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Physics 122 Spring 2017 – Document #05: Cycle 1 Review Sheet
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PHYS 122: Cycle 1 Review Sheet – Part 1
January 18, 2017
From time to time, we will issue “Review Sheets” like this which delineate the main topics
that students will need to know.
What is the points of this course?
An understanding of electrical and magnetic phenomena is a critical component to the development and application of an enormous range of devices in medical technology, communications,
computing, media presentation, etc., etc., all of which are defining components of life in our modern society. A proper technical understanding and application of these devices require an understanding of the physics behind them and the methods of data interpretation that are characteristic
of the physical sciences. In medical technology, as well as in many other fields, we are required
to deal with a wide range of electronic, electrical, and electro-magnetic devices. To fully understand how these devices work, their strengths and limitations, we require a basic and fundamental
understanding of the physical laws of the universe that govern the interactions in electricity and
magnetism.
Electrostatics
In order to understand the function and application of any electric or electro-magnetic device,
we must start with the most fundamental electric interaction and the concept of electric charge.
Everything is built upon this. The sub-discipline of electrostatics means the physics of systems
where the charges in the system are not moving. We begin with Coulomb’s law which tells us how
two simple point charges interact. We showed how Coulomb’s Law can be applied to a system of
several point charges. We invented the concept of the Electric Field which allows us to specify the
electric force (per unit test charge) at any point in space due to a configuration of point charges.
We also introduced the concept of the Electric Potential (i.e. “voltage”) that provides a means of
describing the potential energy (per unit test charge) due to the electric field. Next, we will
introduced the concept of electric current – the motion of charge that is constrained along some
path (like a wire). All of this sets the stage for dealing with our first application of electrostatics:
simple circuits.
When we talk about Electrostatics what we mean is that we are considering a problem where
none of the Central aspect of the problem are (explicitly) time-varying. In other words, we consider
an arrangement of charges where the charges are either pinned into place or are somehow said to
be in equilibrium. Usually in these circumstances a critical calculation is determining the value of
the electric field at all positions. Indeed the general statement of an “electrostatics problem” is a
problem that boils down to “Given the following arrangement of charge, calculate the electric field
everywhere.”
Problems where we must explicitly consider the motion of charges in a system are called electrodynamic problems and are much more difficult to solve.
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The Concept of Charge
The idea of charge is both familiar and mysterious. Everything we do in this course is based
on the property called charge. One important idea to keep in mind: In most materials, which
are made of atoms, there are equal numbers of positive and negative “charged particles” (namely
protons and electrons). In real macroscopic objects that have a non-zero electric charge (be it
positive or negative), this charge is based on the small difference between the number of electrons
and the number of protons in the object. In a negatively charged object, there is a small excess
of electrons. In a positively charged object, there is a small deficit of electrons (resulting in a net
positive charge). We now know that the transfer of charge from one object to another is almost
entirely due to the transfer of (negative) electrons. However, from an electrostatics point of view,
saying that a negative charge −Q was transferred from Body A to Body B is entirely equivalent to
saying that a positive charge +Q was transferred from Body B to Body A. From the point of view
of electrostatics, these are equivalent statements.
Note that this picture implies that charge is neither created no destroyed and that any time a
positive charge is extracted into one location, and equal amount of negative charge must result at
some other location.
Coulomb’s Force Law:
We start with Coulomb’s Law which tells us how two bodies with charges qa and qb interact:
|q||Q|
1
|F~elec | =
4πǫ0
r2
1
where F~elec is the force due to electric charge between any two bodies, k is a constant, ( 4πǫ
=
0
9
2
2
9 × 10 N·m /C in SI units where the charge is given in Coulombs), and r is the distance between
the two charged bodies. Note that the force is radial (i.e. it lines up in the direction between the
two bodies). To determine whether F~elec attracts or repels you need to consider the signs of the
two charges, q and Q. If we define the unit vector r̂ pointing from charge Q to charge q then we
can write the force on charge q in vector form:
qQ
1
r̂
F~qQ ≡ F~(elec on q due to Q) =
4πǫ0 r2
Note that the vectors ~r and r̂ point from charge Q to charge q.
