Brock University
Physics 1P22/1P92
Winter 2015
Dr. D’Agostino
Reading Guide for Ch. 18, Electric Forces and Electric Fields
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The Origin of Electricity
We notice macroscopic manifestations of electricity, which were mysterious back in the early days of
electrical research, but which by now are quite well-understood in terms of microscopic properties
and motions of fundamental electrically-charged particles. This section describes the microscopic
origins of electrical phenomena.
ˆ What is electricity?
– Is electricity a fluid phenomenon? How do we know?
– Is electricity a particle phenomenon? How do we know?
ˆ What is electric charge?
– What is the SI unit for electric charge?
– It seems that the measured values of electric charge are always whole-number multiples
of a smallest unit. Why is this so?
– What is the smallest known unit of electric charge? Where might you find an object
that has exactly this smallest known unit of electric charge?
ˆ What is the concept of net charge (also sometimes known as total charge, and also sometimes
known (possibly confusingly) as simply charge; that is, we sometimes omit the word “net”
even when we mean “net charge”)?
– How can you calculate the net charge of a system of charged particles?
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Charged Objects and Electric Forces
You are no doubt familiar with the slogan, “like charges repel and unlike charges attract.” This
section, and the following two sections, explores this phenomenon qualitatively. In a few sections we
shall discuss Coulomb’s Law, which extends this qualitative discussion and includes our quantitative
understanding of the magnitude of the electric force acting between two charged particles.
The main focus of this section is to explain how charged particles are transferred from one object
to another, so that the net charge of an object can change.
ˆ What is the law of conservation of electric charge?
ˆ How is charge transferred from one object to another?
– That is, how does it happen that two objects with certain net charges, possibly both
zero, end up with different net charges after some process takes place? Describe the kinds
of processes that can produce these results, both macroscopically and microscopically.
– Which kinds of charged particles are typically transferred between solids? How is this
typically accomplished?
– Can other types of charged particles be transferred between objects in other situations?
If so, describe some.
ˆ Electric forces are sometimes called electrostatic forces. What is the significance of this?
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Conductors and Insulators
When electric charge moves through an object, we call this process electrical conduction. An object
might be better or worse than another object at allowing electrical conduction to take place through
it. Objects through which electrical conduction can take place relatively easily are called electrical
conductors, or just conductors for short. Objects through which electrical conduction cannot take
place relatively easily are called electrical insulators.
There is no black-and-white distinction between electrical conductors and electrical insulators.
Some objects are better electrical conductors than others, some objects are better electrical insulators than others.
Nevertheless, as a rough generalization, we often categorize objects as good electrical conductors,
good electrical insulators, or in-between objects.
ˆ Which types of solid objects tend to be good electrical conductors? Give some examples.
ˆ Which types of solid objects tend to be good electrical insulators? Give some examples.
ˆ Explain (at a microscopic level) why good electrical conductors are good. How are good
insulators different from good conductors at a microscopic level?
ˆ If it’s not easy for electric charge to flow through a good insulator, how is it possible to
transfer electric charge between two insulators? Or is it impossible? Explain.
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Charging by Contact and by Induction
Charge transfer by contact is relatively simple to understand, but charge transfer by induction is
quite a bit more complex. It’s worthwhile understanding in detail how charge transfer by induction
takes place, because the same ideas recur in understanding the concept of charge polarization.
Many molecules and substances of interest in chemistry and biology are electrically polarized, so
these ideas come up often.
ˆ Explain how objects can be charged by contact.
ˆ Explain how objects can be charged by induction.
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Coulomb’s Law
Coulomb’s law describes the magnitude and direction of the force exerted by one charged particle
on another charged particle when each particle is at rest relative to the other. For this reason,
it can be said that Coulomb’s law describes electrostatic forces. If there is relative motion between two charges, the situation is quite a lot more complicated, and the force law is also more
complicated.
