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I am currently reading 'Introduction to Superconductivity' by Michael Tinkham. In the second chapter (introduction to electrodynamics of superconductors) Tinkham goes on to describe the idea of the intermediate state. He first uses a slab in a parallel field and the uses a sphere in a field.

In the description of the sphere it turns out(I will not add the derivation for simplicity) that the magnetic field outside a superconducting sphere is:

$$ \textbf{B} = \textbf{Ha} + \frac{H_aR^3}{2} \nabla (\frac{cos(\theta)}{r^2})$$

Where $H_a$ is the external magnetic field strength

R is the radius of the sphere

$\theta$ is the polar angle

Solving for the polar angle of $\textbf{B}$ you find that at r=R

$$(B_\theta)_R=\frac{3}{2}H_asin\theta$$

Which is where the nature of the intermediate state occurs since for different values of $\theta$ on the sphere you will have different field strength, some of which can go over the critical field $H_c$ and cause there to be an intermediate state. This state exists for the range of values $\frac{2}{3}<\frac{H_a}{H_c}<1$.

Now later it does an analysis of the object. It says that the "Volume of the sphere is subdivided into a superconducting and normal laminae".

Question: What is a laminae?

Second part which i really don't understand is the following:

"The flux density in the normal laminae is always exactly $H_c$ and the normal fraction(fraction of normal conducting state on the sphere) is $\rho_n=\frac{B}{H_c}$ where $\textbf{B}$ is the average of $\textbf{h(r)}$ over the laminar structure"($\textbf{h(r)}$ is the manetic field inside the superconductor i think)

This sentence really messes me up. I don't get why the flux density has to be $H_c$ in the normal region and why the normal density is that formula. Furthermore I'm getting really confused by what $\textbf{B}$ actually is.

Please help!

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