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From London Equations to Penetration Depth(Integrate Laplacian)

  1. Mar 11, 2012 #1
    (In SI units)
    Start with London's 2nd equation in Superconductivity, curl J = 1/(μ*λ²), and Ampere's curl B = μ*j.

    Then we curl both side curl curl B = μ* curl J and we do the substitution.
    So curl curl B = 0 - del²B which is the laplacian operator.

    My question is...how to integrate it?

    the equation becomes -del²B = -B/λ²
    And I am not sure how to integrate it to solve for B(x). Am I supposed to?

    I don't understand how the books jump to B(x) = B*exp(-x / λ)

    Please help and thank you very much. Any input is much appreciated too.
     
  2. jcsd
  3. Mar 11, 2012 #2
    what is the solution of the 2nd order ODE:

    [tex]
    y''(x) - \frac{1}{\lambda^2_L} \, y(x) = 0
    [/tex]

    that satisfies the following boundary conditions:
    [tex]
    y(x) \rightarrow 0, x \rightarrow \infty
    [/tex]
    and
    [tex]
    y(0) = y_0
    [/tex]
     
  4. Mar 11, 2012 #3
    OMG it's that easy.
    The general solution is just y = c*exp(x/λ) + c*exp(-x/λ),
    I never thought of del² works as a second derivative. I thought it's a surface integral in 3 dimensional space or something like that, because gradient and curl are generally three dimensional.

    And I have never applied methods to solve differential equations learned from math to physics. Please excuse me as a novice in the field. Thank you very much.
     
  5. Mar 11, 2012 #4
    Actually may I ask what program/software did you use to type/print math operators, symbols, and etc ?
     
  6. Mar 11, 2012 #5
    latex.
     
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