The focal length of a convex lens in air and in a liquid medium are 20 cm and 80 cm, respectively. If the absolute refractive index of lens is 1.5, then the refractive index of liquid is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 4/3 \)
By the lens maker's formula, for a lens of refractive index \( n_l \) in a medium of refractive index \( n_m \),
\( \displaystyle \frac{1}{f} = \left(\frac{n_l}{n_m} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \)
Let \( K = \frac{1}{R_1} - \frac{1}{R_2} \) (the same lens in both cases). In air \( n_m = 1 \):
\( \displaystyle \frac{1}{20} = (1.5 - 1)K = 0.5\,K \)
In the liquid of index \( n \):
\( \displaystyle \frac{1}{80} = \left(\frac{1.5}{n} - 1\right)K \)
Dividing the second by the first:
\( \displaystyle \frac{1.5/n - 1}{0.5} = \frac{20}{80} = \frac{1}{4} \;\Rightarrow\; \frac{1.5}{n} - 1 = \frac{1}{8} \)
\( \displaystyle \frac{1.5}{n} = \frac{9}{8} \;\Rightarrow\; n = \frac{1.5 \times 8}{9} = \frac{4}{3} \)