1. XI. Maxwell’s Equations
Dr. Bill Pezzaglia
Updated: 3/31/2010

2. XI. Maxwell & Electromagnetic Waves2
XI. Maxwell & Electromagnetic Waves
Maxwell’s Equations
Hertz Waves & Poynting
Polarization

3. Light is Electromagnetic Displacement Current Maxwell’s Equations
A. Maxwell’s Equations 3
Light is Electromagnetic
Displacement Current
Maxwell’s Equations

4. 1. Hints that Light is Electromagnetic4
A number of experiments suggested that there was a connection between electricity, magnetism and the phenomena of light.
1834, 1857 Speed of electricity in wire measured to be very fast (close to speed of light)
1850 Speed of light is measured accurately 1850 by Foucault.
1844 Faraday rotates the polarization of light with a magnetic field (implies light has magnetic properties).

5. Charles Wheatstone (1802-1875)
1834 discovery by English physicist Charles Wheatstone that current traveled through long lengths of wire with great velocity – almost 288,000 miles/second
A bit off, it can’t travel faster than speed of light (186, 282 miles/second)
1837 Developed an early telegraph (5 needles)

6. Michael Faraday 6
1844 Faraday rotates the polarization of light using a magnetic field. Suggests light is a transverse magnetic disturbance
1857 Wilhem Weber shows Amp of current is a Coulomb per second, gets characteristic speed of electrical signals to be speed of light.

7. Gustav Kirchhoff (1824-1887) 1857 Telegraphy Equations
Derives (based on earlier work by Faraday & Thomson 1854) that speed of electrical signal in cable should be close to speed of light.

8. 2. Maxwell’s Displacement Current8
1861 Displacement Current: A changing electric field is equivalent to a current
This “displacement current” can hence create a magnetic field
Summarizes all of electromagnetism in 8 equations
James Maxwell ( )

9. a. Restate Faraday’s Law9
In general: The sum of electric field about a closed loop is proportional to the change of magnetic flux passing through the loop
Faraday stated that the voltage (emf=electric field times circumference) in a wire loop was equal to the change in magnetic flux. Maxwell shows that the law is more general. There is an electric field present even if there is no wire!

10. b. Ampere’s circulation Law10
In general: The sum of magnetic field about a closed loop is proportional to the sum of current passing through the loop
This form is really only useful for very symmetric situations, for example where B is constant over the loop. Then you can simply write (where “L” is the perimeter of the path)

11. c. Maxwell’s Displacement Current11
Maxwell makes Ampere’s law look like the complement of Faradays Law. A changing electric flux will generate circular B field.
Note “c” is the speed of light, and Electric Flux is defined:

12. 3. Maxwell’s Equations 12 (a) The General Laws of Maxwell
Gauss’s Law shows that charge is the source of magnetic fields (magnetic flux through a closed surface is proportional to net enclosed charge)
Gauss’s Law for magnetism states that there are no magnetic charges (magnetic flux through a closed surface is zero).
Faraday’s Law: changing magnetic fields create electric fields
Ampere’s Circulation Law: current is the source of magnetic fields. Maxwell adds the “displacement current” to this equation such that changing electric fields create magnetic fields.

13. b. Differential Form of Maxwell’s Equations13
1884 (with Gibbs) Heaviside reorganizes Maxwell’s equations compactly into 4 equations
Oliver Heaviside ( )

14. c. Relativity and Maxwell’s Equations14
1905 Einstein’s Relativity shows that time is the 4th dimension.
In our ordinary “3D” view of the world, electric fields are different than magnetic fields, however we see they are complementary
In “4D” we see that they are both the same thing, i.e. we “unify” electricity with magnetism.
We can write Maxwell’s 4 equations in just 2:

15. Equations predict waves Hertz Experiment Energy in Waves
B. Hertz Waves 15
Equations predict waves
Hertz Experiment
Energy in Waves

