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\begin{document}
\title{Electrodynamics of Superconductors in Weak Fields}
\author{ Amensisa Abdi\\
	 \\         Department of Physics
            \\          University of Cincinnati 
              \\         Cincinnati, Ohio 45221\\[1em]}
\date{Nov 28, 2001 }
\maketitle
\begin{abstract}The discovery of superconductivity in a mercury sample
 by Heike Kamerlingh Onnes in 1911 have led to the rapid development
 and advancement of high $T_{c}$ superconductivity technology [1]. 
The highest $T_{c}$ of superconductors was that of the compound Nb3Ge until
 1986 when two scientists J. G. Bednoz and K. A. Müller
 reported possible superconductivity in a mixture of La and Ba Copper 
Oxides at temperature of 30 K [2]. More recently, scientists have
 reported superconductivity at a temperature as high as 150 K in 
an oxide containing mercury [3]. However, to develop a new technology
 that utilizes superconductors, a complete understanding of the 
macroscopic electrodynamics of superconductors is important. 
Therefore, in this paper, some basic phenomena of superconductors 
and the macroscopic electrodynamic theory of superconductors in weak 
fields are discussed. \\[1em]
\end{abstract}

\pagebreak
\def\baselinestretch{1.1}


\section*{1. Introduction}

The fascinating phenomena of superconductors and their potential 
applications had attracted not only experimentalists for finding 
new superconductors, but also attracted theoreticians in order to 
understand the physics behind superconductivity to describe the 
electrodynamics and thermodynamics properties of superconductivity. 
Intense research has been done toward this both on the microscopic 
and the macroscopic properties of superconductivity.  In general, 
the microscopic properties of superconductivity are described by the 
well-known BCS theory which is developed by Bardeen, Cooper and 
Schriefier in 1957, where as the macroscopic behaviors of 
superconductivity are described by the well-known London theory 
developed by the brothers F. and H. London in 1937[4].

\section*{2. Basic Phenomena of Superconductivity}

Conduction electrons that is free to move through the material
 carry the current in conductors. It is well known that electrons
 have a wave like nature and thus can be represented by a plane wave
 traveling in the same direction. The property of plane wave is that
 it can pass through a perfect crystal without being scattered since 
the atoms in a perfect crystal has a definite equilibrium location in
 regular array. That means, electrons can move through a perfect 
crystal without experiencing any resistance. However, actual crystal 
is never perfect due to defects like missing atoms, presence of 
impurity, atoms out of place, and thermal vibration of atoms. 
Therefore, plane wave moving in a crystal scatters. And electrons
 moving in materials experience resistance.

Generally, resistance of a metal or alloy decreases with 
temperature since the thermal vibration of atoms decrease. 
In 1911, Onnes found that the resistance of a mercury sample fell
 sharply at a temperature below 4.2 K. Soon, this new phenomenon of 
zero dc-resistance at low temperature was found in may other metals. 
The temperature at which superconductivity first occurs is thus termed
 the superconducting transition (or critical) temperature of the
 material and denoted by $T_{c}$. The practical measurement of $T_{c}$ of a
 superconductor is usually made using three basic methods, method 
of resistance measurement, methods of magnetic-permeability 
measurement, and methods of specific heat measurement.

\begin{figure}[htb]
\centerline{\psfig{figure=amen1.ps,height=2.5in}}
\caption{Resistance of a mercury sample at low temperature}
\end{figure}

It has been found that the behavior of zero dc-resistance of 
superconductor is modified significantly at very high frequency 
of alternating current as shown by London in 1940[4]. The two-fluid
 electron model may explain this property of superconductors.
 According to two-fluid electron model, super electrons and normal 
electrons whose relative density depends on temperature carry current 
in superconductors. In dc case, current is total carried by 
superelectrons which does not need lectric field to move. In 
alternating field case, current in superconductors is carried 
by normal electrons in addition to superelectrons and thus resistance.
 When the frequency is in the order of 1013 Hz, the resistance of 
superconductors would the same as the resistance of normal conductor
 independent of temperature.
Onnes also found that the superconductivity could be destroyed when
 magnetic field is applied. The minimum magnetic field needed to 
destroy depends on the shape and orientation of the sample. When a 
cylindrical specimen is placed in a magnetic field in such a way that
 its axis is parallel the field, the transition is sharp and the 
minimum magnetic field needed to destroy superconductivity is 
called the critical magnetic field and denoted by Bc[1]. Within only 
small deviations, the following relation gives the temperature 
dependence of Bc.
\vspace{0.1in}
\begin{eqnarray}
B_{c}(T)&=&B_{co}[1-(\frac{T}{T_{c}})^2]
\end{eqnarray}
\vspace{0.1in}

Where $B_{co}$ is critical field at zero temperature.

\begin{figure}[htb]
\centerline{\psfig{figure=amen2.ps,height=4in}}
\caption{The critical magnetic field as a function of temperature}
\end{figure}

