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# 329lect17 - 17 Magnetization current Maxwells equations in...

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17 Magnetization current, Maxwell’s equations in material media Consider the microscopic-form Maxwell’s equations · D = ρ Gauss’s law · B = 0 ∇ × E = - B t Faraday’s law ∇ × H = J + D t , Ampere’s law where D = o E B = μ o H . Direct applications of these equations in material media containing a colossal number of bound charges is impractical. Macroscopic-form Maxwell’s equations suitable for material media are obtained by first expressing ρ and J above as the macroscopic quantities ρ = ρ f - ∇ · P and J = J f + P t + ∇ × M where 1

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subscripts f indicate charge and current density contributions due to free charge carriers, the term -∇ · P denotes the bound charge density , the term P t denotes the polarization current density due to oscillating dipoles (already discussed in Lecture 11), and ∇× M is a “magnetization current density” also due to bound charges, an e ff ect that we will discuss and clarify later in this section. Using these expressions in Gauss’s and Ampere’s laws · o E = ρ Gauss’s law ∇ × μ - 1 o B = J + o E t , Ampere’s law we obtain · ( o E + P ) = ρ f Gauss’s law ∇ × ( μ - 1 o B - M ) = J f + t ( o E + P ) , Ampere’s law. Now, re-define D and H as D = e E + P = E and H = μ - 1 o B - M = μ - 1 B , 2
respectively, and drop the subscripts f which will no longer be needed. Using these new definitions, the full set of Maxwell’s equations now read as (the same form as before) · D = ρ Gauss’s law · B = 0 ∇ × E = - B t Faraday’s law ∇ × H = J + D t , Ampere’s law with D = E B = μ H , where ρ and J are understood to be due to free charge carriers only.

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