phy

1. I have decided on a change in the grading policy. I did not want to take up class time with extra
quizzes so we only had 4 of the planned 6-7 quizzes. I feel that having 10% of the grade based
on just 4 (really 2 since I drop the 2 lowest) quiz grades is too much. Therefore the homework
will count for 30% of the grade instead of 25% and the quizzes will only count for 5%
I will be skipping Chapters 30 and 31 so that I may cover electromagnetic waves and optics as
these will be covered on the MCAT. In Chapters 32 -34, I will be focusing only on those topics
covered in the MCAT.
I have put some practice problems on line.
I will not be here next week. The last lecture will be given by Professor Rasmussen. However,
you may email me with questions.
The final is Dec 13 at 2 PM. Remember that it will be 40% comprehensive and 60% on material
covered since the last exam. You will be allowed two 8 ½ X 11 sheets of paper for notes (both
sides) and it is open book.
Your grades will be available by Dec 16. You may email me or come by my office in WSTC if
you want to know your grade on your final. I leave Dec 18, so you must contact me by Dec 16
if you want to talk about your grade.
I will have a review session in FN 2.212 the day before the final starting at noon and going until
????. You need to have studied for the exam prior to the session for it to do any good as you
need to know what you don’t understand so I can review it.

2. Eddy Currents
When magnetic field is on, currents (eddy currents) are induced in
conductors so that the pendulum slows down or stops

4. Displacement Current
0 encl
B dl I

 

0
0 0
( ) E
A
q C Ed EA
d

  
    
0
E
c
d
dq
i
dt dt


 
0 displacement current
E
d
d
i
dt


 

5. t
c
c










E
j
B
j
B
0
2
0
2


1
2
0
0
c

 
“displacement
current” of the
electric field
flux as opposed
to conduction
current
!
fields!
magnetic
g
circulatin
to
rise
give
too
fields
electric
changing
but
currents
only
Not
:
tion
generaliza
s
Maxwell'
es
consequenc
reaching
-
far
with
result
a
in
g
culminatin
theory
of
y
consistenc
the
of
analysis
oretical
purely the
a
of
example
great
a
is
equations
Maxwell
the
of
one
tion to
generaliza
Its
currents
steady
-
non
to
applied
once
ory
contradict
becomes
:
currents
varying
-
for time
Law
s
Ampere'
of
Inadequacy
0
0
0
0
t
I
d
I
d
E














s
B
s
B

6. The Reality of Displacement Current
D c
i i

2
0 2
0
2
0
2
inside the capacitor
2
outside capacitor
2
D
c
c
r
B dl rB i
R
r
B i
R
B i
r
 




  
 
 

Field in the region outside of the capacitor exists
as if the wire were continuous within the capacitor

7. 0
2 2
0
1
0
t
t
c c



 

  


  

 
E
B
E
j E
B
B
 
 
0
0 0
0
S
c
S
Q
d
dl d
t
dl I d
t
B d

 


 

   


 
   
 

 
 

 
 

E A
E B A
B E A
A
Maxwell equations in all their consistency and beauty
differential form integral form 1
2
0
0
c

 
Gauss’s Law for E
Gauss’s Law for B
Ampere’s Law
Faraday’s Law
How are these equivalent?

8. Special cases of the more general Stokes' theorem
The Divergence theorem relates the flow (flux) of a vector field through a surface to
the behavior of the vector field inside the surface.
More precisely: the outward flux of a vector field through a closed surface is equal to
the volume integral of the divergence of the region inside the surface.
the sum of all sources minus the sum of all sinks gives the net flow out of a
region.
The left side is a volume integral over the volume V, the right side is the surface integral
over the boundary of the volume V.
The Curl Theorem relates the surface integral of the curl of a vector field over a surface
S to the line integral of the vector field over its boundary,
Use Divergence and Curl Theorems
The left side is a surface integral and the right side is a line integral

9. Equivalence of integral and differential forms of Gauss’s law for electric fields
If  is the charge density (C/m3), the total charge in a volume is the integral over that
volume of 
But from the divergence theorem:
It is often written as where D=ε0E

10. Q
j
 
0
:
with time
changes
inside
charge
the
surface,
closed
rough the
current th
the
of
Because
on
conservati
Charge





















j
j
n
j
t
dV
dA
dV
t
I
t
Q


Something very fundamental

11.   !
!
!
0
1
1
0
)
(
:
y
consistenc
restores
on
modificati
s
Maxwell'
??
t
s
Where'
-
1
0
)
(
:
on
conservati
charge
with
nt
inconsiste
be
would
law
s
Ampere'
Just
0
0
0
2
0
2


















































t
t
t
c
c






j
E
j
E
j
B
j
B
0




 E

12. So the next (?) time you see a shirt that looks like this:
You will know what it means!

13. 0
0 0 0
0
t
t


  
 

  


  

 
E
B
E
E
B j
B
 
 
0
0 0
0
S
c
S
Q
d
dl d
t
dl I d
t
B d

 


 

   


 
   
 

 
 

 
 

E A
E B A
B E A
A
Maxwell equations and electromagnetic waves
differential form integral form 1
2
0
0
c

 
Gauss’s Law for E
Gauss’s Law for B
Ampere’s Law
Faraday’s Law

14. Electromagnetic disturbances in free space
With a complete set of Maxwell equations, a
remarkable new phenomenon occurs:
Fields can leave the sources and travel alone
through space.
The bundle of electric and magnetic fields
maintains itself:
If B were to disappear, this would produce E; if
E tries to go away, this would create B.
So they propagate onward in space.
0
0
2

















B
E
B
B
E
E
t
c
t
No charges and no currents!

