Question

show that the refractive index of the material of giver prism ia given by ​

Answer 1

For any prism, show that refractive index of its material is given by: n or μ=sin(2A)sin(2A+δm) where the terms have their usual meaning.Oct 15, 2019

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Answer 2

Refraction through a prism

Here we will be deriving the formula of refractive index of a glass prism.

Let,

A = Angle of prism (angle b/w two refracting surfaces)

1st Medium is air.

air.2nd Medium is glass.

glass.μ - Refractive index of glass.

If ∠ i1 is in rarer medium then ∠ i2 will also be in rarer medium.

If ∠ r1  is in denser medium then ∠ r2 will also be in denser medium.

n1 and n2 are the normals meeting at O.

In Quadrilateral, AQORA,

∠A + ∠O =180° .........................................Eq.(1)

In ΔQOR,

∠r1  + ∠r2 + ∠O = 180°  ..........................................Eq.(2)

Comparing Eq. (1) and Eq. (2)

∠A + ∠O = ∠r1 + ∠r2 + ∠O

∠A = ∠r1 +∠r2

RS is Emergent Ray.

δ = Angle of Deviation.

= i1 - r1 + i2 - r2

  = i1 + i2 - (r1 + r2)

= i1 + i2 - A

 A = i1 + i2 ......................................Eq.(3)

μ = sin i / sin r   (i ≈ sin i ≈ tan i)

μ = i1/i2

i1 = μr1

lly, i2 = μr2

∴ Eq.(3) becomes :-

     A = μr1 + μr2

         = μ(r1 + r2)

= μ(r1 + r2) - A

  = μ(A) - A

= A(μ-1)

Now, i1 = i2 = i

i1 = i2 = i        r1 = r2 = r

∵ i1 + i2 =   A

  i + i = δm + A   (δm = Angle of minimum deviation)

  2i = A + δm

  i = A + δm/2

and

∵ A = r1 + r2

∴ A = r + r

  A = 2r

  r = A/2

Now, substitute these values in the formula of refractive index which is:-

μ = sin i / sin r

μ   = sin [(A + δm)/2] / sin A/2

Hence, we have derived the formula of refractive index of the prism.