A hollow double concave lens is made of very thin transparent material. It can be filled with air or either of the two liquids \( L_1 \) or \( L_2 \) having refractive indices \( n_1 \) and \( n_2 \) respectively (\( n_2 \gt n_1 \gt 1 \)). The lens will diverge a parallel beam of light if it is filled with
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( L_2 \) inside and immersed in \( L_1 \)
By the lens maker's formula for a lens of refractive index \( n_{\mathrm{in}} \) placed in a medium of index \( n_{\mathrm{out}} \),
\( \displaystyle \frac{1}{f} = \left(\frac{n_{\mathrm{in}}}{n_{\mathrm{out}}} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \)
For a double concave shape, \( R_1 \lt 0 \) and \( R_2 \gt 0 \), so \( \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \) is negative. The lens diverges (\( f \lt 0 \)) only if the first bracket is positive, that is, only if
\( n_{\mathrm{in}} \gt n_{\mathrm{out}} \)
The thin walls do not matter; the filling acts as the lens.
Option (1) Air inside and placed in air — Incorrect. \( n_{\mathrm{in}} = n_{\mathrm{out}} = 1 \), so \( 1/f = 0 \): the beam passes undeviated.
Option (2) Air inside and immersed in \( L_1 \) — Incorrect. \( n_{\mathrm{in}} = 1 \lt n_1 \): the first bracket is negative, so \( f \gt 0 \) and the lens converges.
Option (3) \( L_1 \) inside and immersed in \( L_2 \) — Incorrect. \( n_1 \lt n_2 \): again \( n_{\mathrm{in}} \lt n_{\mathrm{out}} \), so the concave lens converges.
Option (4) \( L_2 \) inside and immersed in \( L_1 \) — Correct. \( n_2 \gt n_1 \): the filling is optically denser than the surroundings, so the concave lens diverges.