1
PH 213 Review Sheet
This term we have rounded off our exhilarating (and exhausting) study of
classical physics with our look at electromagnetism. We began by looking at
electrostatics, the E, and then moved on to electrodynamics, the B. While learning of
these topics in general cases, we intermittently made adaptations in order to discuss the
specific applications towards a study of electric circuits. Finally, lectures briefly touched
on the combination of E and B through Maxwell’s Equations. Instead of taking a
chronological approach in the design of this review sheet, corresponding to the order of
lesson plans in the syllabus, I will follow what seems to me the natural, uninterrupted
flow form electrostatics to electrodynamics and the culmination of the electromagnetic
wave. Circuit theory will then be addressed as a whole body at the end of this review.
I) Electrostatics
We began this term by defining a new physical quantity, electric charge. Just as
more familiar objects from previous terms had the physical quantity of mass, our objects
this term have charges, which will are dealt with similarly. We notice that objects of
opposite (similar) charges will attract (repel), meaning that the force of interaction will be
dependent upon the objects’ charges. Also, we find that as the objects are moved closer
together (further apart) this force increases (decreases), meaning some inverse
proportionality between the force and distance. As can be measured in a lab
environment, we find that F goes like 1/r2. In combination with the charge dependence
and some constant of proportionality, k, we get Coulomb’s Law for the force on particle 1
from 2

QQ
F12  k 1 2 2 r
r12
where
k
1
4 o
.
This electrostatic force acts along the line running directly between the two particles,
with its direction determined by the attractive or repulsive nature of the two charges. In
the case of multiple charges, we simply use Coulomb’s Law in combination with
Newton’s Second Law to find the total force on a particle. Notice that Coulomb’s Law is
formally identical to Newton’s Law of Universal Gravitation as studied in PH 211. The
force due to gravity between two objects had the same dependence upon the objects’
mass as this force has on charge, and both forces vary as 1/r2.
From the discussion of electric forces, we construct a mathematical tool called the
electric field,

 F12
Q
E
 k 2 r ,
Q1
r
where we have used Q1 as a testy particle that can feel the E from the second particle, but
has a small enough charge as not to distort the E. As in the case for the Coulomb force,
the electric field from multiple particles is the superposition of each particle’s individual
electric field. This sum is achieved either as a simple summation in the discrete case, or
as an integration in the continuous case
PH 213 (Spring 2004)
Oregon State University
Vince Rossi
2

E
kr
Qi
2

r  E 
i
i
dQ
kr
2
r .
Some conventions and requirements regarding electric fields must be kept in
mind:
 Electric field lines begin on the positive charged object and terminate on the
negative object, where the number of field lines drawn is proportional to the
magnitude of the electric field
 Field lines are closer together where the electric field is the greatest
 The number of field lines crossing a unit of area perpendicular to the field is
proportional the magnitude of the electric field
 E inside a conductor in the static case is zero
 Any net charge on a conductor distributes itself on the surface (a direct result of
the previous requirement)
 E outside of a conductor is always perpendicular to its surface; otherwise, given
some parallel orientation of the E, charges would redistribute themselves on the
surface until the perpendicular field lines result
Now that we have confirmed that there are electric fields and forces due to charge
distributions, we need to determine how we can find exactly what the total charge is, the
charge distribution, or the electric field given any of those other quantities. We define
the electric flux as the amount of the electric field passing through some unit of area.
Assuming that E is uniform, we define the electric
flux as

 E  E  A  EA  EA cos  .
By Gauss’s Law, we say that the electric flux through a closed surface (Gaussian surface)
is equivalent to the charge enclosed divided by the permitivity
E 

  Q
E  dA  enc .
o
In solving this problem , I recommend solving each half of the equation independently
and then setting the two solutions equal to solve algebraically for the desired term. Of
course given some charge density you would need convert it to total charge as determined
by the object’s symmetry. Typically, the gaussian surface of choice will be cylindrical or
spherical, depending upon the object’s symmetry. Recall how to integrate over threespace in cylindrical and spherical coordinates. Typically, r will be the only component
for which the integral is dependent. For this case, recall how to use the integral over the
remaining two coordinates. For example,
2

