From London Equations to Penetration Depth(Integrate Laplacian)

In summary, the conversation discusses the equations related to superconductivity, specifically London's 2nd equation and Ampere's law, and their application in solving for the second order ODE related to the magnetic field in a superconductor. The solution to the ODE is given as y = c*exp(x/λ) + c*exp(-x/λ), and the use of del² as a second derivative is mentioned. The conversation also mentions the use of latex for typing mathematical equations.
  • #1
calvinjhfeng
32
0
(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.
 
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  • #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]
 
  • #3
Dickfore said:
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]

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.
 
  • #4
Actually may I ask what program/software did you use to type/print math operators, symbols, and etc ?
 
  • #5
latex.
 

FAQ: From London Equations to Penetration Depth(Integrate Laplacian)

1. What are the London equations?

The London equations are a set of equations that describe the behavior of superconductors. They were first proposed by brothers Fritz and Heinz London in 1935 and are based on the idea that when a superconductor is in its superconducting state, it can conduct electricity without any resistance.

2. How do the London equations relate to penetration depth?

The London equations provide a way to calculate the penetration depth of a superconductor, which is the distance that an external magnetic field can penetrate into the superconductor. This depth is an important characteristic of superconductors and is related to their ability to expel magnetic fields.

3. What is the significance of the penetration depth in superconductors?

The penetration depth is a measure of the strength of the superconducting state in a material. The smaller the penetration depth, the stronger the superconductivity. It also determines the critical current, which is the maximum amount of current that a superconductor can carry without losing its superconducting properties.

4. How is the penetration depth calculated using the Laplacian?

The Laplacian is a mathematical operator that is used to describe the curvature of a function. In the case of superconductors, the Laplacian can be used to describe the magnetic field distribution within the material. By integrating the Laplacian, we can calculate the penetration depth of a superconductor.

5. Can the London equations and penetration depth be applied to all superconductors?

The London equations and penetration depth are applicable to most type I superconductors, which are characterized by a single critical temperature and a single critical magnetic field. However, they are not applicable to type II superconductors, which have multiple critical temperatures and critical magnetic fields, and require more complex equations to describe their behavior.

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