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Prove the following formula when refract...

Prove the following formula when refraction takes place at a convex spherical refracting surface and source of light lies in the rarer medium and image formed is real `frac(mu_2)(v)-frac(mu_1)(u)=frac(mu_2-mu_1)(R)` Where the terms have their usual meanings.

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Consider a convex spherical refracting surface of refractive index `mu_(2)` and is placed in a rarer medium of refractive index `mu_(1)`.
Let O be the point object lying on the principal axis in medium. A ray of light from O strikes the convex surface at A and after refraction it bends towards the normal and meets the principal axis at I on producing backwards.
I is the virtual image of the object.
I is the virtual image of the object.

Let `alpha, beta` and `gamma` be the angles made by the incident ray, refracted ray and the normal respectively with the principal axis. Draw AN perpendicular on the principal axis.
From `DeltaAOC,i=alpha+gamma`........i
Since angles `alpha` and `gamma` are small (assumption) so they can be replaced by their tangents. Therefore eqn (i) can be written as
`i=tan alpha+tan gamma`...........ii
Also from `DeltaIC,r=beta+gamma`...........iii
Since angles `beta` and `gamma` small (assumption), so they can be replaced by their tangents.
Hence eq. iii can be written as `r=tan beta +tangamma`
From `DeltaANO" " tan alpha=(AN)/(NI)`.........v
From `DeltaANI" "tan beta=(AN)/(NI)`.........vi
From `DeltaANC" "tan gamma=(AN)/(NC)`.......vii
Using eqns (v) to (vii) in eqns (ii) and (iv) we get
`i=(AN)/(NO)+(AN)/(NC)`...........viii
and `r=(AN)/(NC)+(AN)/(NI)`..........ix
Since aperture of the spherical surface is small (assumption) so pont N lies very close to point.
`:.NO=PO,NC=PC,NI~=PI`
Hence eqn viii and ix can be writen as
`i=(AN)/(PO)+(AN)/(PC)`..........x
and `r=(AN)/(PI)+(AN)/(PC)` .........xi
According to Snell.s law
`(sini)/(sinr)=(mu_(2))/(mu_(1))` or `mu_(1)sini=mu_(2)sinr`
Since angles i and r are small so
`mu_(1)i=mu_(2)" "` (`:.sini=i` and `sinr=r)`...xii
Using eqn (x) and (xi) in eqn (xii) we get
`mu_(1)[(aN)/(PO)+(AN)/(PC)]=mu_(2)[(AN)/(PI)+(AN)/(PC)]` .........xiii
Applying sign conventions we have
`PO=-u,PC=+R,PI=-v`
From eqn. iii
`(mu_(1))/(PO)+(mu_(1))/(PC)=(mu_(2))/(PI)+(mu_(2))/(PC)`
`implies(mu_(1))/(-u)+(mu_(1))/R=(mu_(2))/(-v)+(mu_(2))/R`
`implies(mu_(1))/(-u)+(mu_(1))/v=(mu_(2))/R-(mu_(1))/R`
`implies(-mu_(1))/u+(mu_(2))/v=(mu_(2)-mu_(1))/R`
which is the required expression.
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