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Total Internal Reflection
When light is going from a less dense to a more dense medium, there is always some light refracted and some light reflected.
However, when light is going from a more dense to a less dense medium, there is no refracted light if the angle of incidence is greater than a certain value.
This "special" angle of incidence is called the critical angle of incidence and its size depends on the refractive index of the two media.
The diagrams below illustrate this phenomenon.

The critical angle is the angle of incidence which produces an angle of refraction of 90°.
For glass, in air, the angle is about 42°.

This angle is clearly dependent on the refractive indices of the materials involved.
We will consider the situation in which light meets an interface between glass and air (or a vacuum*).

Applying Snell's law to this situation, we can write
where gna represents the refractive index for light going from the glass to the air.

Therefore, when i=c we can write
because r is 90°
However, you are more likely to see this relation written as
in which case, n represents the absolute refractive index of the glass (that is, the refractive index for light going from air into glass).

* the speed of light in air is very close to its speed in vacuum (108ms-1).

Optical Fibres
Optical fibres are important in various communication systems notably in high speed internet connections where they have largely replaced co-axial cables.

Optical fibres are made from a core (cylinder) of glass of a certain refractive index, n1, surrounded by a "cladding" of lower refractive index, n2.
The diagram below represents a cross-section of a typical fibre.
The diameter of a typical fibre is around 125 microns (1.210-4m)

If the light meets the interface between core and cladding at the critical angle, c, then i is the maximum angle of incidence which will allow transmission of light down the fibre.
We will find an expression for the maximum value of i.

Speed of light in vacuum = c
The refractive index for light going from medium 2 to medium 1 is defined as
also
therefore,
As shown above,
so we can write

and, from Snell's law

Looking at the diagram, it is clear that r+c=90° therefore

so

and, remembering that

which means that the relation we are looking for is

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