3. In free space Maxwell’s equations become

4. . E = 0 In free space Maxwell’s equations become Gauss’s Law ∆

5. . E = 0. B = 0 In free space Maxwell’s equations become No magnetic monopolesGauss’s Law ∆∆

6. . E = 0. B = 0 x E = - ∂B/∂t ∆ In free space Maxwell’s equations become No magnetic monopolesGauss’s Law ∆∆ Faraday’s Law of Induction

7. . E = 0. B = 0 x E = - ∂B/∂t ∆ x B = μ o ε o (∂E/∂t) ∆ In free space Maxwell’s equations become No magnetic monopolesGauss’s Law ∆∆ Ampère’s LawFaraday’s Law of Induction

8. ∆ = ∂/∂x + ∂/∂y + ∂/∂z

9. ∆ ∆= 2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 ∆. ∆

10. Symbols E = Electric field ρ = charge density i = Electric current B = Magnetic field ε o = permittivity J = current density D = Electric displacement μ o = permeability c = speed of light H = Magnetic field strength M = Magnetization P = Polarization

11. x E = – (∂B/∂t) ∆ x B = μ o ε o (∂E/∂t) ∆

12. x E = – (∂B/∂t) ∆ x B = μ o ε o (∂ 2 E/∂t 2 ) ∆ (∂/∂t) x B = μ o ε o (∂E/∂t) ∆

13. x E = – (∂B/∂t) ∆ x B = μ o ε o (∂ 2 E/∂t 2 ) ∆ (∂/∂t) x (∂B/∂t) = μ o ε o (∂ 2 E/∂t 2 ) ∆ x B = μ o ε o (∂E/∂t) ∆

14. x E = – (∂B/∂t) ∆ x B = μ o ε o (∂ 2 E/∂t 2 ) ∆ (∂/∂t) x (∂B/∂t) = μ o ε o (∂ 2 E/∂t 2 ) ∆ x B = μ o ε o (∂E/∂t) ∆

15. x E = – (∂B/∂t) ∆ x B = μ o ε o (∂ 2 E/∂t 2 ) ∆ (∂/∂t) x (∂B/∂t) = μ o ε o (∂ 2 E/∂t 2 ) ∆ x ∆ = – μ o ε o (∂ 2 E/∂t 2 ) ∆ ( x E) x B = μ o ε o (∂E/∂t) ∆

16. x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆

17. x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆ 2 E + ∆-∆ (. E) ∆

18. x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆ 2E2E ∆-

19. x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆ 2E2E ∆- = - μ o ε o (∂E 2 /∂t 2 )

22. x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆ x (x ( ∆ x E) = ∆ 2 E + ∆-∆ (. E) ∆ 2 A ≡ ( ∆-∆.. ) A ≡ ∆ (∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 = - μ o ε o (∂E 2 /∂t 2 )

23. x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆ x (x ( ∆ x E) = ∆ 2 E + ∆-∆ (. E) ∆ 2 A ≡ ( ∆-∆.. ) A ≡ ∆ (∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 = - μ o ε o (∂E 2 /∂t 2 )

24. x (x ( ∆ x E) = ∆ 2 E = ∆- = - μ o ε o (∂E 2 /∂t 2 ) 2 E = ∆ = μ o ε o (∂E 2 /∂t 2 )

25. x (x ( ∆ x E) = ∆ x (- ∂B/∂t) = ∆ -(∂/∂t)( x E) ∆ x (x ( ∆ x E) = ∆ -(∂/∂t) = μ o ε o -(∂/∂t)( x B) ∆ (∂E/∂t) =- μ o ε o (∂E 2 /∂t 2 )

26. x E = – (∂B/∂t) ∆ x B = μ o ε o (∂ 2 E/∂t 2 ) ∆ (∂/∂t) x ∂B/∂t = μ o ε o (∂ 2 E/∂t 2 ) ∆ x ∆ = – μ o ε o (∂ 2 E/∂t 2 ) ∆ x Ex E x B = μ o ε o (∂E/∂t) ∆

