenario, try to predict what would happen after
each step. Explain each in terms of electrons and charging.
1. A rod is rubbed with a material that has a greater affinity for
electrons than the rod does.
2. This rod is brought near a neutral electroscope.
3. This rod touches the electroscope and is removed.
4. A positive rod is alternately brought near and removed.
5. A negative rod is alternately brought near and removed.
6. Finally, you touch the scope with your finger.
Redistributing Charge on Conducting Spheres
Two neutral spheres, A & B, are placed side by side, touching. A negatively
charged rod is brought near A, which induces a charge separation in the “A-B
system.” Some of the valence e-’s in A migrate to B. When the rod is removed and A & B are separated, A is +, B is -, but the system is still neutral.
---
+Q
A
B
-Q
B
A
A is now brought near neutral sphere C, inducing a charge separation on it.
Valence e-’s in C migrate toward A, but since C is being touched on the
positive side, e-’s from the hand will move into C. Interestingly, C retains a
net negative charge after A and the hand are removed even though no charged
object ever made contact with it.
-
+Q
A
C
C
Static Electricity: Shocks
If you walk around on carpeting in your stocking feet, especially in
the winter when the air is dry, and then touch something metal, you
may feel a shock. As you walk you can become negatively charged by
friction. When you make contact with a metal door knob, you
discharge rapidly into the metal and feel a shock at the point of
contact. A similar effect occurs in the winter when you exit a car: if
you slide out of your seat and touch then touch the car door, you
might feel a shock.
The reason the effect most often occurs in winter is because the air is
typically drier then. Humidity in the air can rather quickly rob excess
charges from a charged body, thereby neutralizing it before a rapid,
localized discharge (and resulting shock) can take place.
Care must be taken to prevent static discharges where sensitive
electronics are in use or where volatile substances are stored.
+- # 1
+++++++-+
-+ - - - -+ - -+ - +++++- # 3
+-
Static Electricity: Balloons
Pic #1: If you rub a balloon on your hair,
electrons will be rubbed off your hair onto the
balloon (charging by friction).
#2
Pic #2: If you then place the negatively
charged balloon near a neutral wall, the
balloon will repel some of the electrons near it
in the wall. This is inducing a charge
separation in the wall. Now the wall, while
still neutral, has a positive charge near the
balloon. Thus, the balloon sticks to the wall.
Pick #3: Your hair now might stand up. This is
because it has been left positively charged. As
with the leaves of a charged electroscope, the
light hairs repel each other.
Hanging Balloons
#1
You hang two balloons from the
ceiling and rub them on your hair.
#2
When you move out of the way, the negatively charged balloons
repel each other. On each balloon there are three forces: tension
in the string, gravity, and the electric force. The angle of
separation will grow until equilibrium is achieved (zero net force).
#3
If you move your head close to either
of the balloons, it will move toward
you since your hair remains positively
charged.
Polarization of a Cloud
Lightning is the discharge of static electricity on a
massive scale. Before a strike the bottom part of a
cloud becomes negatively charged and the top part
positively charged. The exact mechanism by which
this polarization (charge separation) takes place is
uncertain, but this is the precursor to a lightning
strike from cloud to cloud or cloud to ground.
One mechanism incorporates friction: when moist, warm air rises, it cools and
water droplets form. These droplets collide with ice crystals and water droplets
in a cloud. Electrons are torn off the rising water droplets by the ice crystals. The
positive droplets rise to the top of the cloud, while the negative ice crystals
remain at the bottom.
A second mechanism involves the freezing process: experiments have shown
that when water vapor freezes the central ice crystal becomes negatively
charged, while the water surrounding it becomes positive. If rising air tears the
surrounding water from the ice, the cloud becomes polarized.
There are other theories as well.
Detailed Lightning Diagrams
Lightning Strikes
The negative bottom part of the cloud induces
a charge separation in the ground below. Air is
normally a very good insulator, but if the charge
separation is big enough, the air between the
cloud and ground can become ionized (a
plasma). This allows some of the electrons in the
cloud to begin to migrate into the ionized air
below. This is called a “leader.” Positive ions
from the ground migrate up to meet the leader.
This is called a “streamer.” As soon as the leader
and streamer meet, a fully conductive path exists
between the cloud and ground and a lightning
strike occurs. Billions of trillions of electrons
flow into the ground in less than a millisecond.
The strike can be hotter than the surface of the
sun. The heat expands the surrounding air;
which then claps as thunder.
