A solid glass sphere of radius 10 cm and refractive index 32 is placed in a medium of refractive index μ=2 . The hemispherical portion of the sphere is silvered. A point object P is placed at distance 30 cm from the centre of sphere. The distance of final image of point object P from center of sphere is 50n11 cm. Find the value of n.

Image formation due to first refraction through unslivered surface. μ1=2,μ2=32u=−20 cm and  r=+10 cm According to refractive surface formula,  μ2v−μ1u=μ2−μ1r⇒32v−2−20=32−2+10=−120∴v=−10 cm  Thus image P1 formed due to refraction through unsilvered surface behaves as object for silvered surface. The silvered surface behaves as concave mirror.  For concave mirror,  u'=−30 cm f'=−102=−5 cmAccording to mirror formula, 1v+1u=1f  ⇒1v−130=−15∴v=−6 cm  The image P2  formed due to reflection through silveredsurface behaves as object for second time refraction  through unsilvered surface. Here, μ1"=32,μ2"=2, u"=−14 cm,r"=−10cmμ2"v"−μ1"u"=μ2"−μ1"r" ⇒2v"−32(−14)=2−32−10=−120−328=−7−15140 ∴v"=−14022×2=−14011 cm  Thus the position of final image P3  from centre of sphere  is =10+14011=110+14011=25011cm=50n11cm ∵n=5