Read the following passage and answer the questions.
In a school workshop, Riya takes a long glass rod of refractive index 1.5. She grinds one end of the rod into a convex spherical surface of radius of curvature 10 cm. The centre of curvature lies inside the glass. She places a small lamp in air on the axis of the rod, facing the curved end. Then she moves the lamp to different distances from this end. For some positions, the light comes to a focus inside the glass. For other positions, the light spreads out inside the rod. Next, she cuts a thin disc from the curved end and polishes its back face flat. This gives her a thin plano-convex lens. From the same glass, she also grinds a thin double convex lens with both faces of radius 10 cm, and a thin double concave lens. Her teacher asks her to explain what she sees with the relation for refraction at a single spherical surface and the lens maker’s formula.
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The thin double convex lens has a shorter focal length than the thin plano-convex lens.
Reason (R): A lens of shorter focal length bends the incident light more.
In the light of the above statements, choose the most appropriate answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Both (A) and (R) are correct but (R) is NOT the correct explanation of (A).
Assertion (A) The thin double convex lens has a shorter focal length than the thin plano-convex lens. — True. By the lens maker’s formula with \( n_{21}=1.5 \), each curved face adds a term \( 1/R \). The flat face adds nothing because its radius is infinite.
\( \begin{aligned} \frac{1}{f_{\text{plano}}} &= 0.5\left(\frac{1}{10}-\frac{1}{\infty}\right)=\frac{1}{20} \\ \frac{1}{f_{\text{double}}} &= 0.5\left(\frac{1}{10}+\frac{1}{10}\right)=\frac{1}{10} \end{aligned} \)
So the plano-convex lens has \( f=20\ \mathrm{cm} \) and the double convex lens has \( f=10\ \mathrm{cm} \).
Reason (R) A lens of shorter focal length bends the incident light more. — True. A shorter focal length means a larger power, so the lens converges or diverges light more.
Why Reason (R) does not explain Assertion (A)?
R starts from the focal length and tells how strongly the lens bends light. It does not say why one lens has the shorter focal length.
The cause is the second curved face. Its radius enters the lens maker’s formula, so \( 1/f \) of the double convex lens is twice that of the plano-convex lens.
NCERT (2026-27):
Refraction by a lens – “So … can be written as”
\( \displaystyle \frac{1}{f}=(n_{21}-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right) \)\( \qquad (9.21) \)
“Equation (9.21) is known as the lens maker’s formula. …”
Power of a lens – “… a lens of shorter focal length bends the incident light more, while converging it in case of a convex lens and diverging it in case of a concave lens.”