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Class_30new

# Class_30new - PHYS809 Class 30 Notes Conservation of energy...

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1 PHYS809Class30Notes Conservation of energy for electromagnetic fields We begin by considering electromagnetic waves propagating in vacuum. The effects of linear media will be considered later. At a fundamental level the waves are due to motion of charged particles. Consider such a particle of charge q . The force of the electric and magnetic fields acting on this particle is the Lorentz force ( ) . q = + × F E v B (30.1) The rate at which this force does work on the particle is . q = v F v E (30.2) By considering all the particles in unit volume, we see that the work done by the fields per unit volume per unit time is , J E (30.3) where J is the current density. This work goes into accelerating the changes and converts energy from the electromagnetic fields into kinetic energy. The rate at which work is done to change the energy of the fields in a fixed volume V is . V dW dV dt = - J E (30.4) We can use a Maxwell equation to eliminate the current density 0 0 1 . V dW dV dt t ε μ = - ∇× - E B E (30.5) Using the identity ( ) , ⋅∇× = ∇⋅ × + ⋅∇× E B B E B E (30.6) This becomes ( ) 0 0 0 1 1 . V dW dV dt t ε μ μ = - ∇⋅ × + ⋅∇× - E B E B E E (30.7) Using another Maxwell equation to eliminate ∇× E , we get ( ) 0 0 0 1 1 .

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Class_30new - PHYS809 Class 30 Notes Conservation of energy...

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