Complex electric field vectors

  1. 1. The problem statement, all variables and given/known data
    I'm trying to figure out the heat dissipation in a volume V due to an incident harmonic electric field.
    I know
    [tex]
    Q = \frac{1}{2}\int_V\mathrm{Re}\left(\mathbf{j}^{*} \cdot \mathbf{E}\right) d^3x
    [/tex]
    [tex]
    = \frac{1}{2}\int_V\mathrm{Re}\left(\left(\sigma \mathbf{E}\right)^{*} \cdot \mathbf{E}\right) d^3x
    [/tex]

    My biggest problem is that I don't know how to evaluate that dot product. If someone could please either explain it or link me to a resource that explains working with complex vectors, I would appreciate it. I haven't taken an EM class yet and I've never had it explained elsewhere how to operate with complex vectors, so I am mostly flailing around with this stuff.
     
  2. jcsd
  3. Redbelly98

    Redbelly98 12,029
    Staff Emeritus
    Science Advisor
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    It depends, is [itex]\sigma[/itex] a scalar or a tensor?

    If it's a scalar, this is pretty straightforward. Since [itex]\sigma \mathbf{E}^* = ( \sigma E_x^*, \sigma E_y^*, \sigma E_z^* ) [/itex], try dotting that with E and see what you get.
     
  4. Sigma (the complex conductivity) is a scalar. Anyway, I think this dot product should be
    [tex]
    \sigma(E_x^2,E_y^2,E_z^2)
    [/tex]
    but it seems weird to have E^2 as opposed to |E|^2 for complex numbers.
     
  5. Redbelly98

    Redbelly98 12,029
    Staff Emeritus
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    Okay, good.

    That's incorrect. Can you show how you came up with that?
     
  6. [tex]
    ( \sigma \mathbf{E})^{*} \cdot \mathbf{E} = ( ( \sigma E_x)^{*}, ( \sigma E_y)^{*}, ( \sigma E_x)^{*}) \cdot (E_x, E_y, E_z) = (( \sigma E_x)^{*})^{*}(Ex) + (( \sigma E_y)^{*})^{*}(E_y) +(( \sigma E_y)^{*})^{*}(E_z)
    [/tex]
    [tex]
    = ( \sigma E_x)(E_x) + ( \sigma E_y)(E_y) + ( \sigma E_z)(E_z) = \sigma (E_x^2 + E_y^2 + E_z^2)
    [/tex]
     
  7. Redbelly98

    Redbelly98 12,029
    Staff Emeritus
    Science Advisor
    Homework Helper

    Okay.

    The way I learned it, a dot product does not involve taking the complex conjugate of the first vector. So actually you would get terms like Ex*·Ex, etc., and you'd end up with σ|E|2.

    Note that heat dissipation Q should be a real number (right?).
     
  8. The expression for energy invloves a conjugate so that energy is real. The dot product is a vector operation and doesn't care if the components are complex or not.
     
  9. lets just go from basics, step by step, assuming everything is complex

    (sigma E)* . E
    where E(vector) and sigma(scalar) are complex

    (sigma)* E* . E = sigma* |E|^2

    or

    sigma* (Ex*, Ey*, Ez*).(Ex, Ey, Ez) = sigma* [|Ex|^2+|Ey|^2+|Ez|^2] = sigma* |E|^2

    must take modulus of the field and its components
     
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