A planoconvex len becomes an optical system of 28 cm focal length when its plane surface is silvered illuminated from left to right as shown in Fig-A.
It the same lens is instead silvered on the curved surface and illuminated from other side as in Fig.B, it acts like an optical system of focal length 10 cm. The refractive index of the material of lens is :

Options

A. 1.55
B. 1.5
C. 1.75
D. 1.51

Answer

A

Explanation

(a) : Case : I

This system behaves like a concave mirror.
If `f_(1)` is focal length of lens then
`(1)/(f_(1))=(mu-1)/(R )`
If `f_(2)` is focal length of plane mirror then
`f_(2)=oo`
If f is equivalent focal length then
`f=-28cm`
Lens is used twice hence equivalent power can be written as follows :
`P=2P_(1)+P_(2)`
`(2)/(28)=2((mu-1)/(R ))" ...(i)"`
Case - II:

This system also behaves like a concave mirror.
If `f_(1)` is focal length of lens then
`(1)/(f_(1))=(mu-1)/(R )`
If `f_(2)` is focal length of mirror then
`f_(2)=(R )/(2)`
If f is equivalent focal length then `P=2P_(1)+P_(2)rArr(1)/(10)=2((mu-2)/(2))+(2)/(R )`
`(1)/(10)=(1)/(28)+(2)/(R )`
`(2)/(R )=(1)/(10)-(2)/(28)=(18)/(280)`
`R=(280)/(9)cm`
From equation `(i) (1)/(28)=2((mu-1)/(280))9`
`mu-1=(5)/(9)`
`mu=1+(5)/(9)=(14)/(9)=1.55`