16. 1. EM Wave Equation 16
(a) Maxwell shows his equations predict that electromagnetic waves can exist in vacuum (note E & B are perpendicular to each other and direction of wave)

17. 1b. Prediction of Electromagnetic Waves17
The Theoretical speed: comes out very close to known speed of light “c”
Magnitude of electric and magnetic fields are simply related by wavespeed:

18. 2. Making EM Waves 18
(a) (1888?) Hertz demonstration that electromagnetic waves can be transmitted and then received. Proves existence of waves with frequencies of 100 million cycles per second.
Heinrich Hertz ( )

19. 2b. Nikola Tesla ( ) 19
1891 (1893?) Chicago World’s fair, demonstrates wireless telegraphy (30 feet)
1894 Lodge transmits 150 yards

20. 2c Guglielmo Marconi ( 20
1899 Marconi “steals” Tesla’s design and broadcasts across the English Channel
1901 Across the Atlantic

21. John Henry Poynting (1852-1914)
3. Poynting’s Theorems 21
Electromagnetic waves can carry:
Energy
Momentum
Angular Momentum (polarization)
This was primarily developed by John Henry Poynting (1884)
John Henry Poynting ( )

22. (a) Energy in Waves 22
We know energy is stored in electric and magnetic fields. The energy density (joules/m3) can be written in the following form remembering that E=cB,
Impedance of free space:
Intensity of wave For sunlight: 1200 W/m2

23. (b) Light Pressure 23
1871 Maxwell deduces that light should exert a pressure, but this was from an “induction” argument, rather than from Lorentz force (as this was before the theory of electron conduction)
1884 Poynting describes momentum of a light wave is given by the “Poynting Vector” divided by c. The magnitude for sunlight is only about 10-4 Pascals!
1900 Lebedev detects light pressure P.N. Lebedev, Ann. Phys. (Leipzig) 6:433 (1901)
1901 Nichols & Hull construct a radiometer which measures light pressure E.F. Nichols and G.F. Hull, Phys. Rev. 13:307 (1901)

24. (c) How does light “push”24
Electrons in material respond to the electric field. The electrical work done on charges is due to intensity of light
Magnetic field pushes the charges in the direction of S:
Light Pressure:

25. Circular Polarization
C. Polarization 25
Linear Polarization
Birefringence
Circular Polarization

26. Linear Polarization 26
[1812 Fresnel develops wave theory of transverse polarized light, well before the electromagnetic nature was known]
light has two perpendicular linear polarizations (electric field) can be horizontal or vertical Hertz shows electromagnetic waves have transverse polarization (equivalent to “light”)

27. Malus’s Law 27
Malus discovered 1808 that light reflected off of glass is polarized. Today we use “linear polarizers”.
1808 Malus’s Law: Linear polarized light passing through a second polarizer tilted at angle  to first will be attenuated:
Hence no light gets through “crossed polarizers” (=±90°)

28. Birefringence 28
1669 Erasmus Bartolinus (Denmark) discovers the birefringence (double refraction) of calcite crystals.
When polarization was understood better, it was realized the two different polarizations took different paths (they are “refracted” differently, or the index of refraction is dependent upon polarization)
Index of refraction: n=c/v, so the different polarizations travel at slightly different speeds.

29. Optical Activity 29
Optically active materials can rotate the polarization
If such a substance is put between “crossed polarizers” (90º angle) you will often see interesting colors.

30. Circular Polarized Light30
Another type of polarized light can be left or right handed circular polarized

31. Quarter Wave Plates 31
A quarter wave plate retards horizontal polarization by 90º to vertical. It can be used to make circular polarized light from linear polarized light.

32. Detecting Circular Polarized Light32
A quarter wave plate will turn circular back into linear, which can be detected by a linear polarizer

33. References 33 http://maxwell.byu.edu/~spencerr/phys442/node4.html

34. Things to Do 34 Find tesla museum stuff
Who first predicted circular polarized light?