Class_35new - PHYS809 Class 35 Notes Energy density and...

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1 PHYS809 Class 35 Notes Energy density and energy flux In looking for solutions to Maxwell’s equation, we found it convenient to consider physical quantities as the real parts of complex expressions, e.g. ( ) ( ) 0 , Re , i t t e ω ⋅ - = k x E x E (35.1) where the complex vector E 0 contains amplitude and phase information. This approach works because Maxwell’s equations are linear in the fields. However quantities such as the energy density and the Poynting vector are quadratic in the fields and some care is need in evaluating them. A simplification arises because we are usually interested in the time averages of such quantities. Consider a product such as ( ) ( ) , , . t t E x D x Since the real part of a complex number z is ( ) * 2, z z + we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * 0 0 0 0 2 2 * * * * 0 0 0 0 0 0 0 0 1 , , 4 1 . 4 i t i t i t i t i t i t t t e e e e e e ⋅ - - ⋅ - ⋅ - - ⋅ - ⋅ - - ⋅ -   = + +   = + + + k x k x k x k x k x k x E x D x E E D D E D E D E D E D (35.2) On time-averaging of a product, the harmonically varying terms go to zero, and we have ( ) ( ) ( ) * * * 0 0 0 0 0 0 1 1 , , Re . 4
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Class_35new - PHYS809 Class 35 Notes Energy density and...

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