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Unformatted text preview: 3 Lorenz gauge and inhomogeneous wave equa- tion Last lecture we found out that given the static sources x y z r- r ρ ( r ) r r O J ( r ) ρ = ρ ( r ) and J = J ( r ) , static fields E =- ∇ V and B = ∇ × A satisfying Electrostatics: (curl-free) ∇ · D = ρ ∇ × E = 0 D = o E Magnetostatics: (divergence-free) ∇ · B = 0 ∇× H = J B = μ o H can be computed using the potentials V ( r ) = ρ ( r ) 4 π o | r- r | d 3 r , the solution of ∇ 2 V =- ρ o . A ( r ) = μ o J ( r ) 4 π | r- r | d 3 r , the solution of ∇ 2 A =- μ o J . 1 • Over the next two lectures we will explain why in case of time-varying sources ∇· D = ρ ∇· B = 0 ∇× E =- ∂ B ∂t ∇ × H = J + ∂ D ∂t ρ = ρ ( r , t ) and J = J ( r , t ) , the full set of Maxwell’s equations (see margin) can be satisfied by E =- ∇ V- ∂ A ∂t and B = ∇× A in terms of delayed or retarded potentials specified as V ( r , t ) = ρ ( r , t- | r- r | c ) 4 π o | r- r | d 3 r , the solution of inhomogeneous wave equation ∇ 2 V- μ o o ∂ 2 V ∂t...
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
- Fall '08