+ +++
+
+
+
+
+
- -- - -
+ +
+
+
+
+ +
+ +
Lightning Rods and Grounding
Discovered by Ben Franklin, a lightning rod is a long, pointed, metal
pole attached to a building. It may seem crazy to attract lightning
close to a susceptible structure, but a lightning rod can afford some
protection. When positive charges accumulate beneath a cloud, the
accumulation is extremely high near the tip of the rod. As a result, an
electric field is produced that is much greater surrounding the tip than
around the building. (We’ll study electric fields in the next unit.) This
strong electric field ionizes the air around the tip of the rod and
“encourages” a strike to occur there.
If a strike does occur, the electricity travels down the rod into a
copper cable that connects the lightning rod to a grounding rod
buried in the earth. There the excess charge is grounded, i.e., the
electrons are dissipated throughout the landscape. By taking this
route, rather than through a building and its wiring, much loss is
prevented.
Van de Graaff
Generator
A Van de Graaff generator consists of a large metal dome attached to a tube, within
which a long rubber belt is turning on rollers. As the belt turns friction between it and
the bottom roller cause the e-’s to move from the belt to the roller. A metal brush then
drains these e-’s away and grounds them. So, as the belt passes the bottom roller it
acquires a positive charge, which is transported to the top of the device (inside the
dome). Here another metal brush facilitates the transfer of electrons from the dome to
the belt, leaving the dome positively charged.
In short, the belt transports electrons from a metal dome to the ground, producing a
very positively charged dome. No outside source of charge is required, and the
generator could even be powered by a hand crank. A person touching the dome will
have some of her e-’s drained out. So, her lightweight, positive hair will repel itself.
Coming close to the charge dome will produce sparks when electrons jump from a
person to the dome.
Internal workings
Detailed explanation
Coulomb’s Law
There is an inverse square formula, called Coulomb’s law, for finding
the force on one point charge due to another:
K q1 q2
F=
r2
K = 9 109 N m2 / C2
This formula is just like Newton’s law of uniform gravitation with charges
replacing masses and K replacing G. It states that the electric force on
each of the point charges is directly proportional to each charge and
inversely proportional to the square of the distance between them. The
easiest way to use the formula to ignore signs when entering charges, since
we already know that like charges repel and opposites attract. K is the
constant of proportionality. Its units serve to reduce all units on the right to
nothing but newtons. Forces are equal but opposite.
+
q1
F
Charges in Motion
r
F
q2
Coulomb's Law Detailed Example
Electric Force vs. Gravitational Force
FE =
K q1 q2
r2
G m1 m2
FG =
r2
K = 9  109 N m2 / C2
G = 6.67  10-11 N m2 / kg2
Gravity is the dominant force when it comes to shaping galaxies and the
like, but notice that K is about 20 orders of magnitude greater than G.
Technically, they can’t be directly compared, since they have different
units. The point is, though, that a whole lot of mass is required to produce
a significant force, but a relatively small amount of charge can overcome
this, explaining how the electric force on a balloon can easily match the
balloon’s weight. When dealing with high-charge, low-mass objects, such
as protons & electrons, the force of gravity is negligible.
Electric Force Example
A proton and an electron are separated by 15 μm. They are released from rest.
Our goal is to find the acceleration each undergoes at the instant of release.
1. Find the electric force on each particle.
1.024 × 10-18 N
2. Find the gravitational force on each particle. A proton’s mass is
1.67  10-27 kg, and an electron’s mass is 9.11  10-31 kg.
4.51 × 10-58 N
3. Find the net force on each and round appropriately. Note that the
gravitational force is inconsequential here. 1.024 × 10-18 N
4. Find the acceleration on each particle.
e-: 1.124 × 1012 m/s2, p+: 6.13 × 108 m/s2
5. Why couldn’t we use kinematics to find the time it would take
the particles to collide? r changes, so F changes, so a changes.
+
15 μm
+
System of 3 Charges
In a system of three point charges, each charge exerts a forces on the
other two. So, here we’ve got a vector net force problem. Find the net
force on charge B. Steps:
1. Find the distance in meters between A and B
using the law of cosines. 0.261947 m
A
+3 μC
2. Find angle B in the triangle using the law of sines.
36.027932 º
3. Find FBA (the magnitude of the force on charge
B due to charge A).
0.786981 N
17 cm
4. Find FBC. 4.591836 N
5. Break up the forces on B into components
and find the net horiz. & vertical forces.
3.78 N (right) , 1.25 N (up)
B
3.98 N at
6. Determine Fnet on B. 18.3 º N of E
+2 μC
115º
14 cm
C
-5 μC
System of 4 Charges
Here four fixed charges are arranged in a rectangle.
Find Fnet on charge D.
Solution:
-16 µC
A
+25 µC
C
767.2 N at 59.6 º N of W
4 cm
B
+9 µC
Link
3 cm
D
-7 µC
Hanging Charge Problem
Two objects of equal charge and mass are
hung from the same point on a ceiling
with equally long strings. They repel each
other forming an angle  between the strings.
