‘Month Of Equations’: What Do London Equations Really Mean?

October 12: London Equations

London equations
London equations

Meaning of 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.

The London equations and terms involved are complicated. Let us try to understand these two beautiful phenomena in a simple way. For that, we start by defining the term superconductivity.

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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. On similar terms I can say that if I start reducing the temperature, the resistance will start decreasing and will reach its minimum only at absolute zero of temperature.

Resistivity vs Temperature for Normal Conductor and Superconductor.
Resistivity vs Temperature for Normal Conductor and Superconductor.

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. It was later discovered that many elements and compounds possess their own critical temperature. This was the discovery of superconductivity.

Several attempts were made to theorize this phenomenon. 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 dominates 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.

People were fascinated by this new type of material. 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 a magnetic field as shown below.

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

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

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 mankind. Just imagine, no resistance, no power loss. The average life of a super-current, if it is allowed to flow, is more than the age of the universe!

For advanced study, see Superconductivity.

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