Past exam of the optics and electrodynamics course of the University of Cambridge / 2018 / 1 / a / Solution

The London penetration depth measures the density-to-mass ratio of carriers participating in superconducting screening. With electron-counted density $n_s$ and the stipulated bare electron mass $m_e$,Using $m_e=9.109\times 10^{-31}\mathrm{kg}$, $e=1.602\times 10^{-19}\mathrm{C}$, ${\mu }_0\simeq 4\pi \times 10^{-7}\mathrm{H}{\mathrm{m}}^{-1}$ and ${\lambda }_L=22\times 10^{-9}\mathrm{m}$ givesThis is the requested conduction-electron density within the assumed local London equation model. More precisely, it is the superconducting carrier density from London screening; identifying it with all conduction Electrons assumes that they participate in screening at the measurement temperature. No temperature-dependent normal-carrier fraction is supplied.

If the condensate is instead counted as Cooper pairs, the pair mass is $2m_e$ and charge magnitude is $2e$, giving pair density $n_{\mathrm{pair}}=n_s/2\simeq 2.9\times 10^{28}{\mathrm{m}}^{-3}$. Keeping Electron and pair counting consistent avoids a spurious factor of two.