When the curved surface of the lens (refractive index μ) 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 I1.
Here, u1=AO=−4cm,v1=AI1=3cm,
μ1=μ, and μ2=1, and R1=∞
∴μ2v−μ1u=μ2−μ1R1 gives 1−3−μ−4=1−μ∞ (i)
When the plane surface of the lens in contact with the table, the image of center of the plane face is formed due to refraction at curved surface. The image of O is formed at I2.
Here, u=AO=−4cm,v=AI2=−25/8cm
μ1=μ,μ2=1, and R2=−R
∴μ2v2−μ1μ2=μ2−μ1R2
Gives 1(−258)−μ−4=1−μ−R
From Eq. (i), μ=4/3, therefore this equation gives
−825+4/34=−(1−43)R−825+13=13R or 175=13R
This gives R=25cm
The focal length (f) of plano-convex lens (R1=R and R2=∞) is 1f=(μ−1)(1R−1∞)=μ−1R=43−125=175⇒f=75cm