1. Exam 2 free response retake: Today, 5 pm room next to my office
Exam 2 free response retake: Today, 5 pm room next to my office *new problem to solve *max 10 points

2. Final EXAM: Friday, May 6th, ME 1130
7:00-9:00 pm

3. Complete online course evaluation and receive extra 2 points on FINAL exam free response

4. Differential Form of Gauss' Law (Sec. 22.8)
Think about a region of space, enclosed by a box.
Divide Gauss' law by the volume of the box:
E || x
Take the limit
of a small box
Work on the left hand side of the equation:
For a general case where E can point in any direction:
GAUSS' LAW
Differential Form
where
“divergence“ (div)

5. Differential Form of Ampere's Law (Sec. 22.9)3
Write I in terms of current density J: 4 2
Divide Ampere's Law by a very small ΔA: 1
Current I
out of the board
In our geometry, n = z

6. Differential Form of Ampere's Law (Sec. 22.9)3
We divided Ampere's Law by a very small ΔA, and got this: 4 2 1
Current I
out of the board
Definition of
derivative!
Now work on the left hand side:
"Crossed derivative"

7. Differential Form of Ampere's Law (Sec. 22.9)3
We divided Ampere's Law by a very small ΔA, and got this: 4 2 1
Current I
out of the board
For a loop in any direction, this can be re-expressed as:
AMPERE'S LAW
Differential Form

8. Curl: Here's the Math
+ ( )
set up the answer
copy 1st two colums

9. Curl: Here's the Math
+ ( )
Blast from the Past!
Lecture 12

10. Maxwell's Equations – The Full Story
Divergence
Curl
GAUSS' LAW
(Magnetism)
FARADAY'S LAW
AMPERE'S LAW
Flux
Circulation

11. Maxwell's Equations – No Charges
In the ABSENCE of
"sources" = charges, currents:
GAUSS' LAW
(Magnetism)
FARADAY'S LAW
AMPERE'S LAW
This says
once a wave
starts,
it keeps going!

12. Maxwell's Equations – No Charges
What happens if we feed one equation into the other?
Use
This

13. Maxwell's Equations – No Charges
What happens if we feed one equation into the other?

14. Maxwell's Equations – No Charges
How do you solve a Differential Equation?
Know the answer! (Ask Wolfram Alpha)
 This is a WAVE EQUATION, with speed c
Using similar ideas, you can show that E obeys the same equation:

15. Wave Description – wavelength: distance between crests (meters)
T – period: the time between crests passing fixed location (seconds)
v – speed: the distance one crest moves in a second (m/s)
f – frequency: the number of crests passing fixed location in one
second (1/s or Hz)
 – angular frequency: 2f: (rad/s)

16. Wave: Variation in Time and Space
After one period T – we will get the point which was at coordinate -lambda
‘-’ sign: the point on wave moves to the right

17. Wave: Phase Shift But E @ t=0 and x =0, may not equal E0
After one period T – we will get the point which was at coordinate -lambda
Two waves are ‘out of phase’
(Shown for x=0)

18. Wave: Amplitude and Intensity
E0 is a parameter called amplitude (positive). Time dependence
is in cosine function
Often we detect ‘intensity’, or energy flux ~ E2.
Intensity I (W/m2):
What if we triple the amplitude of the wave?
Works also for other waves,
such as sound or water waves.

19. Interference
Superposition principle: The net electric field at any location is
vector sum of the electric fields contributed by all sources.
Laser: source of radiation which has the same frequency (monochromatic) and phase (coherent) across the beam.
Two slits are sources of two waves with the same phase and frequency.
We have first considered electric field single charges and then moved to multiple charges.
We have now considered single source of em wave – and now we will consider em field produced by multiple sources. Simplest case – two. Since E for each depends on time situation is not as trivial.
Laser – source of radiation which has basicqally single wave (same frequency and phase across the beam).
Note – more complex, intensity of bands decreases from center to sides.
What can we expect to see on the screen?
Can particle model explain the pattern?

20. Interference: Constructive
Two emitters: E1 E2
Fields in crossing point
Two emitters (radio antennas, or laser beams)
Superposition:
Amplitude increases twice: constructive interference

21. Interference: Energy E1 E2 Two emitters:
What about the intensity (energy flux)?
Energy flux increases 4 times while two emitters produce only
twice more energy
Two emitters (radio antennas, or laser beams)
There must be an area in space where intensity is smaller than that
produced by one emitter

22. Interference: Destructive
Two emitters (radio antennas, or laser beams)
Two waves are 1800 out of phase: destructive interference

23. Two-Slit Experiment with Bullets
Bullets arrive in lumps
We measure the probability of arrival of a lump (bullet)
P1 = probability bullet went through slit 1 in arriving at x
P12 = P1 + P2

24. Two-Slit Experiment with Waves
We measure the Intensity of the wave motion at the detector (related to the square of the wave height)

25. Two-Slit Experiment with Electrons
Electrons arrive in clicks - no “half-clicks”
Clicks come erratically
As detector moves to a different position, rate at which clicks occur is faster or slower
Adding a second detector, we would note that either one or the other would click - never both at once
P12 = |1 + 2|2
There are some points at which very few electrons arrive when both holes are open. Closing one hole increases the number arriving at these places! So closing one hole increases the number arriving from the other. On the other hand, the number arriving at the central max (with both holes open) is more than twice that arriving when a single hole is open. So here, it is as if closing one hole decreases the number arriving from the other!

26. Conclusion
Electrons arrive in lumps like particles and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave. It is in this sense that an electron behaves sometimes like a particle and sometimes like a wave. - R. Feynman Vol 3 Lecture Series

27. Energy and Wavelength Photon: a ‘wave packet’,
EM wave which occupies a short region
Experimental observations:
h = J.s Planck constant
This relationship was first introduced by Max Planck in 1900 to explain the spectrum of black body radiation.
Historic moment: birth of quantum theory
In 1905 Einstein proposed this for explaining photoelectric effect
Nobel price in 1918