A plano-convex lens has thickness 4cm. When places on a horizontal table with the curved surface in contact with it, the apparent depth of the bottom-most point of the lens if found to be 3cm. If the lens is inverted such that the plane face is in contact with the table, the apparent depth of the center of the plane face of the lens is found to be `25//8` cm. Find the focal length of the lens.

Explanation

When the curved surface of the lens (refractive index `mu` ) is in contact with the table, the image of the bottom-most point of lens (in glass ) is formed due to refraction at plane face. The image of O appears at `I_(1)` .
Here, ` u_(1)=AO=-4cm, v_(1)=AI_(1)=3cm,`
`mu_(1)=mu`, and `mu_(2)=1`, and `R_(1)=oo`
`:. (mu_(2))/(v)-(mu_(1))/(u)=(mu_(2)-mu_(1))/(R_(1))` gives `(1)/(-3)-(mu)/(-4)=(1-mu)/(oo)` (i)
When the plane surface of the lens is in contact with the table, the image of center of the plane face if formed due to refraction curved surface. The image of O is formed aaaaaaaat `I_(2)`.
Here, `u=AO=-4cm, upsilon=AI_(2)=-25//8cm`
`mu_(1)=mu, mu_(2)=1`, and `R_(2)=-R`
`:. mu_(2)/v_(2)-mu_(1)/mu_(2)=(mu_(2)-mu_(1))/R_(2)`
Gives `(1)/((-(25)/(8)))-(mu)/(-4)=(1-mu)/(-R)`
From Equ. (i), `mu=4//3`, therefore this equation gives
`-(8)/(25)+(4//3)/(4)=-((1-(4)/(3)))/ (R)-(8)/(25)+(1)/(3)=(1)/(3R)` or `(1)/(75) =(1)/(3R)`
This gives `R=25 cm`.
The focal length (f) of plano-convex lens `(R_(1) =R " and " R_(2)=oo)`
is `(1)/(f) =(mu-1)((1)/(R)-(1)/(oo))=(mu-1)/(R)=((4)/(3)-1)/(25)=(1)/(75)rArr f=75cm`