Electrostatics
Atoms are composed of protons, neutrons, and electrons.
Protons and neutrons are locked within the nucleus.
Objects can become charged by
removing electrons from, or adding electrons to, objects; an excess of electrons results in a
negative charge; an electron deficiency
results in a positive charge
Note: It is the movement of electrons
that results in a non-neutral
charge; protons cannot move, since
they are locked in the nucleus
Materials such as plastic, glass,
and wood are insulators; they do not
conduct electricity because the electrons are tightly bound in the electron
shells of their atoms
Metals are conductors because the electrons are not tightly bound within each
atom; they can fairly easily jump from the electron shell of one atom to the
electron shell of its neighbor; the outermost electrons are called delocalized electrons—that is, not
belonging to any one location
When an insulator such as a plastic
rod is rubbed with fur, the plastic strips electrons from the fur; the rod has
an excess of electrons and becomes negatively charged. The fur consequently
becomes positively charged.
When a plastic rod is rubbed with
silk, silk strips electrons from the rod; The silk has an excess of electrons,
and is negatively charged. The rod has an electron deficiency, so it becomes
positively charged.
In both cases, the charge stays in
one place on the rod. The electrons cannot move along the surface of the rod,
since it is an insulator.
static electricity: electricity that does not move
Results from lab: (Powerpoint
presentation)
·
Like charges repel, opposite charges attract (Law of Electrostatics)
·
Neutral objects are attracted to charged objects
·
After neutral objects touch charged objects,
charge transfer occurs and they acquire like charges, and repel
If
the can touches the rod above, some of the electrons from the rod move onto
the can to neutralize
the positive charges there; the can then has an excess of electrons as does the
rod (the rod becomes a little less negative after touching, but still has
excess electrons), and the objects repel each other
·
If object A has a negative charge and attracts
object B, it does not mean that object B is positively charged—object B could
also be neutral. However, if object B were repelled by object A, then we know
that object B was negatively charged. In summary, attraction is not a valid test for the determination of charge on an
object; only repulsion will definitively determine charge
Ways of charging
an object: (Continued
on PPt)
1)
Friction
(plastic rod, walking across carpet)
2)
Conduction
(charging by contact; acquires same
charge as charging object; see soda can example above
3)
Induction
(acquires a charge opposite to the
charging object); see handout and diagram below
A neg. charged rod The
electroscope The
ground A positively
is held near a neutral is
grounded; the connection
is
charged
electroscope electrons
are repelled broken
electroscope
to
the ground results
Demo: Use
induction spheres and electrophorus; test for charge with electroscope
Force between charges
The force
between charges qualitatively follows the Law of Electrostatics as stated
earlier. In this section we will study the quantitative nature of this force.
Fortunately, the nature of the force is such that if charge is uniformly
distributed on the surface of an isolated object, the influence on another
object some distance away is the same as if the charge were all located at the
center of the object. The spatial nature of the object does not affect the
force, as long as the other object is far enough away. This simplifies matters.
Newton made
this same assumption in his Law of Gravitation, and it works equally well here.
In the late
1700’s French physicist Charles Coulomb did very careful experiments with
charged objects. He was able to discover the effects of charge and distance on
the force. His findings will presented shortly. First, some definitions.
Charge is considered a fundamental
quantity of nature (others: distance, time, and mass). The fundamental unit
associated with charge is the coulomb,
in honor of Charles Coulomb. The amount of charge in a coulomb was defined from
work in current electricity. Later, after electrons were discovered, it was
defined in more detail. It is known now that one coulomb is the charge
contained in 6.4 x 1018 electrons. The charge on one electron is
1
e- = 1.6021 x 10-19 C
Of course, the sign of the electron’s charge is negative. A
proton has the exact amount of charge, but it is positive.
Coulomb’s results mirrored those of
Newton in his
Law of Gravitation. He found that changing the charge had the same effect on
the electric force as the mass on the gravitational force. He also found the
same effect that distance had on the force. His findings are summarized as
Coulomb’s Law: (Powerpoint
presentation)
Coulomb’s Law:
the force between two point charges
is directly proportional to the
product of the charges and
inversely proportional to the square of the distance between them
ex: Find
force between charges of +2.5 C and -1.5 C located 50 cm apart in air.
Note that the force turned out to
be negative. This is because one of the charges was negative and the other
positive. This force will be attractive. So, negative electric forces are
attractive. The force between two positive charges and two negative charges
would come out to be positive, so we can say that positive forces are
repulsive.
Note also that this force is
tremendously large. That is because one coulomb is a very large amount of
charge. It is rather difficult to accumulate a coulomb of charge on an everyday
object. A more common amount of charge, say the amount you might acquire by
walking across a carpet on a dry day, is on the order of a millionth of a
coulomb. The metric prefix for one-millionth is micro-, which is represented by the symbol m. In the problems we do,
many times the charges will be given in microcoulombs.
