In free space Maxwell’s equations become

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

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

. 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

. 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

∆ = ∂/∂x + ∂/∂y + ∂/∂z

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

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

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

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

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) ∆

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) ∆

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) ∆

x (x ( ∆ x A) = ∆ 2 A + ∆-∆ (. A) ∆

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

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

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

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 )

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 )

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

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 )

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) ∆

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) ∆

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

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

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

. 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

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

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

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

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 )

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 )

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

Calculus Differentiation

Calculus Differentiation dy/dx = y

Calculus Differentiation dy/dx = y y = e x

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

dsinx/dx = cosx

and dcosx/dx = - sinx

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

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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/.../zaksiddons.wordpress.com/.../

Problem 3 Plot on graph paper the function y = sinx from x = 0 to x = 360 o y ……………… x 0 -y x

v = √ μ o ε o 1

v = √ μ o ε o 1 v = 3 x 10 8 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.

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

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× electric permittivity of free space, π ≈ , k - Boltzmann's constant, q - charge, B - magnetic induction, Φ - magnetic flux, J - current density, i - electric current, c ≈ m/s - the speed of light, µ 0 = 4π× magnetic permeability of free space, ∇ - del operator (for a vector function V: ∇. V - divergence of V, ∇ ×V - the curl of V).

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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