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Question
At what angle should a ray of light be incident on the face of a prism of refracting angle 60° so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524.
Solution
The incident, refracted and emergent rays associated with a glass prism ABC are shown in the given figure.
Angle of prism, ∠A = 60°
Refractive index of the prism, µ = 1.524
i1 = Incident angle
r1 = Refracted angle
r2 = Angle of incidence at the face AC
e = Emergent angle = 90°
According to Snell’s law, for face AC, we can have:
`(sin "e")/(sin "r"_2) = µ`
`sin "r"_2 = 1/µ xx sin 90°`
= `1/1.524`
= 0.6562
∴ r2 = sin−1 0.6562 ≈ 41°
It is clear from the figure that angle A = r1 + r2
∴ r1 = A − r2 = 60 − 41 = 19°
According to Snell’s law, we have the relation:
µ = `(sin "i"_1)/(sin "r"_1)`
`sin "i"_1 = µ sin "r"_1`
= 1.524 × sin 19°
= 1.524 × 0.3256
= 0.496
∴ i1 = 29.75°
Hence, the angle of incidence is 29.75°.
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