Derive the equation for angle of deviation produced by a prism and thus obtain the equation for refraction for refactive index of material of the prism.

Explanation

Let light ray PQ is incident on one of the refracting faces of the prism. The angles of incidence and refraction at the first face AB are `i_(1) and r_(1)`. The path of the light inside the prism is QR. The angle of incidence and refractive at the second face AC is `r_(2) and i_(2)` respectively. RS is the ray emerging from the second face. Angle `i_(2)` is also called angle of emergence. The angle between the direction of the incident ray PQ and the emergent ray RS is called the angle of devition d. The two normals drawn at the point of incidence Q and emergence R are QN and RN. They meet at point N. The incident ray and the emergent ray meet at a point M.
The deviation `d_(1)` at the surface AB is,
angle `angleRQM = d_(1) = i_(1) - r_(1)`
The deviation `d_(2)` at the surface AC is,
angle `angle QRM = d_(2) = i_(2) - r_(2)`
Total angle of deviation d produced is,
`d = d_(1) + d_(2)`
Substituting for `d_(1) and d_(2)`.
`d = (i_(1) - r_(1)) + (i_(2) - r_(2))`
After rearranging.
`d = (i_(1) - r_(1)) + (i_(2) - r_(2))`
In the quadrilateral AQNR, two of the angles (at the vertices Q and R) are right angles, Therefore, the sum of the other angles of the quadrilatera is `180^(@)`.
`angleA + angleQNR = 180^(@)`
From the triangle `Delta QNR`,
`r_(1) + r_(2) + angleQNR = 180^(@)`
Comparing these two equation (5) and (6) we get,
`r_(1) + r_(2) = A`
Substituting this in two equaiton (4) for angle of deviation,
`d = i_(1) + i_(2) - A`
Thus, the angle of deviation depends on the angle of incidence angle of emergence and the angle for the prism.
Refractive index of the material of the prism:
At minimum deviation, `i_(1) = i_(2) = i and r_(1) = r_(2) r`
Now, the equation (8) becomes, `D = i_(1) + i_(2) - A = 2i - A (or) i = (A + D)/(2)`
The equation (7) becomes, `r_(1) + r_(2) = A rArr = A (or) r = (A)/(2)`
Substituting i and r in Snell.s law, `n = (sin i)/(sin r)`
`n=(sin(A+D)/(2))/(sin((A)/(2)))`
The above equation is used to find the refractive index of the material of the prism.