Find q as a function of m, L and .
Solution: Draw a f.b.d. on one of the
objects, break T into components, and
write net vertical and horiz. equations:
L

L
T
FE
q, m
q, m
mg
T sin( / 2) = FE , T cos( / 2) = mg.
Dividing equations and using Coulomb’s law yields:
mg tan( / 2) = FE = Kq 2 / r 2, where r = 2 L sin( /2). Thus,
2 mg tan( / 2) sin2( / 2)
4
L
q=
K
Point of Equilibrium
Clearly, half way between two equal charges is a point of equilibrium,
P, as shown on the left. (This means there is zero net force on any
charge placed at P.) At no other point in space, even points equidistant
between the two charges, will equilibrium occur.
Depicted on the right are two positive point charges, one with twice
the charge of the other, separated by a distance d. In this case, P must
be closer to q than 2 q since in order for their forces to be the same,
we must be closer to the smaller charge. Since Coulomb’s formula is
nonlinear, we can’t assume that P is twice as close to the smaller
charge. We’ll call this distance x and calculate it in terms of d.
Continued…
x=?
+q
P
+q
+q
P
+2 q
d
Point of Equilibrium
Since P is the equilibrium point, no matter
what charge is placed at P, there should be
zero electric on it. Thus an arbitrary “test
charge” q0 (any size any sign) at P will feel
a force due to q and an equal force due to
2 q. We compute each of these forces via
Coulomb’s law:
K q q0
K (2q)q0
=
2
x
(d - x)2
(cont.)
x
+q
+2 q
d
The K’s, q’s, and q0’s cancel, the latter
showing that the location of P is
independent of the charge placed there.
Cross multiplying we obtain:
(d - x)2 = 2x 2  d 2 - 2 xd + x 2 = 2 x 2
 x 2 + 2 xd - d 2 = 0.
P
Point of Equilibrium
x
(cont.)
P
+q
From x 2 + 2 xd - d 2 = 0,
the quadratic formula yields:
x=
d
-2 d  (2 d)2 - 4(1)(-d 2 )
= -d  d 2
2(1)
+2 q
-2 d 
=
8d2
2
Since x is a distance, we choose the positive root:
x = d ( 2 - 1)  0.41 d. Note that x < 0.5 d, as predicted.
Note that if the two charges had been the same, we would have
 d 2 - 2 xd + x 2 = x 2
 d 2 - 2 x d = 0  d (d - 2x) = 0  x = d/2, as
started with (d - x)2 = x 2
predicted. This serves as a check on our reasoning.
Equilibrium with Several Charges
Several equal point charges are to be arranged in a plane so that another point
charge with non-negligible mass can be suspended above the plane. How might this
be done?
Answer: Arrange the charges in a circle, spaced evenly, and fix them in place.
Place another charge of the same sign above the center of the circle. If placed at the
right distance above the plane, the charge could hover. This arrangement works
because of symmetry. The electric force vectors on the hovering charge are shown.
Each vector is the same magnitude and they lie in a cone. Each vector has a vertical
component and a component in the plane. The planar components cancel out, but
the vertical components add to negate
the weight vector. Continued…
Equilibrium with Several Charges
(cont.)
Note that the charges in the plane are fixed. That is, they are attached somehow in
the plane. They could, for example, be attached to an insulating ring, which is then
set on a table. Regardless, how could the arrangement of charges in the plane be
modified so as to maintain equilibrium of the hovering charge but allow it to hover
at a different height?
Answer: If the charges in the plane are arranged in a circle with a large radius, the
electric force vectors would be more horizontal, thereby working together less and
canceling each other more. The hovering charge would lower. Since its weight
doesn’t change, it must be closer to the plane in order to increase the forces to
compensate for their partial cancellation. If the charges in the plane were arranged
in a small circle, the vectors would be more vertical, thereby working together more
and canceling each other less. The hovering charge would rise and the vectors
would decrease in magnitude. To maximize the height of the hovering charge, all
the charges in the plane should be brought to a single point. Continued…
Credits
www.phys.ufl.edu/~phy3054/elecstat/efield/twopoint/Welcome.html
www.phys.ufl.edu/~phy3054/elecstat/efield/polygon4/Welcome.html
www.eskimo.com/~billb/emotor/belt.html
207.10.97.102/chemzone/lessons/03bonding/mleebonding.htm
chem.ch.huji.ac.il/~eugeniik/instruments/archaic/electroscopes.html
www.physicsclassroom.com/mmedia/estatics/gep.html
www.cutescience.com/files/collegephysics/movies/GroundPositiveRodA.html