Demonstrate that in a prism with small refracting angle `theta` the shaft of light deviates through the angle `alpha ~~(n - 1) theta` regard less of the angle of incidence, provided that the latter is also small.

Explanation

The figure shows the passage of a monochromatic ray thorugh the given prisen, placed in air medium.
From the figure, we have
`theta = beta_(1) + beta_(2)` (A)
and `alpha = (alpha_(1) + alpha_(2)) - (beta_(1) + beta_(2))`
`alpha = (alpha_(1) + alpha_(2)) - theta` (1)
From the snell's law
`sin alpha_(1) = n sin beta_(1)`
or `alpha_(1) = n beta_(1)` (for small angles) (2)
and `sin alpha_(2) = n sin beta_(2)`
or, `alpha_(2) = n beta_(2)` (for small angles) (3)
From Eqns (1), (2) and (3), we get
`alpha = n(beta_(1) + beta_(2)) - theta`
So, `alpha = n (theta) - theta = (n -1) theta` [Using Eqn.A]