Method 1 : When the concave part is filled with water of refractive index 4/3, the optical arrangement is equivalent to concave mirror of focal length F such that
1F=1fw+1fg+1fm+1fg+1fw
1fw=(43−1)(160−1∞)=1180
fw=180 cm
fg=60 cm (calculated earlier)
1F=1180+160+110+160+1180=26180
F=18026cm
X1=R=2F=18026×2=18013=13.85 cm
Method 2 : We use the equation
n2x2−n1x1=n2−n1R
For refraction at the interface ′1′ (air water),
4/3x2−1x1=4/3−1∞....(i)
The image of interface ′1′ is the object for the interface ′2′.
1.5+20−1x1=1.5−4/3+60
X1=36026=13.85
△X=15.0−13.85=1.15 cm
Method 3 : Using lensmaker's formula and the relation
1F=1x2−1x1
fm=180 cm (using lensmaker's formula)
fg=60 cm (using lensmaker's formula)
1−180=1x2−1x1 (for the water lens) ... (i)
1−60=1+20−1x1 (for the glass lens) ... (ii)
(The image by the water lens is the object for the glass lens and if the image by the glass lens is at +20, then the rays will fall normally on the mirror.)
Adding Eqs. (i) and (ii).
1−180+1−60=120−1x1
x1=1803=13.85 cm
△x=15.0−13.85=1.15 cm.