The figure below shows an equiconvex lens (n = 1.5) placed in contact with a thin liquid layer resting on a plane mirror. A small needle, with its tip positioned on the principal axis of the lens, is moved along the axis until its inverted image coincides with the needle tip itself.
When the liquid is present, the distance between the needle and the lens is found to be 50 cm. The experiment is then repeated after removing the liquid, and the distance is observed to be 35 cm. The refractive index of the liquid is-
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1.30
The needle’s image coincides with the needle when rays from its tip leave the lens parallel to the axis, strike the plane mirror normally and retrace their path. So the needle is at the focus of the lens system placed above the mirror.
Without the liquid, the system is the glass lens alone, so its focal length is \( f_1 = 35 \) cm. Using the lens maker’s formula for the equiconvex lens (\( R_1 = +R,\ R_2 = -R \)):
\( \displaystyle \frac{1}{f_1} = (n - 1)\left(\frac{1}{R} + \frac{1}{R}\right) = \frac{2(1.5 - 1)}{R} = \frac{1}{R} \)
\( \displaystyle R = f_1 = 35\,\mathrm{cm} \)
With the liquid, the glass lens and the liquid layer act as two thin lenses in contact, with combined focal length F = 50 cm.
\( \displaystyle \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \)
\( \displaystyle \frac{1}{f_2} = \frac{1}{50} - \frac{1}{35} = \frac{7 - 10}{350} = -\frac{3}{350}\,\mathrm{cm^{-1}} \)
The liquid forms a plano-concave lens: its upper surface follows the lower face of the glass lens (\( R_1 = -35 \) cm) and its lower surface is plane (\( R_2 = \infty \)). Let its refractive index be \( \mu \).
\( \displaystyle \frac{1}{f_2} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) = (\mu - 1)\left(-\frac{1}{35} - 0\right) \)
\( \displaystyle -\frac{\mu - 1}{35} = -\frac{3}{350} \)
\( \displaystyle \mu - 1 = 0.30 \)
\( \displaystyle \mu = 1.30 \)
Note: NTA picked this question from an NCERT (2026-27) exercise and changed the figures. Do not ignore NCERT while practising numericals.
NCERT (2026-27):
Exercises – …
