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Question
A ray of light is incident on a glass prism of refractive index µ and refracting angle A. If it just suffers total internal reflection at the other face, obtain a relation between the angle of incidence, angle of prism and critical angle.
Solution
For surface 1:
Snell's law
sin i = µ sin r1
r2 = ic
In quadrilateral OPQR,
∠OPQ + ∠PQR + ∠QRO + ROP = 360°
90° + ∠PQR + 90° + A = 360°
∠PQR = 180° - A
In ΔPQR,
`r_1 + r_2 + 180^circ - A = 180^circ`
`r_1 + r_2 = A`
`r_1 = A - i_c` ....[∵ r2 = ic]
sin i = µ sin(A - ic) .....(i)
For surface 2:
µ sin ic = sin90°
`mu = 1/sini_c` .....(ii)
`sini = 1/(sini_c) xx sin(A - i_c)` .....[from (i)]
`i = sin^-1((sin(A - i_c))/(sini_c))`
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