The first phenomenological theory for the superconductivity consists of two Londons' equations. After Londons' theory, Bardeen, Cooper, and Schrieffer set up the microscopic theory (BCS theory) for conventional superconductors. The BCS theory states that, below the critical temperature, electrons of a metal are pairing to form quasi-particles, named as Cooper's pairs. The supperconducting state is the condensate of macroscopic number of Cooper's pairs in a unique quantum state. We give a detail derivation of Londons' equations based on the BCS picture, emphasizing the local gauge invariant and taking the variation of the density of Cooper's pairs into account.

Faraday's law of electromotive force describes two different phenomena: (1) changing magnetic flux through a surface induces an electromotive force in the boundary circuit of the surface; (2) a wire moving in a magnetic field is exerted a motional electromotive force. It is known that the fundamental principles supporting both phenomena are the Maxwell equations and the Lorentz formula for the electromagnetic force. However, there are different statements explaining the latter phenomenon. This paper will discuss the motional electromotive force from the viewpoint of constraint on charge trajectory. A charge moving in a predetermined trajectory is exerted a constraint force generally. When the trajectory is time-depedent, the magnetic field extends the constraint force to do work on the charge and leads to the motional electromotive force. It is convenient to describe the energy of charge moving in a predetermined trajectory by electromotive force, as it can avoid the constraint force which is usually unknown. However, strictly speaking, this description is only valid in quasi-static processes. A general concept of the electromotive force for arbitrary processes may be defined, but the generalization is not unique. Here we define the electromotive force as the work done to a positive unit charge when the charge moves along the trajectory instantly. This definition is valid for induced and motional electromotive forces of wires of arbitrary time-depending shape, whatever close or open.