In some ways Coulomb’s law is analogous to Newton’s law of gravity; certainly the structure of
the two formulas is the same. They are both inverse-square laws, which means that the magnitude
of the force that one particle exerts on the other decreases proportionally to the square of the
distance between the two particles. However, the gravitational force between two particles is always
attractive, because there is only one type of mass. Because there are two types of charge, positive
and negative, the electrostatic force between two electrically charged particles is attractive if the
particles have unlike charges, and repulsive if the particles have like charges.
Another important difference is that according to Newton’s law of gravity the gravitational force
between two particles depends only on the distance between the particles, and is independent of the
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relative velocities of the particles, whereas Coulomb’s law is valid only when each charged particles
is at rest relative to the other. As we’ll see later in the course, if there is relative motion between
two charged particles, magnetic forces also arise, so the net force between the two charged particles
is not described just by Newton’s law of gravity and Coulomb’s law.
Another similarity between Coulomb’s law and Newton’s law of gravity is that in each case the
direction of the force is along the line joining the two particles. As we’ll see, magnetic forces are
quite strange by comparison.
The principle of superposition appears in this section, and it appears later in the context of electric
fields as well. A principle of superposition is present in linear theories, such as Newton’s law
of gravity, and Maxwell’s theory of electromagnetism, of which Coulomb’s law forms a part. In
nonlinear theories, such as Einstein’s theory of gravity (which we won’t be learning about in this
course), there is no principle of superposition, which makes nonlinear theories far more difficult to
work with.
ˆ What is Coulomb’s law? How can you use it?
ˆ What is the principle of superposition? How can it be used to help you calculate the force
exerted on a charged particle due to several other charged particles?
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Electric Fields
Coulomb’s law suffers from a difficulty that is shared with Newton’s law of gravity; the latter
difficulty was noticed by Newton’s contemporaries. Newton explained that each object in the
universe that has mass exerts a force on every other object that has mass, and wrote a formula
for the magnitude of the force. Newton’s contemporaries asked him for a mechanism for this force;
they were rooted in a mechanical view of the universe, and they found it difficult to conceive of
two objects exerting forces on each other while “at a distance.” Newton replied, in effect, that he
didn’t know how it happened.
In Newton’s theory of gravity, two objects that have mass attract each other with a gravitational
force simply by virtue of their mass. The strength of the force is proportional to the product of
the two masses, and inversely proportional to the square of the distance between the objects. This
means that if you move two objects apart, the gravitational force between them will decrease; for
example, if you double the distance between the objects, then the force between them will be less
by a factor of 1/4.
Newton’s contemporaries (c 1700) asked him to explain how in the world the attraction could take
place when the objects were very far apart; for example, the Sun and Earth are separated by about
150 million kilometres, but nevertheless there is an attractive gravitational force between them.
How can this be? One can understand that two colliding objects will exert a force on each other,
because they come into contact. How can two distant objects exert a force on each other when
they are so very far from touching?
“Hypotheses non fingo,” said Newton, which has been translated as, “I frame no hypotheses.” A
different person, with today’s sensibilities, might have said, “Look you fools, I’ve just revolutionized
modern science. Isn’t that enough for now? What more do you want from me? Why don’t you go
ahead and figure that one out, I think I’ve done enough for several lifetimes.”
In other words, I don’t know how two objects separated by astronomical distance can exert a force
on each other, and I don’t care to speculate about how, deep down, it happens. I can see the
effects, I can calculate the effects, and that ought to be enough for now.
This phenomenon, that two objects can exert a force on each other although separated by great
distances, came to be known as“action at a distance.” That is, the phrase “action at a distance” is
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a convenient summary of the state of affairs back then: “I know it happens, I can calculate it, but
I just can’t explain how it happens.”
The lack of a mechanical model for the phenomenon of gravity bothered scientists of the day,
but what could they do? Well, Michael Faraday, in the early 1800s, did do something about this
unhappy situation. He invented the field concept, which helped physicists to be a lot more satisfied
with their understanding of the phenomenon. And the concept proved so fruitful that it has become
the central paradigm in modern physics.
Faraday was motivated by the way iron filings align themselves near a magnet:
He imagined that the magnet extended some sort of “tentacles” of force throughout the space
around it, exerting some sort of tension. This “tension” in the space around the magnet is what
causes the iron filings to align.
