Let's consider a type II superconductor as a dissipative medium, so that it possesses a penetration depth (also called skin depth) that can act upon a penetrating time-varying EM field (affecting both $\vec{E}$ and $\vec B$). Now, if this superconductor is below the critical temperature and is under the influence of an applied $\vec B$ field whose magnitude lies in between the two critical magnetic fields, it will have a London penetration depth affecting the $\vec B$ field that penetrates inside it.

Are the two penetration dephs the same? At first glance, by looking at the formulae* of the penetration depths, I'd tend to think not. One depends on the angular frequency of the $\vec B$ field (skin depth) while the other (London penetration depth) doesn't seem to be frequency dependent.

If they are indeed different, what would happen to the magnetic field inside of the superconductor if we apply an external time-varying $\vec B$ field such that $B_{c1}<|\vec B|<B_{c2}$? I'd tend to think it would suffer from both penetration depths effects. Is this correct? One due to being superconductor of type II and the other due to being a dissipative medium.

(*): $\lambda_L = \sqrt{\frac{m}{\mu _0 n q^2}}$ for the London depth and $\delta = \sqrt{\frac{2\rho}{\omega \mu}}\sqrt{\sqrt{1+(\omega \varepsilon \rho)^2} + \omega\varepsilon\rho}$ for the skin depth.


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