Refraction Through Prism
A ray of light suffers two refraction, on passing through a prism and hence deviates through a certain angle from its original path.
(a) Calculation of angle of deviation
In fig. ABC is principle section of a prism with angle of prism = A
A ray of light KL is incident on the face AB of the prism at ∠i1. It bends towards the normal N1O and is refracted along LM at ∠i1. The refracted ray LM is incident at ∠r2 on face AC of the prism. It bends away from normal N2O and emerges along MN at ∠i2. In passing through the prism, ray KL suffers two refractions and has turned through an ∠QPN = δ, which is the angle of deviation.
In Δ PLM, δ = ∠PLM + ∠PML
δ = (i1 – r1) + (i2 – r2)
δ = (i1 + i2) – (r1 + r2) (1)
In OLM, ∠O + r1 + r2 = 180° (2)
In quadrilateral ALOM,
As L + M = 180° (? each angle is 90°)
A + ∠O = 180° (? sum of four angles of a quad. = 360°)
Using eqn. (2),
∠O + r1 + r2 = A + ∠O
r1 + r2 = A (3)
put in (1), δ = (i1 + i2) – A (4)
If µ is refractive index of the material of the prism, then according to Snell’s law,
µ = sin i1/sin r1 = i1/r1 (when angles are small)
∴ i1 = µ r1; similarly, i2 = µ r2
Putting in (4), we get
δ = ( µ r1 + µ r2) – A
δ = µ (r1 + r2) – A
Using (3), we obtain,
δ = µA – A (5)
This is the angle through which a ray deviates on passing through a thin prism of small refracting angle A.
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