51. Two similar thin equi-convex lenses, of focal length f each, are kept coaxially in contact with each other such that the focal length of the combination is $F_1$. When the space between the two lenses is filled with glycerin (which has the same refractive index (= 1.5) as that of glass) then the equivalent focal length is $F_2$. The ratio $F_1 : F_2$ will be ?

51. Two similar thin equi-convex lenses, of focal length f each, are kept coaxially in contact with each other such that the focal length of the combination is $F_1$.  When the space between the two lenses is filled with glycerin (which has the same refractive index (= 1.5) as that of glass) then the equivalent focal length is $F_2$. The ratio $F_1 : F_2$ will be ?

(a) 2 : 1

(b) 1 : 2

(c) 2 : 3

(d) 3 : 4

Answer :  (b) 1 : 2

\( \frac{1}{F_1} = \frac{1}{f} + \frac{1}{f} = \frac{2}{f} \)

\( \frac{1}{F_2} = \frac{1}{f} + \frac{1}{f} + (\mu_{21} - 1)\left( \frac{-2}{R} \right) \)

\( = \frac{1}{f} + \frac{1}{f} - \frac{1}{f} \)

\( = \frac{1}{f} \)

\( \frac{1/F_2}{1/F_1} = \frac{1/f}{2/f} = 1:2 \)

\( F_1 : F_2 = 1 : 2 \)