Derive relation between v,u,R for concave refracting surface for both below cases: 1. when light ray travels from rarer to denser medium and virtual image is formed. 2. when light ray travels from denser to rarer medium and virtual image is formed. please give diagram also...

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2016-05-03T16:19:04+05:30

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Case 1.   
   see diagram. 

angles x =  - MP/R                x+r = - MP/v                   x+i = - MP/uso,   r = MP * [1/R - 1/v]                       i = MP * [1/R  - 1/u]
From Snell's law:       Mu1 *  [1/R - 1/u] = Mu2 *  [1/R - 1/v]
So   Mu2/v - Mu1/u = (Mu2 - Mu1) /R  or,    Mu21 /v - 1/u  = (Mu21 - 1) / R

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case 2:
   the derivation is similar... 

for small angles x, r, i :
x = tan x = MP'/P'C = - MP'/Ri+x = tan i+x = MP'/P'A = - MP'/u
MA'P' = i+x+r-i = x+r = tan (x+r)     = MP'/P'A' = - MP'/v
Snells law:    Mu1 * i  = Mu2 * r
=> Mu1 [1/R - 1/u] = Mu2 [1/R - 1/v]
=> Mu2 /v - Mu1/u = (Mu2 - Mu1)/R         or,  Mu21 /v - 1/u =  (Mu21 - 1)/R
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