**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 i_{1} . QR is the refracted ray inside the prism, which makes two angles r_{1} and r_{2} (inside the prism). RS is the emergent ray at angle i_{2}.

The angle between the emergent ray and incident ray is the deviation ‘d’.

In the quadrilateral AQMR,

∠A + ∠R = 180°

[since N_{1}M are normal]

ie, ∠A + ∠M = 180° .......(1)

In the ∆ QMR,

∴ r_{1} + r_{2} + ∠M = 180° .......(2)

Comparing eq (1) and eq (2)

r_{1} + r_{2} = ∠A ....... (3)

From the ∆ QRT,

(i_{1} – r_{1}) + (i_{2} – r_{2}) = d

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

(i_{1} + i_{2}) – (r_{1} + r_{2}) = d

but, r_{1} + r_{2} = A

∴ (i_{1} + i_{2}) – A = d

(i_{1} + i_{2}) = d + A .......(4)

It is found that for a particular angle of incidence, the deviation is found to be minimum value ‘D’. At the minimum deviation position,

i_{1} = i_{2} =i, r_{1} = r_{2} = r and d = D

Hence eq (3) can be written as,

r + r = A

or r = \(\frac{A}{2}\) .....(5)

Similarly eq (4) can be written as,

i + i = A + D

n = \(\frac{A+D}{2}\) ......(6)

Let n be the refractive index of the prism, then we can write,

n = \(\frac{sin\,i}{sin\,r}\) .....(7)

Substituting eq (5) and eq (6) in eq (7),

n = \(Sin\frac{A+D}{\frac{2}{Sin\frac{A}{2}}}\)

i – d curve:

It is found that when the angle of incidence increases deviation (d) decreases and reaches a minimum value and then increases. This minimum value of the angle of deviation is called the angle of minimum deviation.

3. Refracted ray is parallel to base