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 face of the lens is found to be `25//8` cm. Find the focal length of the lens.

Explanation

Refraction at plane surface:
`mu_1=mu,mu_2=1,u=-4cm,v=-3cm`
`(mu_2)/(v)-(mu_1)/(u)=u`
`(1)/(-3)-(mu)/(-4)=0impliesmu=(4)/(3)`
Refraction at spherical surface:
`mu_1=(4)/(3),mu_2=1,u=-4cm,v=-(25)/(8)cm,R=-R`
`(mu_2)/(v)-(mu_1)/(u)=(mu_2-mu_1)/(R)`
`(1)/((-25)/(8))-((4)/(3))/(-4)=(1-(4)/(3))/(-R)`
`(-8)/(25)+(1)/(3)=(1)/(3R)implies(1)/(3R)=(-24+25)/(75)=(1)/(75)`
Thin lens:
`(1)/(f)=(mu-1)((1)/(R_1)-(1)/(R_2))=((4)/(3)-1)((1)/(25)-(1)/(infty))`
Thick lens ltbtgt Refraction at (1):
`((4)/(3))/(v)-(1)/(infty)=((4)/(3)-1)/(25)impliesv=100cm`
Refraction (2):
`(1)/(v)-((4)/(3))/((100-4))=0`
`v=72cm`
`f=72cm`