High Temperature Superconducting Magnetic Levitation

8 Ë 1 Fundamentals of superconductivity

where∇is the curl operator,Eis the electric field, andBis the magnetic field. We have

휕

휕t

∇×∇×B=

휇 0 nse^2

m

∇×E= −

휇 0 nse^2

m

⋅휕B

휕t

. (1.15)

Because

∇×∇×

휕B

휕t =∇œ∇ ⋅

휕B

휕t−∇

2 휕B

휕t, (1.16)

and∇ ⋅B=0, one obtains

휇 0 nse^2

m

휕B

휕t

=∇^2 휕B

휕t

. (1.17)

Thus,Bmust satisfy this differential equation in superconductors, leading to a
characteristic length scale known as the London penetration depth,휆L,

휆L=œ m

휇 0 nse^2



1 / 2

. (1.18)

One obtain

∇^2 B=^1

휆^2 L

B, (1.19)

or

∇×Js=nse

2

m

B=^1

휇 0 휆^2 L

B. (1.20)

This is the second London equation, and the first London equation can be written as

휕

휕t

Js=nse

2

m

E=^1

휇 0 휆^2 L

E. (1.21)

The second London equation leads to the Meissner-Ochsenfeld effect, and can also be
derived from the BCS theory by assuming that the BCS state is spatially homogeneous.
The London equation predicts that the magnetic field in a superconductor decays
exponentially from the surface into the superconducting material.

1.6 Penetration depth

The London penetration depths휆Ldescribes the distance to which the magnetic
flux penetrates into the superconductor from the surface between air and metal.