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Demonstrate that in a prism with small r...

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.

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In the general case, for the passage of a monochromatic ray through a prism as shwon in the figure of the soln of
`alpha = (alpha_(1) + alpha_(2))` (1)
And from the snell's law,
`sin alpha_(1) = n sin beta_(1)` or `alpha_(1) sin^(-1) (n sin beta_(1))`
Similarly `alpha_(2) = sin^(-1) (n sin beta_(2) = sin^(-1) 9n sin (theta - beta_(1))]` (As `theta = beta_(1) + beta_(2))` (2)
Using (2) in (1)
`alpha = [sin^(-1) (n sin beta_(1)) + sin^(-1) (n sin (theta - beta_(1)))] - theta`
For `alpha` to be minimum, `(d alpha)/(d beta_(1)) = 0`
or, `(n cos beta_(1))/(sqrt(1-n^(2) sin^(2) beta_(1)))-(n cos(theta - beta_(1)))/(sqrt(1-n^(2) sin^(2)(theta - beta_(1)))) =0`
or, `(cos^(2) beta_(1))/((1-n^(2) sin^(2) beta_(1))) = (cos^(2) (theta - beta_(1)))/(1-n^(2) sin^(2)(theta - beta_(1)))`
or, `cos^(2) beta_(1) (1-n^(2) sin^(2) (theta - beta_(1))) = cos^(2) (theta - beta_(1)) (1-n^(2) sin^(2) beta_(1))`
or, `(1-sin^(2) beta_(1)) (1-n^(2) sin^(2) (theta - beta_(1))) = (1-sin^(2) (theta - beta_(1))) (1-n^(2) sin^(2) beta_(1))`
or, `1 - n^(2) sin^(2) (theta - beta_(1)) - sin^(2) beta_(1) + sin^(2) beta_(1) n^(2) sin^(2) (theta - beta_(1))`
`= 1-n^(2) sin^(2) beta_(1) - sin^(2) (theta - beta_(1)) + sin^(2) beta_(1)n^(2) sin^(2) (theta - beta_(1))`
or, `sin^(2) (theta - beta_(1)) -n^(2) sin^(2) (theta - beta_(1)) = sin^(2) beta_(1) (1-n^(2))`
or, `sin^(2) (theta - beta_(1)) (1-n^(2)) = sin^(2) beta_(1) (1-n^(2))`
or, `theta - beta_(1) = beta_(1)` or `beta_(1) = theta//2`
But `beta_(1) + beta_(2) = theta`, so, `beta_(2) = theta//2 = beta_(1)`
Which is the case of symmetric passage of ray.
In the case of symmetric passage of ray
`alpha_(1) = alpha_(2) = alpha'` (say)
and `beta_(1) = beta = theta//2`
Thus the total deviation
`alpha = (alpha_(1) + alpha_(2)) - theta`
`alpha = 2alpha' - theta` or `alpha' = (alpha + theta)/(2)` (1)
But from the Snell's law `sin alpha = n sin beta`
So, `sin .(alpha + theta)/(2) = n sin.(theta)/(2)`
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