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Unformatted text preview: 1 PHYS809 Class 29 Notes Maxwell’s equations and electromagnetic waves In complete vacuum (i.e. no sources present), Maxwell’s equations are 1 0, 0, 0, 0. t t ε μ ∂ ∇⋅ = ∇×  = ∂ ∂ ∇⋅ = ∇× + = ∂ E E B B B E (29.1) Taking the curl of the equations containing time derivatives and using the other two equations, we get 2 2 2 2 2 2 2 2 1 0, 1 0, c t c t ∂ ∇ = ∂ ∂ ∇ = ∂ E E B B (29.2) where c , the speed of electromagnetic waves in vacuum, is 1 . c μ ε = (29.3) Both the electric and magnetic fields obey wave equations, with wave speed c . Scalar and vector potentials Since 0, ∇⋅ = B there is a vector potential such that . = ∇× B A Substituting this into t ∂ ∇× + = ∂ B E gives t ∂ ∇× + = ∂ A E (29.4) Hence there is a scalar potential such that . t ∂ + = ∇Φ ∂ A E (29.5) In complete vacuum (i.e. in the total absence of charges and currents), the other two Maxwell equations 2 1 0, 0, c t ∂ ∇⋅ = ∇×  = ∂ E E B (29.6) give 2 ( ) 2 0, t ∂ ∇ Φ + ∇⋅ = ∂ A (29.7) and 2 2 2 2 2 1 1 . c t c t ∂ ∂Φ ∇ = ∇ ∇⋅ + ∂ ∂ A A A (29.8) We are free to choose the gauge. Once possibility is the Coulomb gauge 0. ∇⋅ = A In this case the scalar potential is a solution of Laplace’s equation and enters in a source term in a wave equation for the vector potential. If we use the Lorentz gauge 2 1 0, c t ∂Φ ∇⋅ + = ∂ A (29.9) then both potentials are solutions of wave equations 2 2 2 2 1 0, c t ∂ Φ ∇ Φ  = ∂ (29.10) and 2 2 2 2 1 0. c t ∂ ∇ = ∂ A A (29.11) Green functions for the wave equation When sources are included, the wave equations for the potentials in the Lorentz gauge are 2 2 2 2 2 2 2 2 1 , 1 ....
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.
 Fall '11
 MacDonald

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