Ray Optics and Optical Instruments — Exam Practice

Answer Diamond has a very high refractive index (≈2.42), giving it a small critical angle. Light entering the diamond undergoes total internal reflection repeatedly off its many cut faces before finally emerging, producing far more brilliance than glass, whose lower refractive index (larger critical angle) lets more light escape directly.

Answer With the object beyond C, rays from its tip reflect off the mirror and converge to form a real, inverted, diminished image between F and C.

Using similar triangles formed by the incident and reflected rays with the principal axis, and applying the New Cartesian sign convention (distances against incident light are negative), the geometric relations combine to give 1/v + 1/u = 1/f, with f = R/2.

Answer (a) Applying the single-spherical-surface refraction formula (n₂/v − n₁/u = (n₂−n₁)/R) once at each face of a thin lens, and combining the two results (since the image from the first surface becomes the object for the second), gives: 1/f = (n₂₁−1)(1/R₁ − 1/R₂).

(b) Taking R₁ = +20 cm, R₂ = −30 cm (standard convex lens convention): 1/f = (1.5−1)(1/20 − 1/(−30)) = 0.5 × (1/20 + 1/30) = 0.5 × (5/60) = 1/24.
f = 24 cm.

Solution P = 1/f, with f in metres: f = 0.5 m, so P = 1/0.5 = 2 D.

Solution sinθ_c = 1/n (n = refractive index of the denser medium relative to the rarer one) — a fixed property of the specific pair of media involved, not of the angle at which light happens to strike.

Solution In myopia, distant objects focus in front of the retina. A diverging (concave) lens spreads the rays slightly before they enter the eye, pushing the focus back onto the retina.

Solution Both are converging (convex) lenses — the objective forms a real, magnified image, which the eyepiece then magnifies further as a simple magnifying glass.

Solution u = −20 cm, f = +15 cm (convex mirror). 1/v = 1/f − 1/u = 1/15 − (−1/20) = 1/15 + 1/20 = 7/60 → v ≈ +8.57 cm.

m = −v/u = −(8.57)/(−20) ≈ 0.43. Positive v means the image is virtual (behind the mirror); positive, fractional m means it's erect and diminished — exactly as expected for a convex mirror, which always produces this type of image.

Solution TIR requires both conditions: travelling from denser to rarer medium, and an angle of incidence exceeding the critical angle for that pair of media.

Solution m = f_o/f_e = 100/5 = 20.

L = f_o + f_e = 100 + 5 = 105 cm.