Question

The following graph represent id curve of a optical instrument placed in air.

1. Name the device which give the above i-d curve.

2. Obtain an expression for deviation produced by such a device.

3. What is the relevance of the value ‘D’? Arrive at an expression for refractive index in terms of this value from (b).

4. How is the deviation affected if the above arrangement is immersed in a liquid of refractive index less than that of the above device.

Answer 1

1. Prism.

2. Refraction through a prism:

ABC is a section of a prism. AB and AC are the refracting faces, BC is the base of the prism. ∠A is the angle of prism. Aray PQ incidents on the face AB at an angle i₁ . QR is the refracted ray inside the prism, which makes two angles r₁ and r₂ (inside the prism). RS is the emergent ray at angle i₂ .

The angle between the emergent ray and incident ray is the deviation ‘d’. In the quadrilateral AQMR, 

∠A + ∠R = 180°

[since N₁ M are normal]

ie, ∠A + ∠M = 180°..........(1)

In the ∆ QMR, 

∴ r₁ + r₂ + ∠M = 180° ...... (2)

Comparing eq (1) and eq (2)

r₁ + r₂ = ∠A ......(3)

From the ∆ QRT,

(i₁ – r₁) + (i₂ – r₂) = d

[since exterior angle equal sum of the opposite interior angles]

(i₁ + i₂) – (r₁ + r₂) = d
but, r₁ + r₂ = A
∴ (i₁ + i₂) – A = d

3. At minimum deviation D = 2i – A, r₁ = r₂ = r

i = \(\frac{A+D}{2},r = \frac{A}{2}\)

n = \(\frac{sin\,i}{sin\,r}\) = \(Sin\frac{A+D}{\frac{2}{Sin\frac{A}{2}}}\)

4. Deviation decreases.