0
0
  sin dd  4
in spherical coordinates where the transformation from Cartesian to spherical differentials
is
dxdydz  r 2 sin drdd .
Since the electrostatic force is conservative (i.e. dependence upon position is 1/r2
and path independent) we can define an electrostatic potential energy. This result can
also be seen again as analogous to the gravitational force and gravitational potential
energy. If we place a positive charge in a between an upper plate of positive charge and a
lower plate of negative charge, we will watch it “fall” towards the lower plate. The
positively charged particle has a lower potential energy at the lower plate due to the
PH 213 (Spring 2004)
Oregon State University
Vince Rossi
3
repulsive and attractive forces acting upon it—objects always move towards a position of
minimum potential unless confined otherwise. This picture looks no different than that of
a pinecone falling form a pine tree to the ground, and we know the pinecone has potential
energy, so by analogy the positive particle must as well. The electrostatic potential
energy is defined as
U 12  k
Q1Q2
r12
in a two particle system, recalling that energy is not a vector quantity and therefore lacks
directionality. We define similarly the electric potential
Vba  k
Q
,
r
which is to E what U is to F. Again, the electric potential due to many charges, or a
continuous charge distribution, is found through summation and integration, respectively.
Integration is again with respect to dQ, where some charge density may need to be
substituted instead, changing the integration to be with respect to some spatial coordinate.
We can move between F, U, E, and V as follows


QQ
Q
F  k 1 2 2 r   /  Q2  E  k 21 r
r12
r12
  /  r12
U k
Q1Q2
r12
  /  r12
  /  Q2  V  k
Q1
r12
where we can also determine the electric field directly from the negative gradient of the
electric potential


E  V
where in Cartesian coordinates

d
d
d

x 
y 
z .
dx
dy
dz
II) Electrodynamics
We then begin to consider the situation of moving charges, where we define an
electric current as the change in charge with respect to time. More specifically, electric
current is a measure of the amount of charge to pass by some point over some time
interval
I
dQ
.
dt
That charges are moving requires there be some electric potential causing them to move
towards the lower potential. We can no longer say the E inside of a conductor is zero for
the dynamic case, as we had for the static case. Much as an electric field was caused by
stationary charges, the magnetic field, B, is caused by currents and acts only on moving
charges. The magnetic force acting on a moving charge, or current in a wire, is
dependent upon the particle’s velocity, or the current, as well as the length of interaction
and the magnitude and direction of the magnetic field. For the case of current in a wire
 


F  Il  B  F 
and for moving particles
PH 213 (Spring 2004)


 Idl  B

 
F  qv  B ,
Oregon State University
Vince Rossi
4
where the directions of the magnetic field, magnetic force, and current/velocity are
mutually orthogonal (as defined by the cross product).
We are given two tools for determining the magnetic field. For cases with high
symmetry, we can use a Ampere’s Law, which is analogous to Gauss’s Law in
electrostatics. Ampere’s Law says that the magnetic flux through some closed loop is
equivalent to the product of permeability with the enclosed current
 
 B  dl  o I enc .
Again, I recommend solving each side individually and then putting them back together
to find a particular solution. Typically, you will pull B out of the integral after making
some symmetry argument, where integration with respect do dl is just the length of the
closed loop (i.e. the circumference in the case of a circular loop). In particular, remember
the symmetry of the ideal solenoid and of the toroid and how their B’s look. In the
situation where no particular symmetry lends to the problem, we then use the Biot-Savart
Law

 o I dl  r
o I dl sin 
dB 
 dB 
2
4 r
4
r2
where B is again mutually orthogonal with the direction of the current and the radial
component marking the distance between the current distribution and the point of
interest. To find the total B, we integrate both sides, which is essentially saying we find
the B due to each infinitesimal piece of wire through which the current runs, and sum
them all together. This is the superposition of B due to each piece of the wire.
The magnetic flux is defined
 
B  B A
for a uniform B, or
B 


 B  dA
for a nonuniform B or varying A. A change in the flux through some closed loop causes
and induced EMF in the wire loop. Given a loop of N turns,
  N
d B
,
dt
where the magnetic flux can change either to a changing B, changing the area of the loop
in the field, or by changing the orientation of the loop in B. In general, Faraday’s Law is
written