15. Generating Electromagnetic Radiation
Heinrich Hertz was the first person to produce
electromagnetic waves intentionally in the lab
Oscillating charges in the LC circuit were sources of electromagnetic waves
Marconi – first radio communication.
Radio transmitter- electric charges oscillate along the antennae
and produce EM waves. Radio receiver – incoming EM waves induce
charge oscillations and those are detected

16. Plane EM waves
A simple plane EM wave
Wavefront – boundary plane between the
regions with and without EM disturbance
We will first show that such a plane
EM wave satisfies Maxwell equations
First, we will see if it satisfies Gauss’s laws for E and B fields

17. Consider Faraday’s Law
B
d
dl
dt


  
E
Circulation of vector E around loop efgh equals to -Ea
dl Ea

  
 E
Rate of change of flux through the surface
bounded by efgh is d=(Ba)(cdt)
B
d
Bac
dt


Hence –Ea=-Bac, and
E cB


18. Now consider Ampere’s Law
0 0
E
d
dl
dt
 


 
B
Circulation of vector B around loop efgh equals to Ba
B dl Ba

 

Rate of change of flux through the surface
bounded by efgh is d=(Ea)(cdt)
E
d
Eac
dt

 0 0
B cE
 

0 0
1
c
 
 299,792,458 /
c m s


19. Key Properties of EM Waves
The EM wave in vacuum is transverse; both E and B are perpendicular
to the direction of propagation of the wave, and to each other.
Direction of propagation and fields are related by
There is definite ratio between E and B; E=cB
The wave travels in vacuum with definite and unchanging speed c
Unlike mechanical waves, which need oscillating particles of a medium
to transmit a disturbance, EM waves require no medium.
k E B
  
 

20. Plane waves
waves
transverse
-
n
propagatio
of
direction
the
lar to
perpendicu
are
fields
The
:
fields
dependent)
-
(
dynamic
for
0
1
0
0
0
0
)
,
(
),
,
(
:
direction
-
in
only
space
in
change
that
fields
varying)
-
(time
dynamic
in
interested
now
be
would
We
2
t
B
E
t
E
c
z
B
y
B
t
B
z
E
y
E
x
B
x
E
z
E
y
E
x
E
t
x
t
x
x
x
x
x
y
z
x
y
z
x
x
z
y
x













































B
E
B
B
E
E
0
0
2

















B
E
B
B
E
E
t
c
t
EM Wave Equation

21. 0
0
2

















B
E
B
B
E
E
t
c
t
t
E
c
x
B
x
B
z
B
B
B
t
t
B
x
E
y
E
x
E
t
B
x
E
z
E
E
E
B
E
t
x
t
x
y
z
z
x
z
y
x
y
y
z
x
z
y
x
x












































2
y
z
1
other!
each
lar to
perpendicu
are
and
0
,
0
dependent)
-
(
Dynamic
0
0
0
,
0
choose
us
Let
0
),
,
(
),
,
(
B
E
B
B
E
E

22. Plane waves
0
1
:
for
same
the
and
0
1
:
for
equation
wave
the
derive
we
1
1
equations
two
Combining
2
2
2
2
2
2
2
2
2
2
2
2














































t
B
c
x
B
B
t
E
c
x
E
E
t
E
t
c
x
B
t
t
B
x
x
E
x
t
E
c
x
B
t
B
x
E
z
z
z
y
y
y
y
z
z
y
y
z
z
y
sinusoidal waves

23. General (one-dimensional) wave equation
0
1
)
(
),
(
)
(
),
(
2
2
2
2
2
2
2
2
2
2





























t
y
v
x
y
vt
x
f
v
t
y
vt
x
f
v
t
y
vt
x
f
x
y
vt
x
f
x
y
direction
-
negative
in the
propagates
direction
-
positive
in the
propagates
functions)
arbitrary
are
and
(
)
(
)
(
:
waves
g
propagatin
are
solutions
General
n
propagatio
wave
the
of
speed
the
is
motion)
wave
string
in the
nt
displaceme
(e.g.
function
wave
the
is
Here
direction
-
along
n
propagatio
wave
general
a
describes
0
1
2
2
2
2
2
x
g
x
f
g
f
ct
x
g
vt
x
f
y
v
y
x
t
y
v
x
y











24. wave
netic
electromag
an
is
Light
in vacuum
light
of
speed
the
is
This
m/s
10
3
1
speed
with
waves
EM
of
n
propagatio
the
describe
0
1
0
1
equations
our
So
8
0
0
2
2
2
2
2
2
2
2
2
2

















c
t
B
c
x
B
t
E
c
x
E
z
z
y
y