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Derive an expression for refraction at a...

Derive an expression for refraction at a single spherical surface, i.e., a relation between u, v, R, `n_(1)` (rarer medium) and `n_(2)` (denser medium), where the terms have their usual meaning.

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In the figure, SPS. represents a convex spherical surface of radius of curvature R with its centre of curvature at C. O is a point object and I is its real image formed due to refraction by the surface.

In `DeltaCOA`, i is the exterior angle,
So `i=alpha+gamma" "......(1)`
In `DeltaACI,gamma` is the exterior angle,
So `gamma=r+beta`
`r=gamma-beta" "......(2)`
By Snels.s law,
`(sini)/(sinr)=(mu_(2))/(mu_(1))`
`mu_(1)sini=mu_(2)sinr`.
Since Since aperture is small so `anglei` and `angler` will be small.
So,
`:.sini=iandsinr=r`
`mu_(1)i=mu_(2)r`
`impliesmu_(1)(alpha+gamma)=mu_(2)(gamma-beta)`
[Using (1) and (2)]
`mu_(2)beta+mu_(1)alpha=(mu_(2)-mu_(1))gamma" ".......(3)`
So, `alpha=tanalpha=(h)/(-u),beta=tanbeta=(h)/(v),`
`gamma=tangamma=(h)/(R)`
Substituting the above values in equation (3),
`mu_(2)(h)/(v)+mu_(1)(h)/(-u)=(mu_(2)-mu_(1))(h)/(R)`
or `(mu_(2))/(v)-(mu_(1))/(u)=(mu_(2)-mu_(1))/(R)`
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