Note now that the sign of force indicates the direction. If the value is positive, the force is
repulsive (a push along r̂ in the direction from Q to q). If the force is negative, this indicates an
attractive force (a pull opposite the direction r̂.) Note that since any forces will add (by superposition) we can calculate the net electric force on any one charge due to a collection of point charges
by simply summing up the forces, one pair at a time.
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The Electric Field:
We introduce (by invention) the Electric Field for two reasons:
• as a convenient mathematical device for keeping track of the influence of electrical charges
at any location in space, and
• as a means for explaining the somewhat mysterious “spooky action at a distance” that describes the electric force. By considering the field, we explain the interaction as between
particles and fields that fill all of space. In other words, we say that the first charge (called
the “source charge”) creates the field and the second charge (called the “test charge”) experiences a force due to that field at that location.
We define the Electric Field as the net electric force that would be experience by any small
hypothetical test charge qtest per unit test charge. Since the force is a vector, and since we can
calculate such a (hypothetical) force at any position in space, we call the Electric Field a vector
field.
We use the word “field” to indicate that we are talking about something that is a function of 3dimensional position in space. In other words, the electric field is defined as a vector that depends
~ r which we decode as “The electric Field E
~ as a function of the vector
on position. We say E(~
position ~r. If we use the variable ~r to represent the position vector then the electric field is a vector
function of a vector position.
Coulomb’s Law: The Electric Field due to a Point Charge
For a point charge Q the vector field as a function of position ~r relative to the point charge is:
1 Q
r̂
4πǫ0 r2
where now r is the length of the position vector ~r and r̂ is the unit vector that points radially out
from the point particle.
~ r) =
E(~
We note that this expression is just a re-articulation of Coulomb’s Law. Indeed, for the remainder of the course we will consider the expression for the electric field as what we mean by
Coulomb’s Law.
Continues next page....
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The Principle of Superposition:
The magnitude of the Electric Field calculated at Position P resulting from a single point
charge Q is given by Coulomb’s Law as follows:
1
Q
~ rP Q ) =
E(~
r̂ P Q
4πǫ0 rP2 Q
~ rP Q ) mean the electric field ON
This expression is what we mean by Coulomb’s Law. Here E(~
the Position at Point P DUE to the point charge Q. The vector ~rP Q points from the charge Q to the
position P . In other words ~rAB means the position vector that arrives ON Point P due to originating
Point Q. We will always stick with the notation of two subscripts where the first subscript is the
location where the force (or the field) is evaluated and the second subscript is where the charge
that is causing/creating the force/field is located.
Note that in order to calculate the field at Point P we need to ignore any point charge that
is located exactly at position P . In other words, when we calculate a force on a charge at some
position, we always calculate the electric field due to all of the other charges in the problem,
ignoring the field due to the point charge at that location.
The direction of the field due to any point charge depends upon the sign of the points source
charge. Again, since any forces will add in accordance with the Principle of Superposition we
can calculate the net Electric Field at any point as equal to the vector sum of the individual Electric
Fields due to each of the point charge in the system: For example the total electric field at some
point P is given by
~P = E
~P1 + E
~P2 + E
~ P 3 ... + E
~PN
E
~ P i is the field at point P due to the i-th point charge:
where E
Qi
1
~Pi =
r̂ P i
E
4πǫ0 rP2 i
Where ~rP i is the vector that points from the i-th charge Qi to the position at point P .
You should be able to solve any problem where you are asked to calculate the electric field
at any point in space given any distribution of point charges. When you do this, be sure to take
advantage of any symmetry that you can exploit to simplify problems. For example, if you need
to calculate the electric field at some point P , and there is a positive point source +q at a distance
d to the left of P and a second positive point source +q a distance d to the right of P , then by
symmetry the contributions to the net electric field at point P due to these two charges will cancel
each other out to zero. You do not have to calculate explicitly the contributions to the field due to
these two charges if you can instead invoke symmetry.