27. x E = – (∂B/∂t) ∆ x B = μ o ε o (∂ 2 E/∂t 2 ) ∆ (∂/∂t) x ∂B/∂t = μ o ε o (∂ 2 E/∂t 2 ) ∆ x ∆ = – μ o ε o (∂ 2 E/∂t 2 ) ∆ x Ex E x B = μ o ε o (∂E/∂t) ∆

28. 2 E = ∆ - μ o ε o ∂E 2 /∂t 2 2 E = ∆ (1/c 2 )∂E 2 /∂t 2 2 Ψ = ∆ - (ώ 2 /c 2 ) Ψ ώ = 2πωΨ = ψ e -iώt E → Ψ 2 ψ= ∆ - (ώ/c) 2 ψ ώ/c = 2π/λ c = ωλ 2 ψ= ∆ - (2π/λ) 2 ψ p = h/λ

29. ώ/c = 2π/λ 2 ψ= ∆ - (2π/λ) 2 ψ p = h / λ p / h= 1/λ 2 ψ= ∆ - (2πp/h) 2 ψ 2 ψ = ∆ - (p 2 /ħ 2 ) ψ

30. 2 ψ = ∆ - (p 2 /ħ 2 ) ψ E = T + V E = p 2 /2m + V 2m ( E – V ) = p 2 2 ψ = ∆ - ( 2m /ħ 2 )( E – V ) ψ 2 ψ ∆ + ( 2m /ħ 2 )( E – V ) ψ = 0

31. . E = 0 ∆. B = 0 ∆ x E = - ∂B/∂t ∆ x B = μ o ε o (∂E/∂t) ∆ In free space Maxwell’s equations become Ampere’s LawFaraday’s Law of Induction

32. Symbols E = Electric fieldElectric field ρ = charge density i = electric currentelectric current B = Magnetic fieldMagnetic field ε o = permittivitypermittivity J = current density D = Electric displacement μ o = permeabilitypermeability c = speed of light H = Magnetic field strengthMagnetic field strength M = MagnetizationMagnetization P = PolarizationPolarization

33. . E = 0 ∆. B = 0 ∆ ∆ x E = - (∂B/∂t) ∆ x B = μ o ε o (∂E/∂t) Maxwell’s Equations

34. . E = 0 ∆. B = 0 ∆ x E = - ∂B/∂t ∆ x B = (∂E/∂t) ∆

37. x (x ( ∆ x E) = ∆ x (- ∂B/∂t) = ∆ -(∂/∂t)( x E) ∆ x (x ( ∆ x E) = ∆ -(∂/∂t) = μ o ε o -(∂/∂t)( x B) ∆ (∂E/∂t) == - μ o ε o (∂E 2 /∂t 2 )

38. x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆ x (x ( ∆ x E) = ∆ 2 E + ∆-∆ (. E) ∆ 2 A ≡ ( ∆-∆.. ) A ≡ ∆ (∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 = - μ o ε o (∂E 2 /∂t 2 )

39. x (x ( ∆ x E) = ∆ 2 E = ∆- = - μ o ε o (∂E 2 /∂t 2 ) 2 E = ∆ = μ o ε o (∂E 2 /∂t 2 )

40. Calculus Differentiation

41. Calculus Differentiation dy/dx = y

42. Calculus Differentiation dy/dx = y y = e x

43. Calculus Differentiation dy/dx = y y = e x e ix = cosx + i sinx

44. dsinx/dx = cosx

45. and dcosx/dx = - sinx

46. dsinx/dx = cosx and dcosx/dx = - sinx thus d 2 sinx/dx 2 = -sinx

47. Maxwell took all the semi-quantitative conclusions of Oersted, Ampere, Gauss and Faraday and cast them all into a brilliant overall theoretical framework. The framework is summarised in Maxwell’s Four Equations

49. These equations are a bit complicated and we are not going to deal with them in this very general course. However we can discuss arguably the most important and at the time most amazing consequence of these equations.

