Refraction — Set 7

Correct Answer: B. total internal reflection inside the diamond

• **total internal reflection inside the diamond** = diamond has n ≈ 2.42, giving a very small critical angle of about 24°; most light entering the diamond undergoes multiple total internal reflections before exiting through the top facets, creating brilliance. • **Critical angle ~24°** — because this angle is so small, almost any ray inside diamond hits a facet at i > 24° and undergoes TIR; gem cutters shape facets to maximise this effect and direct light upwards through the table facet. • 💡 Wrong-option analysis: only absorption of light: absorption would darken the diamond, not create sparkle — diamonds are transparent and sparkle because light exits, not because it is absorbed; only scattering by dust: scattering by dust produces a diffuse haze; it cannot create the precise directionality and brilliance of diamond sparkle; diffraction around the diamond: diffraction occurs at edges and produces faint coloured fringes (iridescence at very small scales) — it cannot explain the broad brilliant sparkle of a cut diamond.

Correct Answer: A. 24 degrees

• **24 degrees** = sin c = 1/n = 1/2.42 ≈ 0.413; c = arcsin(0.413) ≈ 24°. • **Smallest critical angle** — diamond's critical angle (~24°) is the smallest among common transparent materials; this means TIR occurs for nearly all directions of light inside the gem, producing maximum brilliance. • 💡 Wrong-option analysis: 60 degrees: sin 60° = 0.866 would require n = 1/0.866 ≈ 1.15, much less dense than diamond; 42 degrees: sin 42° ≈ 0.67 would require n ≈ 1.5, which is the refractive index of common glass, not diamond; 10 degrees: sin 10° ≈ 0.174 would require n = 1/0.174 ≈ 5.75 — no known transparent material has such a high refractive index.

Correct Answer: D. a little before actual sunrise

• **a little before actual sunrise** = the atmosphere acts like a medium of gradually increasing density from top to bottom; sunlight curves downward as it enters denser layers, making the sun appear above the horizon even when it is geometrically still below — extending the day by about 2 minutes. • **Advanced sunrise, delayed sunset** — the same effect makes the sun visible for a few minutes after it has actually set below the geometric horizon; the total lengthening of the day due to atmospheric refraction is approximately 4 minutes. • 💡 Wrong-option analysis: only in winter: atmospheric refraction operates throughout the year — the effect is slightly stronger in winter due to denser cold air, but it is not limited to winter; only at noon: at noon the sun is nearly overhead and the path through the atmosphere is shortest, so refraction is minimal — the effect is most pronounced at sunrise and sunset; only after sunset: atmospheric refraction makes the sun visible both a little before actual sunrise AND a little after actual sunset — restricting it to 'only after sunset' misses half the effect.

Correct Answer: D. slightly higher than their true position

• **slightly higher than their true position** = starlight bends downward as it enters progressively denser atmospheric layers; the observer's eye traces the ray back in a straight line and perceives the star at an elevated apparent position above its true geometric position. • **Refraction bends starlight earthward** — the bending is greatest near the horizon (~0.5°) and zero at the zenith; stars near the horizon appear significantly elevated; this is why star atlases include an atmospheric refraction correction. • 💡 Wrong-option analysis: lower than their true position: refraction bends light toward Earth (downward), so the apparent ray reaching the eye comes from above the true position — stars appear higher, not lower; at random positions only: atmospheric refraction is systematic and predictable (largest near horizon, zero at zenith) — not random; exactly at true position always: stars appear at their true positions only at the zenith where the ray passes through the atmosphere vertically without bending; for all other positions refraction shifts the apparent position upward.

Correct Answer: C. 41 degrees

• **41 degrees** = apply Snell's law: sin r = sin i / n = sin 60°/1.33 = 0.866/1.33 ≈ 0.651; r = arcsin(0.651) ≈ 41°. • **Towards the normal** — going from air into water the ray bends towards the normal; r = 41° < i = 60°, consistent with entering a denser medium. • 💡 Wrong-option analysis: 60 degrees: r = i = 60° only if n_water = n_air (no refraction); since n_water = 1.33 ≠ 1, refraction occurs and r ≠ 60°; 75 degrees: r = 75° > i = 60° would mean the ray bends away from the normal — that happens going from denser to rarer, not air to water; 30 degrees: r = 30° would require n = sin60/sin30 = 0.866/0.5 = 1.73 — too high for water (n = 1.33).