 E  dl  
d B
.
dt
III) Maxwell’s Equations and Electromagnetic Waves
As should be apparent by this last equation, we can relate magnetic fields to
electric fields. More specifically, we can claim that a change in a magnetic field creates
an electric field. We will find that the converse can be said as well. If we define a
displacement current
ID 
d E
dQ
dE
dEA
 o A
 o
 o
,
dt
dt
dt
dt
we can write the generalized form of Ampere’s Law equation incorporating both
displacement and conduction currents

PH 213 (Spring 2004)
 
d E
B  dl   o I enc   o  o
  o ( I   o I D ) enc .
dt
Oregon State University
(1)
Vince Rossi
5
If we try to use Gauss’s Law for magnetism as we did for electric fields, we would find
that there is never a net enclosed magnetic field


 B  dA  0 .
(2)
This is due to the fact that magnetic monopoles do not exist—all field lines originate and
terminate on the magnetic object, such that all lines leaving the Gaussian surface will
always reenter it. Equations (1) and (2), along with the general form of Faraday’s Law


 E  dl  
d B
,
dt
(3)
and Coulomb’s Law


 E  dA 
Qenc
o
,
(4)
comprise the set of Maxwell’s Equations, defining the relationships between E and B.
This coupled representation of E and B is an electromagnetic wave—light. Some
properties of electromagnetic waves are as follows:
 E, B, and v (the velocity at which the wave propagates) are all three mutually
orthogonal
 E and B alternate in direction, therefore they alternate between some maximum
and minimum amplitudes, passing through zero amplitude
 E and B are in phase such that E = 0 when B = 0
 For r relatively large, field lines are relatively flat, i.e. plane waves (this is what
allowed us to work with ray diagrams in our study of geometric optics in PH 212
in place of dealing with the wave nature of light)
 Electromagnetic waves are transverse waves of fields not matter (i.e. not
mechanical waves) and can therefore propagate in vacuum
 Accelerating electric charges give rise to EM waves
Much more can be said on this topic, nut I will have to leave you to study further on your
own. Be aware that Maxwell’s equations change from those given above (1-4) in
vacuum. Perhaps review the mathematical forms of waves studied in PH 212 so that you
can put E and B in such sinusoidal forms.
IV) Circuit Theory
As I have tried to be complete in my review of the above sections, I must
unfortunately be brief in the review of circuit theory. Know Ohm’s Law (fundamental to
circuit analysis) and how to apply Kirchoff’s Junction and Loop rules. The foundations
of Kirchoff’s rules are the conservation of charge and conservation of energy. When
dealing with resistors, recall that their function is to dissipate energy, therefore as current
passes through them, there will be a voltage drop. When an uncharged capacitor is in
parallel with a resistor in a circuit that is just turned on, all of the current will flow to the
capacitor since its impedance increases with charge. When a capacitor is uncharged, it
acts as a short in the circuit. When a capacitor is fully charged though, it will act as an
open circuit, such that no current will pass through it. Recall that a capacitor stores
energy in its electric field and that an inductor stores energy in its magnetic field. When
we have an ideal LC circuit, the current alternates and the energy stored in the circuit
oscillates between the capacitor and the inductor, being a maximum in one when it is zero
in the other. However, in real circuits, there is resistance and the circuit will be damped.
My last bit of advice on circuits would be to study with an E-Fundies graduate if you’re
PH 213 (Spring 2004)
Oregon State University
Vince Rossi
6
not one yourself, and to use whatever method of circuit analysis you are most
comfortable with and understand.
************************************************************************
It has been my pleasure working with you this year, and I hope your experience in
physics has been invigorating and inspiring, despite its overwhelming workload. If you
have completed your physics requirements, but are interested in learning more, consider
taking the modern physics course (PH 314), which deals with special relativity and
quantum mechanics. If your interest is not great enough to inspire you to take more
courses or take on more work, the knowledge you have obtained in this sequence should
be more than adequate for you to understand any of the popular books on physics that
you can find at the local bookstore or library. Congratulations on your completion of one
of the most difficult introductory sequences here at OSU, and good luck on the final.
PH 213 (Spring 2004)
Oregon State University
Vince Rossi