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When light propagates from air into glass or from glass in to air it may
change its direction of travel. Snell's law reveals the relationship between the
directions of travel in the two media.

n_{1}sinθ_{1 }= n_{2}sinθ_{2}.

Consider light propagating in glass with index of refraction n_{1 }=
1.5 towards a glass-air boundary. If the angle the light makes with the normal
to the boundary in the glass is θ_{1}, then
the angle it makes in the air is given by

sinθ_{2 }= (n_{1}/n_{2})sinθ_{1
}= 1.5 sinθ_{1}.

If sinθ_{1 }> (1/1.5) = 2/3, or
θ_{1 }> 41.8^{o}, then sinθ_{2}
is greater than 1 and there is no solution for θ_{2}.

The angle θ_{c} for which sinθ_{c
}= n_{2}/n_{1} = 1 is called the **critical angle**.

For angles
θ_{1} greater than the critical angle there
exists no solution for θ_{2}, and there is no
refracted ray.

The incident light is totally reflected, obeying the law of
reflection. If n_{2 }= 1.5 and n_{1 }= 1 then the critical angle
is θ_{c }= 41.8^{o}

**Total internal reflection**
occurs only if light travels from a medium of high index of refraction to a
medium of low index of refraction.

Summary: Let light travel from medium 1 into medium 2
and let n_{1 }> n_{2}.

Then the critical angle θ_{c} is given by sinθ_{c }= n_{2}/n_{1}.

For angles greater than the critical angle the incident light is totally
reflected, obeying the law of reflection.

A light source is at the bottom of a pool of water (n = 1.33). At what minimum angle of incidence will a ray be totally reflected at the surface?

Solution:

- Reasoning:

The angle θ_{c}for which sinθ_{c }= n_{2}/n_{1}= 1 is the critical angle. For angles greater than the critical angle there is no refracted ray. - Details of the calculation:

For total internal reflection sinθ ≥ sinθ_{c }= 1/1.33, θ ≥ 48.7^{o}.

Light enters through the side of a water-filled aquarium. It is
internally reflected at the critical angle θ_{c} at the top surface.
What is θ_{i} in terms of θ_{c} and n_{water}?

**Discuss this with your fellow students in the discussion forum!
Snell's law also applies to the ray entering the aquarium.**

An ordinary glass mirror consists of a reflective metallic coating on the back of a sheet of glass. This is not the only way to make a mirror. Total internal reflection can be exploited to make a perfectly reflecting mirror using only glass, with no metal backing. It is possible to use prisms of various shapes to reorient images.

**Right-Angle Prism**

The prism is used like a
mirror. If the refractive index of the prism is large enough, we
obtain Total Internal Reflection (TIR) and the prism acts like a
mirror of 100% reflectance.

How large must the refractive index be?

We want the critical angle θ_{c
}to be smaller than 45 degrees. Since for a glass prism in
air sinθ_{c} = 1/n, we need n > 1/sin(45^{o}) or n >
√2
= 1.414.

**Dove Prism**

This prism is used to "flip" an image without reorienting the optical axis.

**Penta Prism**

Like the Right-Angle Prism, this
prism reorients the optical axis by 90^{o}. However, the Penta Prism
does not flip the image. Penta Prisms are used in 35-mm
Single-Lens-Reflex (SLR) cameras to produce an upright image in the
viewfinder.

Assume
you are using total internal reflection to make a corner reflector. If
there is air outside and the incident angle is 45.0^{o}, what
must be the minimum index of refraction of the material from which the
reflector is made?

Solution:

- Reasoning:

For total internal reflection we need sinθ_{ }≥ sinθ_{c }= n_{2}/n_{1}. Here n_{2}= 1. Therefore sinθ_{c}= 1/n. - Details of the calculation:

We need sin(45^{o}) > 1/n, or n > 1/sin(45^{o}) or n > √2 = 1.414.

Additional information: The Physics Classroom: Refraction and the Ray Model of Light Lessons 3