It is quite evident that whenever we are using the terms country, state, region, language etc. we indirectly refer certain boundaries either they are natural or man-made. In scientific problems also, many times we come across boundaries. In mathematics, certain problems are defined as boundary value problems. One of the significant features of nanotechnology is the interface (boundary) property. Physically, boundaries possess high potential. Generally, boundaries are thought in terms of “limitation” or “restriction” (think about boundaries between countries). Interestingly, boundaries have continuity property too. Normally, we miss the continuity aspect of boundaries. Here I tried to explore this aspect of boundaries in certain fields.
First, let us look into the social boundaries. Social boundaries, here it is referred to the boundaries between countries and between states.
Culture: Culture is the one which is always continuous across any social boundaries either man-made or geographical. One can examine this with the boundaries between states or countries. Let us take India and Pakistan or India and Nepal. Across the border, people from the two countries have cultural similarities.
Language: Let us take Tamilnadu-Kerala boundary. Here, some components of language is continuous across the boundary. For example, Wayanad district of Kerala is geographically isolated from Tamilnadu by Western Ghats. Even then, the tribal people of Wayanad speak Malayalam in Tamil slang with many ancient Tamil words. It is much different from conventional Malayalam used in other parts of Kerala. Also, in Kanyakumari district, Tamil is spoken in Malayalam slang with certain Malayalam words.
Let us take coastal area of Orissa and Andhra Pradesh. The mother tongue of majority of coastal fishermen of southern and central Orissa is Telugu not Oriya! Here language is continuous!
One can extend these things further.
Let us turn our attention to science.
In Electromagnetic theory: Consider the boundary between two dielectric media. When an electromagnetic wave passes across this boundary, Maxwell’s theory tells that normal component of the magnetic field and the tangential component of the electric field is continuous!
In Polycrystals: We all know that all metals and alloys are polycrystals and contain numerous grains with different orientations separated by grain boundaries. In the case of Diffusion induced grain-boundary motion (DIGM), the normal component of the stress tensor across the grain boundaries is continuous in order to maintain the system to be in elastic equilibrium.
In Magnetic materials: Ferromagnetic materials contain numerous magnetic domains with different orientations of magnetization separated either by Bloch walls or Neel walls (boundaries). Interestingly, phase angle and the normal component of the magnetization across the bloch wall (boundary) are continuous!
Electronics: In PN junction diodes, a boundary is formed between P type and N type material. The charge density, electric field and electrostatic potential across the potential barrier are contiuous!
In Semiconductor-metal Junction: At the juncture of Semiconductor-metal (normally called ohmic contact) the electric potential across the boundary is continuous!
In Sound: During the propagation of sound waves across the two different media, the particle velocity and partial pressures across the boundary are continuous.
Vapour pressure: The origin of vapour pressure is essentially linked with boundary. Consider a liquid gas interface. Due to evaporation, the liquid at the interface become vapour, making the mass transport of the liquid across the liquid-gas interface continuous.
In human physiology: Brain is made up of special type of cells, called neurons. These neurons form a collection of tissues based depending on their functions. Since, neuron contains many ions, it is electrically active. Across the tissue boundaries, the tangential component of the electric and magnetic fields and their curls are continuous!
Surprises:
In optics: When a ray light passes through the dielectric boundary (air to glass) refraction occurs. Refraction is basically a continuous property of the light wave across the boundary. But problem comes when light travels from a denser medium to a rarer medium (glass to air) where beyond a particular angle, all the light gets reflected into the denser medium (total internal reflection). Means no refraction at all! No continuity… But physics didn ’t abandon me! Yes, even in the case of total internal reflection, some waves are continuous across the interface. Those waves are called evanescent waves!
Quantum tunnelling: Many problems in quantum mechanics have boundaries with certain potentials (consider “particle across a potential barrier” problem). Unless the potential is infinity, the wave functions of the particles are always continuous across the boundaries. One striking example is quantum tunnelling. Though the particle has a lesser energy than the potential at the boundary, the continuity of wave function assures a certain probability across the boundary which is classically impossible.
This is very similar to that of evanescent waves. Infact the solutions to Schrodinger wave equation in this case are called evanescent wave solutions! Hence, the tunnelling property is not a new phenomenon which is basically a property of waves at boundaries.