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Electric Field Lines:
We can graphically represent the electric field two ways:
• As a set of vectors spread across all of space (the magnitude and direction of each vector
defined at each point), or
• As a set of field lines, representing the direction of the net electric force. The relative density
of the field lines scales as the magnitude (strength) of the field.
Here are some rules for Electric Field Lines that we can infer from their definition:
• Field Lines start on positive charges.
• Field Lines stop on negative charges.
• Field Lines never cross.
Force on a Point Charge due to the Electric Field:
Remember that a charge q is placed (embedded within) a given Electric Field will experience
a net electric force as given by:
~
F~elec = q E
~ is the electric field at the position of charge q. Sometimes we are just given the value
where E
~ Sometimes we need to calculate it. In this case, we calculate the field due to all of the
of E.
other charges in the system not including the charge q. It’s important to remember that when you
calculate the electric field that will influence any given charge, you do not not calculate the field
due to that particular charge. A point charge feels the field due to the other charges. It does not
feel it’s own field.
We typically call the charge that experiences the forces the “test” charge and we call all of the
other charges that create the electric field the “source” charges. In this nomenclature, the “source”
charges create the electric field, and then the “test” charge experiences a force due to the electric
field.
Continues next page....
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Insulators vs. Conductors:
When we expand our concept of electrostatics from the simple idea of ideal point charges to
actual physical materials we have to take into account the (sometimes very complex) interactions
between the charges in a given material and the fields. We start with two “ideal” cases corresponding to the extremes in the idea of charge mobility:
• An Ideal Insulator is a materials where charges in that material cannot and do not move.
Good examples of materials that act like Ideal Insulators include glass, most plastics, air,
etc. If a charge is places on or in or withing an object that is made of insulating materials,
that charge will generally stay put.
• An Ideal Conductor on the other hand corresponds to a material where charges are very free
to move quickly and can re-arrange themselves in response to any and all applied electric
fields. Most metals, slightly impure water, and the planet earth as a whole can be regarded
as nearly ideal conductors.
Note that although we usually treat materials as either “ideal” conductors or “ideal” insulators,
in fact most materials can be one or the other depending on the circumstances, in particular the
temperature of the material and the strength of the applied electric field. For example, the air is
generally a very good insulator, unless the local electric field gets large enough to begin to ionize
the molecules, in which can it can rather dramatically become a good conductor. This happens for
example during lightning strikes.
Remember this important idea: If I tell you in a problem where the charge is located explicitly
and/or I state/imply that the materials involved are insulators, then you generally do not need to
worry about charges moving and you can calculate the electric field explicitly from the initial
charge distributions. However, if I tell you that there is a conductor somewhere in the problem
you may have to consider the impact of the conductor on the answer, and you may or may not be
able to figure out where all of the charges located because the conductor will act so as to rearrange
charges inside the conductor and on the surface of the conductor. This will have an impact on the
electric field. This is true even if the conductor is electrically neutral on average. Specifically:
• In any electrostatic problem, for any position that is within the material of any ideal conductor the electric field at that position is precisely and always zero. (If it were not, charge would
flow until it re-arranged itself so that the field was zero inside the conductor). Memorize this
critically important fact. The electric field inside the material of any ideal conductor is always exactly zero. If I give you a solid gold ball and ask you to calculate the field anywhere
since the surface of that ball, the answer is zero.
• In any static problem, any excess charge that is placed on a conductor always appears on the
outside surface of the conductor (if it did not, it would create a field inside the conductor
which would violate the previous condition).
• In any static problem, the electric field is always perpendicular to the surface of the conductor at the surface. (If it were not, charge would flow along the surface). In other words
Electric field lines always enter/leave the surface of a conductor at a right angle relative
to that surface.
