prism and refractive index of prism

 PRISM:

 

A prism is a transparent medium separated from the surrounding medium by at least two plane surfaces which are inclined at a certain angle in such a way that, light incident on one of the plane surfaces emerges from the other plane surface.

Consider a triangular glass prism. It contains two triangular bases and three rectangular plane lateral surfaces. These lateral surfaces are inclined to each other. Let us consider that triangle PQR represents outline of the prism where it rests on its triangular base. 

    Let us assume that a light ray is incident on the plane surface PQ of a prism at M as shown in figure. Draw a perpendicular to the surface at M. It becomes a normal to that surface. The angle between the incident ray and normal is called angle of incidence (i₁). The ray is refracted at M. It moves through prism and meets the other plane surface at N and finally comes out of the prism. The ray which comes out of the surface PR at N is called emergent ray. Draw a perpendicular to PR at point N. The angle between the emergent ray and normal is called angle of emergence (i₂ ).The angle between the plane surfaces PQ and PR is called the angle of the prism or refracting angle of prism A and the angle between the incident ray and emergent ray is called angle of deviation(d).

 TO FIND THE REFRACTIVE INDEX OF A PRISM EXPERIMENTALLY:

     Take a prism and place it on the white chart in such a way that the triangular base of the prism is on the chart. Draw a line around the prism (boundary) using a pencil. Remove the prism.

It is a triangle. Name its vertices as P, Q, and R.[for many prisms the triangle formed is equilateral]. The refracting surfaces could be rectangular in shape. Find the angle between PQ and PR . This is the angle of the prism (A). Mark M on the side of triangle PQ and also draw a perpendicular to PQ at M. Place the centre of the protractor at M and along the normal. Mark an angle of 300 and then draw a line up to M. This line denotes the incident ray. This angle is called angle of incidence. Note it in a table

Now join the points M and N by a straight line. The line passing through the points A, B, M, N, C and D represents the path of light when it suffers refraction through the prism. Extend both incident and emergent rays till they meet at a point ‘O’. Measure the angle between these two rays. This is the angle of deviation. It is denoted by a letter- ‘d’. Note it in table. Repeat this procedure for various angles of incidence such as 40⁰,50⁰ etc. Find the corresponding angles of deviation and angles of emergence and note them in table.

You will notice that the angle of deviation decreases first and then increases with increase in the angle of incidence.

 

Now try to plot a graph

Take angle of incidence along X- axis and the angle of deviation along Y- axis. Using a suitable scale, mark points on a graph paper for every pair of angles. Finally join the points to obtain a graph (smooth curve).

Draw a tangent line to the curve, parallel to X- axis, at the lowest point of the graph. The point where this line cuts the Y- axis gives the angle of minimum deviation. It is denoted by D. Draw a parallel line to y-axis through the point where the tangent touches the graph. This line meets x-axis at a point showing the angle of incidence corresponding to the minimum deviation. If you do the experiment with this angle of incidence you will get an angle of emergence equal to the angle of incidence.

Derivation of formula for refractive index of a prism

From triangle OMN, we get

d = i₁- r₁ + i₂ – r₂      

d = (i₁+i₂) – (r₁+r₂) ––—— (1)

From triangle PMN, we have

A + (900-r₁) + (900-r₂) = 1800

By simplification, we get

r₁ + r₂ = A ———(2)

From (1) and (2), we have

d = (i₁+i₂) – A

A+d = i₁+i₂ ––——(3)

This is the relation between angle of

incidence, angle of emergence, angle of deviation and angle of prism.

From Snell’s law, we know that n₁ sin i = n₂ sin r

Let n be the refractive index of the prism.

Using Snell’s law at M, with refractive index of air

n₁ =1; i = i1 ; n₂ = n ; r = r₁ , gives

sin i₁ = n sin r₁ ———(4)

similarly, at N with n₁ = n ; i = r₂ ; n₂ = 1 ; r = i₂ , gives

n Sin r₂ = Sin i₂ ———(5)

We know that at the angle of minimum deviation (D), the angle of incidence is equal to the angle of emergence i.e., i₁ = i₂.

You will note that MN is parallel to the side QR, actually ray MN is parallel to the base of the prism.

When i₁ = i₂, angle of deviation (d) becomes angle of minimum deviation (D).

Then equation (3) becomes

A+D = 2i₁

or i₁ = (A+D)/2

When i₁ = i₂ then, it is clear that r₁ = r₂

So from equation (2) we get,

2r₁ = A

or r₁ = A/2

Substituting i₁ and r₁ in (4) we get

Sin{(A+D)/2)} = n. Sin(A/2)

n = Sin(A+D/2)/Sin(A/2)

 This is the formula for the refractive index of the prism.

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