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Deduce the relation between n,u,v, Q, R for refraction at a spherical surface, where the symbols have their usual meaning. |
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Answer» Solution :OM = u = object distance MI = v = image distance MC = R = radius of curvature Angle i = angle of incidence Angle r = angle of refraction ON = INCIDENT RAY NI = refracted ray  NC=normal & `n_1,n_2` are the refractive indices From the figure for SMALL angles tan`angleNOM=(MN)/(OM)`, tan `angleNCM=(MN)/(MC), tan angleNCM=(MN)/(MI)` In the triangle NOC, `hati` = exterior angle = sum of the interior opposite angles `anglei=angleNOC + angleNCO=angleNOM+angleNCM=(MN)/(OM)+(MN)/(MC)` Similarly , `angleNCM = angleCNI + angleNIC = angleCNI+angleNIM` `angler=angleCNI=angleMCN-angleNIM=(MN)/(MC)-(MN)/(MI)` From Snell.s law `n_1` sin i = `n_2` sin r Forsmall angles `n_1 i = n_2 r` Substituting the VALUES of i and r we get `n_1((MN)/(OM)+(MN)/(MC))=n_2((MN)/(MC)-(MN)/(MI))` `n_1/(OM)+n_2/(MI)=(n_2-n_1)/(MC)` APPLYING sign convention, OM=-u , MI=v , MC=R `n_1/v-n_2/u=(n_2-n_1)/R` This is the required expression. |
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