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Write Spin Force as Curl?

  1. Feb 26, 2013 #1
    1. The problem statement, all variables and given/known data
    I put this in the math forum because although it's for my EM waves class, it's a math question.

    Show that the spin force can be written as:

    [itex]F_{spin}=\frac{-1}{2}Im(\alpha)Im(E\cdot\nabla E^{*})=\nabla\times L_s[/itex]

    Find [itex]L_s[/itex].

    Where [itex]\alpha[/itex] is complex. I'm using [itex]E^{*}[/itex] to denote the complex conjugate of [itex]E[/itex]. Also, since these are all vectors, I'm omitting the arrow notation atop the vector quantities.

    2. Relevant equations

    [itex]Im(z)=\frac{1}{2i}(z-z^{*})[/itex]

    3. The attempt at a solution

    From the relevant equations:
    [itex]Im(\alpha)=\frac{1}{2i}[\alpha-\alpha^{*}][/itex]
    [itex]Im(E\cdot\nabla E^{*})=\frac{1}{2i}[E\cdot\nabla E^{*}-(E\cdot\nabla E^{*})^{*}][/itex]

    Substituting in,
    [itex]F_{spin}=\frac{1}{8}[\alpha-\alpha^{*}][E\cdot\nabla E^{*}-(E\cdot\nabla E^{*})^{*}]=\nabla\times L_s[/itex]

    Here, in order to make a curl appear, I'd like to apply the identity:
    [itex]\nabla\times(A\times B)=A(\nabla\cdot B)-B(\nabla\cdot A)+(B\cdot\nabla)A-(A\cdot\nabla)B[/itex]

    However, I'm not sure what the quantity [itex][(E\cdot\nabla E^{*})^{*}][/itex] looks like... I don't know how to conjugate this and I'm stuck here for the moment.
     
    Last edited: Feb 26, 2013
  2. jcsd
  3. Feb 26, 2013 #2

    haruspex

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    If E is a vector then I'm not sure what ∇E means. ∇.E would be a scalar, making E.(∇.E) problematic. Do you mean ∇×E?
     
  4. Feb 26, 2013 #3
    E is the electric field vector. it is a function of position and time, so that's just its gradient vector, also a function of position and time.
     
  5. Feb 26, 2013 #4

    haruspex

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