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• In any static problem, the surface of any conductor represents an equipotential. That is to
say that the voltage that is measured on the surface of a conductor is the same at any point
on that surface. See more about voltage further on.
• The presence of charges outside but in the vicinity of a neutral conductor will induce charge
on the surface of the conductor. What this means is that the charges move around so as
to ensure that there is no electric field inside the conductor. Usually this involves negative
charges in the conductor moving closer to nearby positive charges outside the conductor, and
vice versa. As a result, a positive charge will attract a neutral conductor. A negative charge
will also attract a neutral conductor. Be sure you understand this.
• The “ground” is a term that represent the earth as a whole, a huge conductor that is so large
that any charge taken or given has negligible effect on the net neutrality of the earth – the net
charge is spread over a very large area. In other words, we treat the ground as a conductor
that always maintains zero net electric charge.
Continues next page....
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Continuous Charge Distributions:
We can generalize from discrete point charges to a continuous source of charge in one, two, or
three dimensions: We define the concepts of charge density as calculated for lines, surfaces, and
volumes:
Dimensions
Term
Symbol Units
Example
0-D
Charge
q
C
point charge
1-D
Linear Charge Density
λ
C/m
line charge
2
2-D
Surface Charge Density
σ
C/m sheet charge
3-D
Charge Density
ρ
C/m3
“potato”
Calculating the Electric Field for Continuous
Sources: Direct Integration with Coulomb’s Law
Although we have not worked problems to calculate the electric field due to continuous sources,
you should be familiar with the general approach to doing this: namely we break the continuous
source into a large number of infinitesimal charge bits, dq, and we integrate over all space the
contribution to the electric field due to each of these little bits. In the language of calculus:
~ r) =
dE(~
and so
1 dq
r̂
4πǫ0 r2
1
~ r) =
E(~
dE =
4πǫ0
allspace
Z
~′
Z
allspace
dq ′ ′
r̂
r′2
where the “primed” coordinates correspond to the quantities that are integrated over.
Note that the vector ~r′ points from the position of the charge element dq ′ to the position where
the field is calculated ~r. Note that as a rule we have to break down the element of charge so as
to parameterize this in terms of coordinates of integration. For example, in the case of a threedimensional charge density ρ we could write:
dq = ρ dV = ρ dx dy dz
Continues next page....
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The Electric Field due to Special Charge Distributions
In class we indicated that by doing direct integration of Coulomb’s Law we can in principle
derive the Electric Field for certain familiar arrangements. Specifically:
• The Electric Field due to a Point Charge Q (or any spherically symmetric compact charge
distribution with net charge Q) is given by:
Q
1
~
r̂
E=
4πǫ0 r2
where r̂ points radially away from the point charge Q. This is just Coulomb’s Law.
• The Electric Field for a Line Charge of infinite length, with linear charge density λ is given
by:
1
λ
~
E=
r̂
2πǫ0 r
where r̂ points radially away from the line.
• The Electric Field for a Sheet Charge (a plane of charge extending infinitely) with surface
charge density σ:
~ = σ r̂
E
2ǫ0
where r̂ points directly perpendicular away from the plane.
All students should commit to memory these three expressions for the electric field for
these three cases: (1) the field due to a Point Charge of value Q, (2) the field due to a Line
Charge of linear charge density λ, and (3) the field due to a Sheet Charge of surface charge
density σ.
Continues next page....
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The Electric Potential (also called “Voltage”)
We started a discussion of the topic of Electric Potential, which I tend to simply call “voltage”.
The electric potential is defined as the potential energy per unit positive test charge. We can
calculate the electric potential by calculating the path integral corresponding to the (negative) work
done in moving a charge through a field:
Z
~
~ · dℓ
∆V = −
E
path
Since the electric field is conservative this integral only depends on the initial and final positions. In other words, the electric potential is position-dependent only and also path-independent.
