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Confusion about a derivation

  1. Sep 2, 2014 #1
    In Griffiths section 4.4.3, he derives the energy in a dielectric system as
    W=0.5∫D.Edτ.
    Part of the derivation involves the relation
    0.5Δ(D.E)=0.5Δ(εE2)=ε(ΔE).E=(ΔD).E
    for infinitesimal increments, using DE. Now the part 0.5Δ(εE2)=ε(ΔE).E loses me so I was wondering if anybody could explain it. Thanks.
     
  2. jcsd
  3. Sep 2, 2014 #2

    AlephZero

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    If you can't "see" it, write out the components. I'm not going to do it in full but this shows what's happening:

    ##\frac 1 2 \nabla(\mathbf{D}.\mathbf{E}) = \frac 1 2 ( \frac{\partial}{\partial x}(D_x.E_x) \cdots) = \frac 1 2 \epsilon ( \frac{\partial}{\partial x}(E_x.E_x) \cdots) = \frac 1 2 \epsilon(2\frac{\partial E_x}{\partial_x}E_x \cdots) ## etc.
     
  4. Sep 2, 2014 #3
    Hmm I was using deltas not nablas...

    Anyway I think I've worked it out. Effectively we have d(E2) and because d(E2)/dE=2E, d(E2)=2EdE. The fact the book used deltas blinded me from this...

    Only issue now is understanding why εEΔE=εEE - how do I know they have the same direction? Note E is just the field due to some free charge distribution ρf and ΔE is just the change in E due to the addition of an amount Δρf of the free charge.
     
    Last edited: Sep 2, 2014
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