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.
To get the relation between the object distance and the image distance for the curved end, Snell’s law \(n_1\sin i=n_2\sin r\) is written as \(n_1 i=n_2 r\). This step is allowed because
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → the aperture of the surface is small, so the angles are small
For a small angle \( \theta \) in radian, \( \sin\theta \) is nearly equal to \( \theta \). Snell’s law can be written with the angles themselves only when the angles of incidence and refraction are small.
Option (1) the aperture of the surface is small, so the angles are small — Correct. When the curved surface is small compared with the object and image distances, every ray meets it at a small angle. Then \( \sin i\approx i \) and \( \sin r\approx r \).
Option (2) the light passes from a rarer medium into a denser medium — Incorrect. The relation for a spherical surface holds for light going either way. The direction of travel does not make the angles small.
Option (3) the lamp is a point object placed on the principal axis — Incorrect. A point object on the axis only fixes where the rays start. A wide surface would still give large angles at its edges.
Option (4) the radius of curvature is small compared with the length of the rod — Incorrect. The length of the rod plays no part in refraction at the curved end. What matters is the size of the surface compared with the distances.
NCERT (2026-27):
Refraction at a spherical surface – “… As before, we take the aperture (or the lateral size) of the surface to be small compared to other distances involved, so that small angle approximation can be made.”
Refraction at a spherical surface – “Now, by Snell’s law n1 sin i = n2 sin r or for small angles n1i = n2r”