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In free space Maxwell’s equations become In free space Maxwell’s equations become .E =0 ∆ Gauss’s Law In free space Maxwell’s equations become =0 ∆ Gauss’s Law ∆ .E .B =0 No magnetic monopoles In free space Maxwell’s equations become =0 ∆ Gauss’s Law x E = - ∂B/∂t ∆ Faraday’s Law of Induction ∆ .E .B =0 No magnetic monopoles In free space Maxwell’s equations become ∆ =0 ∆ Gauss’s Law x E = - ∂B/∂t ∆ Faraday’s Law of Induction .B =0 No magnetic monopoles ∆ .E x B = μo εo (∂E/∂t) Ampère’s Law = ∂/∂x + ∂/∂y + ∂/∂z ∆ ∆ = ∂/∂x + ∂/∂y + ∂/∂z 2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 ∆= ∆. ∆ E ρ i B εo J D μo c H M P = = = = = = = = = = = = Symbols Electric field charge density Electric current Magnetic field permittivity current density Electric displacement permeability speed of light Magnetic field strength Magnetization Polarization ∆ x E = – (∂B/∂t) ∆ x B = μoεo (∂E/∂t) ∆ x E = – (∂B/∂t) ∆ x B = μoεo (∂E/∂t) ∆ (∂/∂t) x B = μoεo (∂2E/∂t2) ∆ x E = – (∂B/∂t) ∆ x B = μoεo (∂E/∂t) ∆ (∂/∂t) x B = μoεo (∂2E/∂t2) x (∂B/∂t) = μoεo (∂2E/∂t2) ∆ ∆ x E = – (∂B/∂t) ∆ x B = μoεo (∂E/∂t) ∆ (∂/∂t) x B = μoεo (∂2E/∂t2) x (∂B/∂t) = μoεo (∂2E/∂t2) ∆ ∆ x E = – (∂B/∂t) ∆ x B = μoεo (∂E/∂t) ∆ (∂/∂t) x B = μoεo (∂2E/∂t2) ∆ x (∂B/∂t) = μoεo (∂2E/∂t2) ∆ x( x E) = – μoεo (∂2E/∂t2) ∆ ∆- x A) = 2A + ( ∆ ∆ ∆ x( . A) ∆ 2A ∆- 2E + + ( ( ∆ ∆ ∆- x A) = ∆ ∆ ∆ x( . . A) E) ∆ ∆- 2A ∆- x A) = 2E + ( ∆ ∆ ∆ x( . A) ∆ ∆- 2A ∆- x A) = 2E + ( ∆ ∆ ∆ x( . A) = - μo εo (∂E2/∂t2) ∆ ∆ ∆ . )A ≡ x E) = ∆- ∆ x( . 2A + ( . A) (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 2E + ( ∆ ∆ ∆ ≡( ∆- ∆ 2A x A) = ∆ ∆ x( . E) = - μo εo (∂E2/∂t2) ∆ ∆- ∆ ∆ . )A ≡ x E) = ∆- ∆ x( . 2A + ( . A) (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 2E + ( ∆ ∆ ∆ ≡( ∆- ∆ 2A x A) = ∆ ∆ x( . E) = - μo εo (∂E2/∂t2) ∆ ∆- x E) = 2E ∆- ∆ ∆ x( 2E == - μo εo (∂E2/∂t2) == μo εo (∂E2/∂t2) ∆ x (- ∂B/∂t) = -(∂/∂t)( x E) x E) = -(∂/∂t)( x B) ∆ ∆ ∆ x( ∆ x E) = ∆ x( ∆ ∆ -(∂/∂t) = μo εo (∂E/∂t) = - μo εo (∂E2/∂t2) ∆ x E = – (∂B/∂t) ∆ x B = μoεo (∂E/∂t) ∆ (∂/∂t) x B = μoεo (∂2E/∂t2) x ∂B/∂t = μoεo (∂2E/∂t2) ∆ ∆ x xE = – μoεo (∂2E/∂t2) ∆ ∆ x E = – (∂B/∂t) ∆ x B = μoεo (∂E/∂t) ∆ (∂/∂t) x B = μoεo (∂2E/∂t2) x ∂B/∂t = μoεo (∂2E/∂t2) ∆ ∆ x xE = – μoεo (∂2E/∂t2) ∆ ∆ 2E = - μo εo ∂E2/∂t2 ∆ 2E 2) ∂E2/∂t2 (1/c = E →Ψ Ψ = ψ e -iώt ∆ 2Ψ ∆ 2 ψ= - (ώ/c)2 ψ ∆ 2 ψ= - (2π/λ)2 ψ ώ = 2πω = - (ώ2/c2) Ψ c = ωλ ώ/c = 2π/λ p = h/λ 2 ψ= - (2π/λ)2 ψ ώ/c = 2π/λ ∆ p=h/λ 2 ψ= 2ψ - (2πp/h)2 ψ = - (p2/ħ2) ψ p / h= 1/λ ∆ ∆ 2ψ = - (p2/ħ2) ψ ∆ E =T+V E = p2/2m + V 2m ( E – V ) = p2 ∆ 2ψ ∆ 2ψ = - (2m /ħ2)( E – V ) ψ + (2m /ħ2)( E – V ) ψ = 0 In free space Maxwell’s equations