The absence of any magnetic flux in superconductivity is other
 fundamental property of superconductors (The Meissner Effect)[1]. 
Thus, superconductor behaves not only like perfect conductor of infinite conductivity but also like perfect diamagnetic material of B=0. The infinite conductivity of superconductors is taken for grant to explain electrodynamics of superconductors till Meiser and Ochsenfeld (1933) measured the field surrounding a superconductor and concluded that the magnetic induction field B inside a superconductor is zero.

\begin{figure}[htb]
\centerline{\psfig{figure=amen3.ps,height=3in}}
\caption{The Meissner effect of a superconductor}
\end{figure}

\section*{3. The London Theory of Superconductors }
Before the discovery of the Meissner effect in superconductors,
 the electrodynamic behaviors of superconductors is analyzed with 
the use of a simple free electron model, in which electrons 
accelerate without any resistance under applied electric field. E. Becker, Heller, and Sauter argued that if the electrons encounter no resistance, an applied electric filed would accelerate them steadily according to Newton’s law [4]:
\vspace{0.1in}
\begin{eqnarray}
m^{*}\frac{dv_{s}}{dt} &=& e^{*}E
\end{eqnarray}
\vspace{0.1in}

Where m* and e* are the effective mass and charge of the 
superelectrons respectively.  The supercurrent density denoted by Js 
can be written in terms of the mean local velocity Vs as

\vspace{0.1in}
\begin{eqnarray}
J_{s}&=&n_{s}e^{*}v_{s}
\end{eqnarray}
\vspace{0.1in}

Where $n_{s}$ is the superelectrons number density. Substituting 
eq. (3) into eq. (2) and taking linear approximation, we may
 obtain

\vspace{0.1in}
\begin{eqnarray}
\mu_{o}\lambda_{L}^{2}\frac{\partial{J_{s}}}{\partial{t}} &=& E
\end{eqnarray}
\vspace{0.1in}

Where $\lambda_{L}$ is a parameter of length dimension defined by

\vspace{0.1in}
\begin{eqnarray}
\lambda_{L}&=&\sqrt{\frac{m^{*}}{\mu_{o}n_{s}e^{*}^{2}}}
\end{eqnarray}
\vspace{0.1in}

By taking the curl of both sides of eq. (4) and using Maxwell 
equations, we get

\vspace{0.1in}
\begin{eqnarray}
\mu_{o}\lambda_{L}^{2}\frac{\partial}{\partial{t}}(\nabla \times  J_{s})
 &=& -\frac{\partial{B}}{\partial{t}}
\end{eqnarray}
\vspace{0.1in}

Using Maxwell equations and assuming any flux density in 
superconductors is due to current (no displacement current), 
we obtain

\vspace{0.1in}
\begin{eqnarray}
\lambda_{L}^{2}\frac{\partial}{\partial{t}}[\nabla \times ( \nabla \times 
 J_{s})] &=& -\frac{\partial{B}}{\partial{t}}
\end{eqnarray}
\vspace{0.1in}

Noting divergence of B is zero and using vector identity eq. (7) can
 be rewritten as:

\vspace{0.1in}
\begin{eqnarray}
\lambda_{L}^{2}\nabla^{2}\frac{\partial{B}}{\partial{t}} &=& 
-\frac{\partial{B}}{\partial{t}}
\end{eqnarray}
\vspace{0.1in}

Substituting eq. (3.7) into eq. (3.5), we can obtain

\vspace{0.1in}
\begin{eqnarray}
B&=& -\mu_{o}\lambda_{L}^{2}\nabla\times J_{s}
\end{eqnarray}
\vspace{0.1in}

The set of equations (4) and (9) describes the electrodynamic 
behaviors of superconductor according to London theory. 
Equation (4) is called the first London theory, describing the
 resistanceless properties of a superconductor, and equation (9)
 is called the second London equation, describing the Messner effect.
 Equation (9) can be also derived from more fundamental idea if
 superelectron fluid could be treated as a quantum mechanical 
quantity [4].






\section*{4. Conclutions }
Superconductivity occurs in a metal or alloy below critical 
temperature. Superconductors are not only perfect conductors 
in which dc-resistance is zero but also they are perfect diamagnetic 
in which B field is zero. London electrodynamics theory developed by 
the brothers F. and H. London describes superconductors in week 
fields. That means London theory is in consistence with the
 experimental results zero dc-resistance and Meissner effect. 



\begin{thebibliography}{99}
\bibitem{refA} Grosso, Giuseppe and Pastori Parravicini, Giuseppe, Solid State 
Physics (2000).
\bibitem{refB} J.G.\ Bednorz and K.A.
\ M\"uller, Z.\ Phys.\ {\bf 64}, 189(1986).

\bibitem{refC} A. Serway, Raymond and S. Faughn, Jerry, College Physics (1999)
\bibitem{refD} Zhou, Shu-Ang, Electrodynamics of Solids and Microwave 
Superconductivity (1999)

\end{document}