Diamagnetism — Question 9

Question 9

Flux freezing and equilibrium flux expulsion predict different histories

A simply connected sample initially contains uniform induction Bi through an interior material loop of area A. Compare two idealizations after cooling in the maintained applied field: a perfect conductor with E=0, and a reversible Meissner state with negligible penetration depth and interior B=0. No trapped vortices are present.

1. Use Faraday law to determine the final loop flux in each model.

2. If cooling takes time tau, find the average electric circulation needed for the Meissner flux change, using the initial flux as positive.

Question 9 - Solution

Flux freezing and equilibrium flux expulsion predict different histories

(a) Method and calculation

$$E=0\Rightarrow\frac{d\Phi}{dt}=-\oint\mathbf E\cdot d\boldsymbol\ell=0\Rightarrow\boxed{\Phi_{\mathrm{conductor}}=B_iA}.$$$$B_{\mathrm{Meissner}}=0\Rightarrow\boxed{\Phi_{\mathrm{Meissner}}=0}.$$

(b) Interpretation and further calculation

$$\int_0^\tau\oint\mathbf E\cdot d\boldsymbol\ell\,dt=-\Delta\Phi=B_iA.$$$$\boxed{\left\langle\oint\mathbf E\cdot d\boldsymbol\ell\right\rangle=B_iA/\tau}.$$

Physical check

Zero resistance alone preserves the initial flux; equilibrium expulsion requires a transient electric circulation while the flux changes.