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Unformatted text preview: 1 PHYS809 Class 31 Notes 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 = + P E J B (31.2) Using Maxwells 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 the electromagnetic field....
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