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Question
For any prism, prove that :
'n' or `mu = sin((A + delta_m)/2)/sin(A/2)`
where the terms have their usual meaning
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Solution
In the figure, a ray of light PQ is incident at an angle i on the face AB of prism ABC. This ray is
refracted along QR at an angle r. This reflected ray is incident on the face AC at an angle r ' and
emerges along RS at an angle i '.
In ΔQDR
`delta = (1 - r )+ (i' + r')`
`= (i + i') - (r + r')` ....(i)
In Quad. AQER, A + E = 180° ...(ii)
In ΔQER r + r' + E = 180° ...(iii)
r + r' = A [From eq ii and iii]
Putting value r + r' inequation (i)
`delta = i + i' - A`
In the position of minimum deviation condition
`i = i', r = r', delta = delta_m`
So r + r' = A
2r = A
or `r = A/2`
`delta_m = 2i - A`
`i = (A + delta_m)/2`
Putting value of i and r from (v), (vi), in Snell’s law,
`n = sini/sin r`
`m = (sin ((A+delta_m)/2))/(sin (A/2))`
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