Assuming that all the charge carriers are superelectrons, there is no power dissipation inside the supercon ductor, and so Poynting s theorem over a volume V may be written x hdv (53.5) where the left side of the expression is the power flowing into the region. By taking the time derivative of the energy density and appealing to Faraday's and Ampere's laws to find the time derivatives of the field quantities, ve find that the only way for Poynting's theorem to be satisfied is if (53.6) This relation, known as the first London equation(after the London brothers, Heinz and Fritz), is thus necessary if the superelectrons have no resistance to their motion Equation (53. 6)reveals that the superelectrons' inertia creates a lag between their motion and that of an pplied electric field. As a result, a superconductor will support a time-varying voltage drop. The impedance associated with the supercurrent is therefore inductive and it will be useful to think of A as a kinetic inductance created by the correlated motion of the Cooper pairs If the first London equation is substituted into Faraday's law,VxE=-(aB/ar), and integrated with respect to time, the second London equation results V×(AJ.)=-B (53.7) where the constant of integration has been defined to be zero. This choice is made so that the second London equation is consistent with the Meissner effect as we now demonstrate. Taking the curl of the quasi-static form of Amperes law, VX H=J,, results in the expression V2B=-kV X J,, where a vector identity, VxVX C v(V. C)-V2C; the constitutive relation, B=kH; and Gauss's law, V.B=0, have been used By now appealing to the second London equation, we obtain the vector Helmholtz equatio V-B B=0 where the penetration depth is defined n*(q) From Eq. (53.8), we find that a flux density applied parallel to the surface of a semi-infinite superconductor will decay away exponentially from the surface on a spatial length scale of order A. In other words, a bulk superconductor will exclude an applied flux as predicted by the Meissner effect The London equations reveal that there is a characteristic length A over which electromagnetic fields can change inside a superconductor. This penetration depth is different from the more familiar skin depth of electromagnetic theory, the latter being a frequency-dependent quantity. Indeed, the penetration depth at zero temperature is a distinct material property of a particular superconductor Notice that A is sensitive to the number of correlated electrons( the superelectrons)in the material. As previously discussed, this number is a function of temperature and so only at t=o do all the electrons that usually conduct ohmically participate in the Cooper pairing. For intermediate temperatures, 0< T<T, there c2000 by CRC Press LLC© 2000 by CRC Press LLC Assuming that all the charge carriers are superelectrons, there is no power dissipation inside the supercon￾ductor, and so Poynting’s theorem over a volume V may be written (53.5) where the left side of the expression is the power flowing into the region. By taking the time derivative of the energy density and appealing to Faraday’s and Ampère’s laws to find the time derivatives of the field quantities, we find that the only way for Poynting’s theorem to be satisfied is if (53.6) This relation, known as the first London equation (after the London brothers, Heinz and Fritz), is thus necessary if the superelectrons have no resistance to their motion. Equation (53.6) reveals that the superelectrons’ inertia creates a lag between their motion and that of an applied electric field. As a result, a superconductor will support a time-varying voltage drop. The impedance associated with the supercurrent is therefore inductive and it will be useful to think of L as a kinetic inductance created by the correlated motion of the Cooper pairs. If the first London equation is substituted into Faraday’s law, ¹ ¥ E = –(]B/]t), and integrated with respect to time, the second London equation results: (53.7) where the constant of integration has been defined to be zero. This choice is made so that the second London equation is consistent with the Meissner effect as we now demonstrate. Taking the curl of the quasi-static form of Ampère’s law, ¹ ¥ H = Js , results in the expression ¹2 B = –mo ¹ ¥ Js , where a vector identity, ¹ ¥ ¹ ¥ C = ¹(¹ • C) – ¹2 C; the constitutive relation, B = moH; and Gauss’s law, ¹ • B = 0, have been used. By now appealing to the second London equation, we obtain the vector Helmholtz equation (53.8) where the penetration depth is defined (53.9) From Eq. (53.8), we find that a flux density applied parallel to the surface of a semi-infinite superconductor will decay away exponentially from the surface on a spatial length scale of order l. In other words, a bulk superconductor will exclude an applied flux as predicted by the Meissner effect. The London equations reveal that there is a characteristic length l over which electromagnetic fields can change inside a superconductor. This penetration depth is different from the more familiar skin depth of electromagnetic theory, the latter being a frequency-dependent quantity. Indeed, the penetration depth at zero temperature is a distinct material property of a particular superconductor. Notice that l is sensitive to the number of correlated electrons (the superelectrons) in the material. As previously discussed, this number is a function of temperature and so only at T = 0 do all the electrons that usually conduct ohmically participate in the Cooper pairing. For intermediate temperatures, 0 < T < Tc , there - —× ¥ ( ) = Ú Ú E H dv w t dv V V ¶ ¶ E J = ( ) ¶ ¶t L s — ¥ (LJ B ) = - s —- = 2 2 1 B B 0 l l º m = ( ) m L o o m n q * * * 2