51. physics.hmc.edu

54. image at: www.irregularwebcomic.net/1420.htmlwww.irregularwebcomic.net/1420.html

55. Feynman on Maxwell'sContributions "Perhaps the most dramatic moment in the development of physics during the 19th century occurred to J. C. Maxwell one day in the 1860's, when he combined the laws of electricity and magnetism with the laws of the behavior of light.

56. As equations are combined – for instance when one has two equations in two unknowns one can juggle the equations and obtain two new equations each involving only one of the unknowns and so solve them.

57. . Let’s take a very simple example y = 4x and y = 3 + x

58. . Let’s take a very simple example y = 4x and y = 3 + x thus 4x = 3 + x

59. . Let’s take a very simple example y = 4x and y = 3 + x thus 4x = 3 + x 3x = 3

60. . Let’s take a very simple example y = 4x and y = 3 + x thus 4x = 3 + x 3x = 3 x =1 and y = 4

61. . Let’s take a very simple example y = 4x and y = 3 + x thus 4x = 3 + x 3x = 3 x =1 and y = 4 Check by back substitution

65. at: zaksiddons.wordpress.com/.../zaksiddons.wordpress.com/.../

66. Problem 3 Plot on graph paper the function y = sinx from x = 0 to x = 360 o y 0 15 30 45 60 75 90 0 ……………… 360 0 x 0 -y x

67. v = √ μ o ε o 1

68. v = √ μ o ε o 1 v = 3 x 10 8 m/s

69. As a result, the properties of light were partly unravelled -- that old and subtle stuff that is so important and mysterious that it was felt necessary to arrange a special creation for it when writing Genesis. Maxwell could say, when he was finished with his discovery, 'Let there be electricity and magnetism, and there is light!' " Richard Feynman in The Feynman Lectures on Physics, vol. 1, 28-1.

70. Symbols E = Electric fieldElectric field ρ = charge density i = electric currentelectric current B = Magnetic fieldMagnetic field ε o = permittivitypermittivity J = current density D = Electric displacement μ o = permeabilitypermeability c = speed of light H = Magnetic field strengthMagnetic field strength M = MagnetizationMagnetization P = PolarizationPolarization

77. LAWDIFFERENTIAL FORM INTEGRAL FORM Gauss' law for electricity Gauss' law for magnetism Faraday's law of induction Ampere's law NOTES: E - electric field, ρ - charge density, ε 0 ≈ 8.85×10 -12 - electric permittivity of free space, π ≈ 3.14159, k - Boltzmann's constant, q - charge, B - magnetic induction, Φ - magnetic flux, J - current density, i - electric current, c ≈ 299 792 458 m/s - the speed of light, µ 0 = 4π×10 -7 - magnetic permeability of free space, ∇ - del operator (for a vector function V: ∇. V - divergence of V, ∇ ×V - the curl of V).

78. at: www.physics.hmc.edu/courses/Ph51.htmlwww.physics.hmc.edu/courses/Ph51.html

82. Faraday's Law of Induction The line integral of the electric field around a closed loop is equal to the negative of the rate of change of the magnetic flux through the area enclosed by the loop.line integralelectric fieldmagnetic flux This line integral is equal to the generated voltage or emf in the loop, so Faraday's law is the basis for electric generators. It also forms the basis for inductors and transformers.generated voltageemfelectric generatorsinductorstransformers Application to voltage generation in a coilGauss' law, electricityGauss' law, magnetismFaraday's law Ampere's law In d e x M a x w el l' s e q u at io n s c o n c e pt s GoBackGoBack Maxwell's Equations HyperPhysicsHyperPhysics***** Electricity and Magnetism Electricity and Magnetism R Nave