Correct Answer: C. n = c/v

• **n = c/v** = absolute refractive index is defined as the ratio of the speed of light in vacuum to the speed in the medium: n = c/v; since v ≤ c in all physical media, n ≥ 1 always. • **Standard definition** — this definition makes vacuum the reference with n = 1; all other media have n > 1; the larger the n, the slower light travels in that medium. • 💡 Wrong-option analysis: n = 1/(c - v): this has units of s/m (inverse of speed), not dimensionless — it cannot be the definition of refractive index; n = v/c: this gives the reciprocal (v/c ≤ 1), meaning denser media would have smaller values — this is the inverse of the refractive index; n = c + v: adding two speeds gives a speed (m/s), which is dimensionally incompatible with the dimensionless refractive index.

Correct Answer: B. 49 degrees

• **49 degrees** = apply Snell's law for air to glass: sin i = n × sin r = 1.5 × sin 30° = 1.5 × 0.5 = 0.75; i = arcsin(0.75) ≈ 49°. • **i > r** — since light enters a denser medium (glass), the incident angle is larger than the refraction angle (49° > 30°), which is consistent with bending towards the normal. • 💡 Wrong-option analysis: 40 degrees: sin 40° ≈ 0.643 ≠ 0.75; if i = 40° then n = sin40/sin30 = 0.643/0.5 = 1.286 — not 1.5; 60 degrees: sin 60° = 0.866 would give n = 0.866/0.5 = 1.73 — too dense; not matching n = 1.5; 30 degrees: i = r = 30° would mean n = 1 (same medium) — but glass has n = 1.5 ≠ 1.

Correct Answer: A. air

• **air** = air has a refractive index of approximately 1.0003 for visible light, the lowest among the options; vacuum has n = 1 exactly, and air is the closest to vacuum. • **Comparison** — water n ≈ 1.33, glass n ≈ 1.5, diamond n ≈ 2.42; air at n ≈ 1.0003 is nearly transparent to all wavelengths and slows light the least. • 💡 Wrong-option analysis: water: water has n ≈ 1.33 — significantly higher than air; light slows to ~75% of c in water; glass: glass has n ≈ 1.5 — higher than both air and water; light slows to ~67% of c; diamond: diamond has n ≈ 2.42 — the highest of all common transparent materials; light slows to only ~41% of c inside diamond.

Correct Answer: C. 0 degrees

• **0 degrees** = a ray along the normal means angle of incidence i = 0°; Snell's law gives n1 sin 0° = n2 sin r, so sin r = 0 and r = 0°; the ray continues straight without bending. • **No deviation** — at normal incidence the ray is not deflected regardless of the refractive indices; speed and wavelength change at the boundary, but direction does not. • 💡 Wrong-option analysis: equal to critical angle: the critical angle is a specific non-zero angle (e.g. ~42° for glass); r = 0° at normal incidence is far from the critical angle; cannot be defined: the angle of refraction is perfectly well defined at normal incidence — it equals 0° and is easy to calculate; 90 degrees: r = 90° is the grazing-refraction condition (Snell's law at the critical angle for TIR), not what happens at normal incidence.

Correct Answer: B. 1.5

• **1.5** = use n = c/v = 3.0×10^8 / 2.0×10^8 = 1.5; this is the refractive index of common glass or a similar medium. • **v = c/n** — re-checking: v = c/n = 3.0×10^8/1.5 = 2.0×10^8 m/s, consistent with the given speed; glass reduces light speed to two-thirds of c. • 💡 Wrong-option analysis: 1.0: n = 1.0 would mean v = c (speed in vacuum), but the given speed is 2.0×10^8 m/s < c — the medium is not vacuum; 2.0: n = 2.0 would give v = c/2.0 = 1.5×10^8 m/s — but the given speed is 2.0×10^8 m/s, so n = 2.0 is incorrect; 0.67: n = 0.67 < 1 is physically impossible for a standard transparent medium — no real material slows light to give n < 1.