London Equations: The Electrodynamics of Superconductors

As far as the Equations of the Month series is concerned,till now, we have covered different fields in physics. So today, its the turn of another such booming field, Condensed Matter Physics. Moreover, we will specifically deal with Superconductivity today. And, the equations we are going learn about today are the London Equations.

Meaning of London Equations:

The Equation on the left side says that inside a superconductor, a current will flow without any resistance. This current will never die out and will keep flowing while the second equation points out the peculiar property of expulsion of magnetic field by a superconductor. No magnetic field lines pass through a superconductor.

London Equations

Without a doubt, the equations and the terms involved are quite complicated. So, Let us try to understand these two beautiful phenomena in a simple way. For that we start with defining the term superconductivity.

You know the basic property of conductors. Suppose I have a wire through which the current is flowing. If I increase the temperature, the amount of resistance will increase. Why? Because the electrons gain more energy and the rate of collision increases thus hindering their free flow in the wire. Also, On the similar terms I can say that if I start reducing the temperature, the resistance will start decreasing and will reach its minima only at absolute zero of temperature.

London Equations
Resistance vs Temperature for Normal Conductor and Superconductor.

Discovery of Superconductivity:

Now comes the twist. It was a fine day on April 8, 1911 and Dutch Physicist, K.Onnes was working in his lab. He observed something really strange. So strange that at first he thought it was an error in his own experiment. He discovered that when mercury was cooled below 4.2 K, its electrical resistance suddenly disappeared. That temperature, below which the resistance becomes zero is now known as the critical temperature. Subsequently, it was later discovered that many elements and compounds possess their own critical temperature. So, This was the discovery of superconductivity.

London's first Equation:

Several attempts were made to theorize the phenomenon of superconductivity. One such attempt to formulate the mathematics of superconductivity was made by the London brothers. They assumed that there are two types of electrons inside a metal: normal electrons and super electrons. Below 4.2 K, the number of super electrons dominate the normal electrons and thus they flow with zero resistance. This is exactly what the first equation says: The rate of change of current density ( current density is the current flowing per unit area) only depends on the applied electric field and the number of super electrons. It has no resistance term as in the case of normal conductors.

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Meissner's Effect and London's Second Equation:

Soon after the discovery, Many independent groups of scientists started their research in superconductivity (that's how science works). Soon a major new discovery was made which showed that superconductors are much more than just zero resistance materials. In 1933, Meissner and Ochsenfeld discovered that when a superconductor is placed in a weak magnetic field below its critical temperature, it ejects magnetic field as shown below.

The Meissner Effect (expulsion of magnetic field from a superconductor)

This is what the second equation says. The second equation is a differential equation of second order having exponentially decaying solution. This means that the magnetic field lines die out exponentially. Also, the amount of penetration is given by λ which is very small. This effect is known as the Meissner effect. The two equations together are called London Equations . They perfectly describe the electrodynamics of superconductivity.

Author's Message:

Superconductivity is a very interesting branch of Condensed Matter Physics. The problem is that it is only attainable at very low temperatures. If we develop a material whose critical temperature is near to the room temperature, believe me, it will be the turning point in the history of the mankind. Just imagine, no resistance, no power loss. The average life of a super-current, if we allow it to flow, is more than the age of the universe! And to do all this, the understanding of London equations is must.

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