Like Potential Energy, the electric potential is always defined a a change from one point to the
next. The choice of a
bf reference point (typically to define zero voltage) is arbitrary. For convenience we often chose a
reference point to represent an electric potential of zero volts correspond to a special position. For
example:
• we sometimes define zero volts as where we find most negative charge, or
• we sometimes define zero volts as the electrically neutral “ground”, or
• we sometimes define zero volts as the electric potential corresponding to the point at infinity.
This is the usual choice for potentials associated with discrete point charges.
More about Voltage
Unfortunately, the terminology is much more confusing than it needs to be. Remember! All
these words all mean (pretty much) the same thing:
• Electric Potential
• Potential
• Potential Difference
• Voltage
• Voltage Drop
• “emf” (I really do not like this term! Better would be “induced voltage”)
In lecture tend to use the term “voltage” instead of Electric Potential because I find that otherwise I confuse Electric Potential with Potential Energy. Yeesh.
The hardest part of understanding voltage is understanding that it is implicitly (or sometimes
explicitly) measured with respect to some reference point. You can’t just have voltage all by itself.
Voltage is always measure with respect to the reference point. So the reference point needs to be
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clearly defined. The choice of the reference point is arbitrary but there is usually a natural best
choice for each problem to make it easy to solve. Always ask yourself, “voltage with respect to
what”.
In circuits we often talk about the “voltage across” a component such as resistor or capacitor.
This is the voltage on one side of the component with respect to the voltage on the other side.
The Relationship between Electric Field and Voltage:
We can measure the voltage difference between two points by calculating the path integral:
Z
~
~ · dℓ
∆V = −
E
path
In many problems, we chose a specific reference point where we (usually) select a particular
value of the voltage. Then other voltages are defined relative to this reference value:
V (~r) − V (~rref ) = −
Z
~
r ′ =~
r
~′
~ r′ ) · dr
E(~
~
r ′ =~
rref
~ ′ so that this path integral reduces
~ is aligned with dr
For some interesting field geometries, E
to a simple regular integral:
Z r
E(r′ ) dr′
V (r) − V (rref ) = −
rref
where E may depend upon the position.
By the way, don’t get hung up over the negative sign in the definition of voltage. Just ask
yourself this: will a positive test charge do positive work (i.e, move in opposition to the direction
of force) as it is displace along the path from initial to final position? If positive work must be
done, the voltage at the final position will be positive relative to the starting reference point.
Equipotential Surfaces:
Remember that the voltage is a scalar field while the electric field is a vector field.
If we consider the set of points in space that represent a particular value of the voltage, then
these points make a surface called an “equipotential surface”. We showed in class and in lecture
that equipotential surfaces are found to be perpendicular to electric field lines. This makes sense
because moving a charge perpendicular to electric field lines does no work and so the voltage does
not change.
Voltage between two charged plates:
If we have two uniformly charged plates one positive, one negative, the field between the two
plates is uniform. Calling this field E0 we can say:
σ
σ
~
=
|E| ≡ E0 = 2
2ǫ0
ǫ0
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We can calculate the voltage as a function of position z between the two plates (where z is the
distance from the negative plate in the direction of the positive plate:
Z z
~ ′ ) · d~z′
E(z
V (z) − V (zref ) = −
zref
V (z) − 0 = −
Z
z
−E0 dz ′
0
σ
V (z) = E0 z =
z
ǫ0
Here the negative sign inside the integral comes from the dot product and indicates that moving
in opposition to the electric field vector.
Voltage for a point source:
If we want to calculate the electric potential due to a point source, we calculate this with respect
to the point at infinity:
Z r
1 q
Edr′ =
V =−
4πǫ0 r
∞
The electric potential (voltage) at any point due to several point charges is just the algebraic
sum of the potential for each charge at that point.
Potential Energy for point sources:
We can calculate the potential energy required to bring a point charge to a position where the
voltage is given:
U = qV
For two point charges this gives:
1 q1 q2
4πǫ0 r
For more charges, the potential energy in the total system is the potential energy associated
with each pair of charges.
U=