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Unformatted text preview: Homework 14 Answers, 95.657 Fall 2011, Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Problem 1 Jackson 6.9 Discuss the conservation of energy and linear momentum for a macroscopic system of sources and electromagnetic fields in a uniform, isotropic medium described by a permittivity ε and a permeability μ . Show that in a straightforward calculation the energy density, Poynting vector, field-momentum density, and Maxwell stress tensor are given by the Minkowski expressions: u = 1 2 (ϵ E 2 +μ H 2 ) S = E × H g =μ ϵ E × H T ij =[ϵ E i E j +μ H i H j − 1 2 δ ij (ϵ E 2 +μ H 2 )] What modifications are made if ε and μ are functions of position? SOLUTION: As derived in class, the energy density and energy flow in linear, low-dispersion, low-loss materials are given by: u = 1 2 ( H ⋅ B + E ⋅ D ) and S = E × H For linear material, B = μ H and D = ε E so these become: u = 1 2 ( ϵ E 2 +μ H 2 ) and S = E × H The field-momentum density is given by: g = S...
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