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Class_31new - PHYS809 Class 31 Notes Momentum and the...

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1 PHYS809Class31Notes Momentum and the electromagnetic stress tensor We now consider conservation of momentum. Let P part be the total momentum of all the particles in a fixed volume V . Using the Lorentz force on a single particle of charge q , ( ) , q = + × F E v B (31.1) the rate of change of the momentum is 3 . part V d d x dt ρ = + × P E J B (31.2) Using Maxwell’s equation to eliminate the charge and current densities, this is 3 . part V d d x dt t = ∇⋅ + ∇× - × P D E D H B (31.3) Using the identity ( ) , t t t - × = - × + × D B B D B D (31.4) we get ( ) ( ) 3 . part V d d x dt t t = ∇⋅ + ∇× × - × + × P B E D H B D B D (31.5) Using the Maxwell equations 0, t ∇× + = B E (31.6) and 0, ∇⋅ = B (31.7) we get ( ) ( ) 3 3 . part V V d d d x dt dt d x + × = ∇⋅ + ∇⋅ - × ∇× - × ∇× P D B E D H B B H D E (31.8) We associate the second term on the left side of the equal sign with the rate of change of momentum of the electromagnetic field. The density of electromagnetic momentum is then 3 .
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