Question

For any prism, prove that :

'n' or `mu = sin((A + delta_m)/2)/sin(A/2)`

where the terms have their usual meaning

Answer 1

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))`