Might not gravity act in the same sort of way? And electrical forces too? Perhaps all forces arise
in the same way: whatever the source of the force is (mass in the case of gravity, electric charge
in the case of electric forces, and electric currents in the case of magnetism), the source creates a
“tension” of some sort in space. Then an object that is able to respond to the force simply “feels”
the “tension” in the “tentacles” at its location.
The collection of tentacles under tension is what we call a field. So the collection of gravitational
tentacles is called a gravitational field, the collection of electric tentacles is called an electric field,
the collection of magnetic tentacles is called a magnetic field, and so on. Each force has its own
type of field.
So this is Faraday’s vision, and the solution to the action-at-a-distance conundrum: Each source
sets up a field in space, and then objects respond to the field present right at their locations.
Maxwell put mathematical muscle behind Faraday’s field concept (c 1865), and used his new equations of electrodynamics to make the startling conclusion that light is an electromagnetic phenomenon. Experiments by Hertz about 20 years later strongly supported this hypothesis, and by
now the evidence piled up in a century and a half is overwhelming.
But it wasn’t until about 1900, when vectors came into common use, that mathematicians created
a gadget called a vector field. Roughly speaking, a vector field is the assignment of an arrow at each
point in space. Mathematical vector fields are the current state of the art mathematical models
for physical force fields; at each point in space, the direction of the modelling arrow indicates
the direction of the force field, and the length of the arrow indicates the strength of the physical
field.
You can make a connection between the lines of force favoured by Faraday and our current understanding of vector fields by imagining a tiny motorcycle moving along each line of force. At each
point in space, the beam of the motorcycle’s headlight points in the direction of the vector field
arrow at that point. (For all you calculus lovers, we would say that at each point on the line of
force, the corresponding vector field arrow is tangent to the line of force.) Or you can imagine that
each iron filing in the magnetic field pattern is a little arrow.
It’s fashionable nowadays to eschew mechanical pictures/models of microscopic physics, such as
tentacles under “tension” to describe such things as forces. It’s much more popular to take a more
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abstract approach, and to think of a field in terms of its mathematical model: as the assignment of
an arrow (which amounts to a collection of numbers if you think of a vector as a list of components)
at each point of space.
However, it’s worth considering that it was Faraday’s intuition that came first, and Maxwell’s
equations came later. That is, creative scientists often think in terms of images of some sort
(Einstein reported that he often operated with his kinesthetic sense); once they have created a useful
idea, they often then express the idea more abstractly, mathematically, and often never report their
original images. Schools tend to teach the finished, mathematical product, emphasizing the logical
connections between the new idea and the existing scientific complex of ideas. The generative
aspect of the idea is almost never discussed in school; unfortunately, this means that not only are
high-school and undergraduate students typically not trained to be creative, but they do not even
become aware of the creative process in science either. Poincaré said that, “Science is no more a
collection of facts than a house is a heap of stones.” Unfortunately, mathematics and science are
typically taught in schools as if they were just heaps of stones.
The field perspective gives one a quite dynamic picture of space. Although space may be empty of
air far from the earth, nevertheless it is roiling with all kinds of stuff, and so is hardly a vacuum:
fields of all kinds are present. In very recent times (starting in the 1930s), with the development of
quantum field theory, our understanding of all fundamental particles and interactions is in terms
of fields. Every fundamental particle (electron, quark, neutrino, etc.) is considered to be an
excitation (a “quantum”) of a certain type of field. So besides the “force” fields (electric, magnetic,
gravitational, nuclear (strong and weak)), we also have “matter” fields. From the field perspective,
an electron is nothing more than an excitation of the electron field, an up quark is an excitation of
the up-quark field, and a photon is an excitation of the electromagnetic field.
Einstein’s theory of special relativity is both a theory and a meta-theory; the latter aspect means
that it specifies restrictions of various kinds that all valid physical theories must satisfy. One of
the restrictions is that information should not be transmitted (in any way whatsoever) at a speed
greater than a certain limiting speed, which happens to be the speed of light in vacuum. The
field concept is just the thing needed to make sure that this restriction is implemented in physical
theories.