1
microcoulomb = 1 mC = 1 x 10-6 C
ex. Find
the force between a charge of - 3.2 mC and -1.6 mC when they are separated
by a distance of 24 cm.
Try-It-Yourself:
1.
Find the force between charges of +3.0 C and -1.2 C
when separated by 12 m.
2. Find the force
between charges of -4.2 mC and -6.8 mC when separated by 18 cm.
3.
The electric between two charges objects is +2.4 N. The
charge on one object is
+8.2 mC and
the charges are separated by 32 cm. Find the other charge.
Answers: 1.
F = -2.25 x 108 N
2. F
= 7.9 N
3. q2
= +3.3 x 10-6 C
A few other things of note about the electric force. It
follows the inverse-square law, which means that if the distance between the
charges is doubled, the force between them is 1/22 or ¼ of the
previous value. If the distance is tripled, the force drops to 1/9, etc. This
is similar to the gravitational force. As it turns out, the inverse square law
is obeyed by many forces and energies is physics. The electric force is also a
non-contact force (similar to gravity). It acts over space.
In terms of
strength, the electric force is stronger than gravity. For example, the electric force between two
protons is much stronger than the gravitational force between them. However, it
is much easier to acquire large masses than large charges, so in the
large-scale universe gravity is dominant. It acts over much longer ranges that
the electric force. On the microscale, in atoms and molecules, the electric
force is supreme.
Electric Fields
As
discussed earlier, the electric force is a non-contact force. A charged object
can affect another charge over a distance. Many scientists, including Newton, had difficulty
with this concept. In the mid-1800’s British physicist Michael Faraday
introduced the concept of a field to
try to make it conceptually easier.
A charge
placed in space affects that space around it. It sets up an electric field.
An electric field is said to exist in a
region of space if an electric charge in the
region is subject
to an electric force
The “electric charge in the region” is called a test charge, and is taken to be a
infinitesimally small positive charge +q. The direction of the electric field lines created by a
charged object will be the trajectory of the test charge when placed in the
region around the charged object.
Strength of the Field
If a test
charge is placed in an electric field, it will have a force exerted on it by
the field. The magnitude of this force can be used to measure the strength of
the field, or the electric field intensity.
The electric
field intensity is defined to be the force that a unit charge “feels” when
placed in the field.
ex. A
charge of 3.0 x 10-8 C experiences a force of 6.0 x 10-4
N when placed in an
electric field. Calculate the
electric field intensity.
Try-It-Yourself:
1.
A charge of 5.0 mC experiences a force of
0.0025 N when placed in an electric field. Find the electric field intensity.
2.
A charge of 6.4 mC is placed in an electric
field of intensity 75 N/C. Find the force on the charge.
Answers: 1.
E = 5.0 x 102 N/C
2. F = 4.8 x 10-4 N
Mapping Electric Field Lines
The
electric field around a single charged object can be easily mapped by
considering the path that a small positive test charge would take when placed
near the object. But what happens when there are more than one charged object
in a region?
As it turns
out, we can use the same technique of plotting the trajectory of a test charge
q+. Consider a situation in which there is a positive charge and a
negative charge in the same region. A test charge is placed in the space
between the objects.
What path does the charge q+ take when released?
These will be the electric field lines. Let’s draw them in on the diagram
below.
Field lines around unlike charges:
Note the direction of the field lines. This will be from the
positive charge towards the negative charge, since that is the path of q+.
For this reason, we say (by definition)
·
Electric field lines originate on positive
charges and terminate on negative charges
An electric field always ends on some negative charge. It
never simply stops at a point in space. It always has a beginning and an end.
We assume that the lines from +Q end on some negative charge not in the
picture. Likewise, we assume the lines ending on –Q originated on some positive
charge off the page.
What about the field around two objects of the same charge?
Again, consider the path of a charge q+. Draw the field lines on the
diagram below:
Field lines around like positive charges:
Again, we assume that the field lines end on some negative
charges not in the picture. One more thing about electric field lines:
- Field lines never intersect. They can approach each asymptotically, but they never touch.
The region in the
middle is kind of a “no-man’s-land”. A positive charge could be in unstable
equilibrium there, but it would be difficult for it to remain so.
Potential Difference
Consider a
sphere on which there is an even distribution of negative charge.. Let’s place
positive charge q1 and q2 in the electric field generated
by the sphere:
Each of the charges feel an attractive force. If released,
they would accelerate towards the sphere, doing work along the way. Thus, they
have energy. Specifically, they have electric
potential energy. If the charge q1 is to be moved farther away
from the sphere, say to the location of q2, work must be done on it.
The work done on the charge will increase its potential energy. So, the charge
q2 has a higher electric potential energy than q1. The
amount of work that would have to be done on charge q1 to move it to
q2’s location will equal the difference in potential energy of the
two locations. We can then speak of an
electric potential energy difference, or simply potential difference, for short.
