From London Equations to Penetration Depth(Integrate Laplacian)

[calvinjhfeng]

(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.

 

[Dickfore]

what is the solution of the 2nd order ODE:

<br /> y&#039;&#039;(x) - \frac{1}{\lambda^2_L} \, y(x) = 0<br />

that satisfies the following boundary conditions:
<br /> y(x) \rightarrow 0, x \rightarrow \infty<br />
and
<br /> y(0) = y_0<br />

 

[calvinjhfeng]

what is the solution of the 2nd order ODE:

<br /> y&#039;&#039;(x) - \frac{1}{\lambda^2_L} \, y(x) = 0<br />

that satisfies the following boundary conditions:
<br /> y(x) \rightarrow 0, x \rightarrow \infty<br />
and
<br /> y(0) = y_0<br />

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.

 

[calvinjhfeng]

Actually may I ask what program/software did you use to type/print math operators, symbols, and etc ?

 

[Dickfore]

latex.