become ∆ x E = - ∂B/∂t ∆ Faraday’s Law of Induction ∆ =0 ∆ .E .B =0 x B = μo εo (∂E/∂t) Ampere’s Law E ρ i B εo J D μo c H M P = = = = = = = = = = = = Symbols Electric field charge density electric current Magnetic field permittivity current density Electric displacement permeability speed of light Magnetic field strength Magnetization Polarization Maxwell’s Equations ∆ .E = 0 ∆ .B = 0 x E = - (∂B/∂t) x B = μoεo (∂E/∂t) ∆ ∆ ∆ x E = - ∂B/∂t ∆ =0 ∆ .E .B =0 x B = (∂E/∂t) ∆ x (- ∂B/∂t) = -(∂/∂t)( x E) x E) = -(∂/∂t)( x B) ∆ ∆ ∆ x( ∆ x E) = ∆ x( ∆ ∆ -(∂/∂t) = μo εo (∂E/∂t) = = - μo εo (∂E2/∂t2) ∆ ∆ . )A ≡ x E) = ∆- ∆ x( . 2A + ( . A) (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 2E + ( ∆ ∆ ∆ ≡( ∆- ∆ 2A x A) = ∆ ∆ x( . E) = - μo εo (∂E2/∂t2) ∆ ∆- x E) = 2E ∆- ∆ ∆ x( 2E == - μo εo (∂E2/∂t2) == μo εo (∂E2/∂t2) ∆ Calculus Differentiation Calculus Differentiation dy/dx = y Calculus Differentiation dy/dx = y y = ex Calculus Differentiation dy/dx = y y = ex eix = cosx + i sinx dsinx/dx = cosx dsinx/dx = cosx and dcosx/dx = - sinx dsinx/dx = cosx and dcosx/dx = - sinx thus d2sinx/dx2 = -sinx 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 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. physics.hmc.edu image at: www.irregularwebcomic.net/1420.html 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. 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. . Let’s take a very simple example y = 4x and y = 3 + x . Let’s take a very simple example y = 4x and y = 3 + x thus 4x = 3 + x . Let’s take a very simple example y = 4x and y = 3 + x thus 4x = 3 + x 3x = 3 . Let’s take a very simple example y = 4x and y = 3 + x thus 4x = 3 + x 3x = 3 x =1 and y = 4 . 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 at: zaksiddons.wordpress.com/.../ Problem 3 Plot on graph paper the function y = sinx from x = 0 to x = 360o y 0 x -y 0 15 30 45 60 75 900 ……………… 3600 x v = 1 √ μoεo v = 1 √ μoεo v = 3 x 108 m/s 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. E ρ i B εo J D μo c H M P = = = = = = = = = = = = Symbols Electric field charge density electric current Magnetic field permittivity current density Electric displacement permeability speed of light Magnetic field strength Magnetization Polarization LAW DIFFERENTIAL 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, ε0 ≈ 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, µ0 = 4π×10-7 - magnetic permeability of free space, ∇ - del operator (for a vector function V: ∇. V - divergence of V, ∇×V - the curl of V). at: www.physics.hmc.edu/courses/Ph51.html Maxwell's Equations M a x w el l' s e q u at io n s c o n

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