For example, suppose that the Sun were to suddenly change its position; in the action-at-a-distance
approach, there is no natural way to explain why there should be a time delay in the transmission
of the effect to a distant object, such as earth. In the field approach, there is a natural way to
explain the time delay: the motion of the Sun would create ripples in the field, which would move
with a characteristic speed. Only when the ripples reached Earth would the earth “notice” that
there had been a change.
Ripples in electromagnetic fields have been created, observed, measured, studied, and understood
in tremendous detail. These ripples are called electromagnetic waves, which include visible light,
X-rays, microwaves, radio waves, and so on.
However, gravitational waves have not yet been observed, as they are expected to be extremely
weak. Devices have been constructed to attempt their observation, but no luck yet. Nevertheless,
the field concept has been very fruitful in describing gravitational phenomena as well.
A final note: When I was a child, there was a science fiction TV program called Lost in Space,
http://www.imdb.com/title/tt0058824/. Practically every episode of this program featured
what they called “force fields,” but which have nothing to do with the technical meaning of the word
as currently understood in modern physics (and as I’ve tried to describe here). Rather, what the
TV program called a force field manifested as an invisible barrier. I recall the actors in the program
walking along and then suddenly recoiling when they encountered one of these invisible barriers,
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then exclaiming, “We’ve just run into a force field!” If you’ve run into the science-fiction force field,
note that it does not correspond to our current picture of fields in physics or mathematics.
ˆ Can you calculate the magnitude of an electric field due to a single charged particle (a “point
charge”)?
ˆ Can you use the principle of superposition to calculate the net electric field due to several
charged particles?
ˆ Can you calculate the force acting on a charged particle due to an electric field?
ˆ Can you calculate the electric field inside a parallel-plate capacitor?
ˆ Can you visualize the electric field pattern for a single point charge?
ˆ Can you visualize the electric field pattern for a parallel-plate capacitor?
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Electric Field Lines
A popular way to visualize an electric field is to imagine that at every point of three-dimensional
space there exists an arrow; the length of the arrow represents the magnitude of the electric field at
this point, and the direction of the arrow indicates the direction of the electric field at this point.
It would be unhelpful to visualize the entire electric field in this way, because then all of the arrows
would merge together into one mess, so typically one draws only selected arrows. The idea is to
use enough arrows to give one a good sense for the field, but not so many that it’s confusing.
An alternative representation is to use electric field lines (also called lines of force in the old days)
to represent an electric field.
ˆ Can you visualize the electric field pattern for a single point charge?
ˆ Can you visualize the electric field pattern for an electric dipole?
ˆ Can you visualize the electric field pattern for a parallel-plate capacitor?
ˆ Do you understand that more complex configurations of charged particles produce more
complex electric field patterns?
ˆ Have you played with an applet to explore electric field patterns?
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The Electric Field Inside a Conductor; Electrostatic Shielding
When a good electrical conductor is placed in a region of space in which there is a static (i.e.,
constant in time) electric field, there are some interesting results (make sure you can follow the
reasoning in the textbook that explains why this occurs):
ˆ Any excess charge on the conductor is found at the surface.
ˆ The electric field inside the conductor is zero. If the conductor has a hollow space within it,
then the electric field is also zero within the hollow space; this is called electrostatic shielding.
ˆ The electric field just outside the surface of a conductor is perpendicular to the surface.
It takes a very short time after the conductor is placed in the region of space containing an electric field for charges to move and then “settle down” into equilibrium; typically this takes place
within a small fraction of a second. After this settling-down period, the properties listed above are
present.
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These conclusions are not valid if there are steady currents flowing, as might happen if there is a
static electric field driving a current around an electric circuit. Situations in which current flows
will be discussed later in the course.
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Gauss’s law
OMIT this section on Gauss’s law. We won’t discuss it in lectures or tutorials, and it won’t appear
on tests and exams.
Of course, if you are a physics major or otherwise enthusiastic, you’ll want to work through this
section eventually, maybe on a rainy day in the summer.
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