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Homework Help: Skin depth question?

  1. Mar 26, 2009 #1
    hi there, im stuck on this question can some give me a hint on how to solve the question:

    the question is:

    Calculate the skin depth for copper at 1.0 GHz . σ = 5.8x10^7 S/m, εr=1, μr=1.

    the formula i think that i need to use is

    tan δ = σ / (ω)(ε0)(εr)

    ω = 2nf = 2 x n x 1 x 10^9
    ε0 = 8.85 x 10^-12
    er = 1
    where do i put μr?

    im missing a another step but i dont no what, can some give me a hint please?

    the answer to the queation is
    tan δ >>1, d = 2.1um
  2. jcsd
  3. Mar 27, 2009 #2
    You've got to understand what skin depth is and you'll get this question. When you solve Maxwell's equations in a conducting material, you get a complex wavenumber (i.e. a complex dielectric constant) and that imaginary part of the wavenumber leads to a decay of the wave in the conductor. Remember from electrostatics that E-fields should be zero inside a conductor? Well, for time-dependent fields in "good conductors" the E-fields damp out pretty darn quickly. Copper is such a conductor. Start with Maxwells equations in vacuum (your dielectric constant and mag. permeability are identically 1).

    [tex]\nabla \times \vec{B} = \frac{4 \pi}{c} \vec{J} + \frac{1}{c} \partial_t \vec{E} \hspace{1cm} \nabla \times \vec{E} = -\frac{1}{c} \partial_t \vec{B}[/tex]

    Assume your E,B fields have time dependence of the form
    [tex]e^{-i \omega t}[/tex]

    And that for the conductor the current density is proportional to the E-field
    [tex]\vec{J}=\sigma \vec{E}[/tex]

    Take the curl of both equations and combine to get a modified wave equation for E. It looks like
    [tex]\nabla^2 \vec{E} + k^2 \vec{E} = 0[/tex]

    Where our k is
    [tex]k=\frac{w}{c}\sqrt{1+\frac{i4 \pi \sigma}{\omega}}[/tex]

    For "good conductors" (like copper) we have [tex]\sigma \gg \omega[/tex] This amounts to killing the displacement current term in the first place. Or here our k becomes

    [tex]k = \frac{\omega}{c}\sqrt{i4 \pi \frac{\sigma}{\omega}} = \frac{1}{c}\sqrt{i4 \pi \sigma \omega} = \frac{(1+i)}{c}\sqrt{4 \pi \sigma \omega}[/tex]

    Be sure of how your text/course defines the skin depth. Sometimes it's 1/Im(k) sometimes its 1/[2Im(k)] where Im(k) is the imaginary part of k. Be careful here - I've used Gaussian units throughout so just adapt the technique to your units which are probably SI.
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