Potential Difference: the work done per unit charge as a charge
is moved
between two locations in an electric field;
denoted by V
The unit of potential difference is derived from the unit of
work (joule) and the unit of charge (coulomb). The unit is a joule/coulomb, or
J/C. The J/C has been further defined. It is called a volt, in honor of Italian scientist Alessandro Volta (1745-1827),
the inventor of the electric battery.
1
volt =
1 V = 1 J/C
We can make
a comparison between electric potential energy and gravitational potential
energy. In each case, the actual amount of potential is not important (indeed,
it may be very difficult to know). We need only to know the difference of
potential energies between two locations.
In each case, the base level, or zero point, can be chosen arbitrarily.
In the gravitational case we often choose ground level to be at zero potential
energy. Similarly, we compare the electric potential energies of electrons with
those of the electrons in the earth. We define the electrons in the ground as
having zero electric potential., Thus, electrons that are “grounded” have no
potential energy. The potential difference of electrons in a battery is the
difference in electric potential energy of those electrons as compared to the
electrons in the ground.
Distribution of Charge
British
physicist Michael Faraday did a famous experiment called the ”ice-pail
experiment”, using a metal ice pail that you might see holding a bottle of
champagne in a fancy restaurant. He placed an electroscope inside the pail and
then deposited a static charge on the pail, both on the inside and the outside.
He discovered that all charge on the inside moved to the outside surface. No
charge remained on the inside of the conductor. From this and other
experiments, he was able to make a number of conclusions:
·
All static charge on a conductor resides on its
surface
·
No electric field can exist inside a conductor
·
There is no potential difference between any two
points on a conducting surface
·
If charge is put on a spherical surface, it will
distribute evenly
·
If charge is put on a non-spherical surface, it
will congregate at points of highest curvature
·
Points of highest curvature will have the
greatest charge density
An easy way
to prove the second point is to get a small battery-operated radio, a shoebox,
and some aluminum foil. Wrap the box completely in foil, including the cover.
Tune the radio to your favorite music station, turn the volume up rather loud,
and place the radio inside the box. As you close the box, the music will
disappear! This is because the radio signal cannot penetrate the foil. The
electric field of the signal cannot exist inside the conductor, and so it
cannot reach the radio. All you hear is a faint hiss due to the workings of the
radio.
The fourth
point above is easily understood if you remember that like charges repel each
other. The charges will settle so that they are as far away from each other as
possible, and so they will distribute themselves evenly on a spherical surface.
If the surface is not spherical, however, the charge distribution is not
uniform.
The charge density at the points is sometimes high enough to
create an electric field that
will ionize the air around the
point; charge then leaks off the point to neutralize the ionized air; in this
way the conductor is neutralized as a result (Ionization causes positive
particles to move one way, and negative particles to move the other way
In dry air, it takes 30,000 V to ionize 1 cm of air; if
there is a ground within that 1 cm, a
“wire” of ionized air is created;
charge will move on the wire; spark occurs
Demo: Tesla coil,
walking across carpet, lightning
Only at sharp points will the electric field intensity be
enough to ionize the air; charge will
leak off here
Demo:
Electrostatic whirl
At tips of ship masts, and at the ends of wings and tail of
aircraft, a glow discharge
sometimes occurs; “St. Elmo’s Fire”
Sometimes aircraft drag “pig-tails” or have sharp points on
the aircraft so that the charge
can leak off’ lightning rods leak
charge from a house or barn to make it less susceptible to lightning; if
lightning does strike, it is carried to ground; lightning rods do not attract
lightning, they try to prevent it by leaking off charge
Capacitors
Capacitors are parallel plates on which charge can be
deposited; separated by an insulator
called a dielectric; for now, air will be the dielectric; charge can be
stored on the plates for later use; the better the dielectric, the more charge
that can be stored
The charge Q that
is stored on a capacitor, in general, is directly proportional to the
potential difference V between the plates
Q
= k V , where k = constant
Q
= CV C =
capacitance, measured in farads
1
farad = 1 F = 1 C/V
A capacitance of 1 farad is enormous; typical values of
capacitance are on the order of
picofarads (10-12) or
microfarads
ex. How much
charge is stored on a 2.0 μF capacitor that is connected across a 12-V
battery?
Q = CV = (2.0 x 10-6 F)(12 V)
= 2.4 x 10-5 C
The electric field between plates : + + +
+ + + + +
+ + +
+ + +
+ + +  |
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The field between
the plates is very uniform, as long as it is away from the ends of the plates.
The strength of the field depends on the voltage across the plates and the
distance between them. Specifically,
V
E =
d
ex. Two parallel
plates of a capacitor are 1.5 mm apart. If a potential difference of 6.0 V is
placed across the capacitor, find the electric field between the
plates.
V
6.0 V
E = = =
4000 V/m
d
0.0015 m
You may recall that earlier we expressed
electric field intensity in units of N/C. It can be shown the V/m and N/C are
equivalent ; that is, you can
derive one from the other (Try it!).
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