Refraction — Set 6

Correct Answer: D. refractive index

• **refractive index** = by Snell's law, for a given pair of media and fixed wavelength, the ratio sin i / sin r is constant and equals the relative refractive index of the second medium with respect to the first. • **Snell's constant** — this constant ratio was historically called 'Snell's constant'; its modern name is the relative refractive index n21; it equals n2/n1 = v1/v2. • 💡 Wrong-option analysis: wavelength ratio: wavelength ratio λ1/λ2 also equals the refractive index (since frequency is constant), but sin i/sin r itself is defined as the refractive index — 'wavelength ratio' is not the standard name given to this ratio in optics; power of lens: power of a lens (in diopters) is 1/focal length — it is not related to sin i/sin r at a flat boundary; critical angle: the critical angle is a specific value of i (not a ratio of sines) at which the refracted angle equals 90°.

Correct Answer: B. no unit (dimensionless)

• **no unit (dimensionless)** = refractive index n = c/v is the ratio of two speeds; since both c and v have units of m/s, the units cancel and n is a pure dimensionless number. • **Dimensionless ratio** — alternatively, n = sin i/sin r is the ratio of two pure numbers (sines), also confirming it is dimensionless; typical values are 1.33 for water and 1.5 for glass. • 💡 Wrong-option analysis: ohm: the ohm (Ω) is the unit of electrical resistance — it has nothing to do with optical refraction; meter: meter is the SI unit of length — refractive index is a ratio of speeds (or sines), not a length; second: second is the SI unit of time — speed ratios cancel the time units too, leaving no unit.

Correct Answer: C. n = real depth / apparent depth

• **n = real depth / apparent depth** = from the apparent depth formula, apparent depth = real depth/n; rearranging gives n = real depth / apparent depth; since real > apparent for water seen from air, n > 1, which is correct. • **Practical use** — by measuring the apparent and real depths experimentally, one can determine the refractive index of any transparent liquid without sophisticated equipment. • 💡 Wrong-option analysis: n = apparent depth / real depth: this inverts the formula and would give n < 1 for all real cases — but refractive index of water is 1.33, so this is wrong; n = real depth + apparent depth: adding depths gives a length (in cm or m), not a dimensionless refractive index — dimensionally incorrect; n = real depth - apparent depth: subtracting depths also gives a length — dimensionally incorrect for a dimensionless quantity.

Correct Answer: C. 1.33

• **1.33** = use n = real depth / apparent depth = 24 / 18 = 1.33; this is approximately the refractive index of water. • **24/18 = 4/3 ≈ 1.33** — simplifying the fraction 24/18 = 4/3 ≈ 1.333, confirming the medium is water or a water-like liquid. • 💡 Wrong-option analysis: 1.20: n = 1.20 would give apparent depth = 24/1.20 = 20 cm, not 18 cm — inconsistent with the data; 0.75: n = 0.75 < 1 is physically impossible for a real medium in normal optics — no standard material has n below 1 for visible light; 2.00: n = 2.00 would give apparent depth = 24/2.00 = 12 cm, not 18 cm — inconsistent with the given data.

Correct Answer: C. sin c = n2/n1

• **sin c = n2/n1** = at the critical angle, the refracted ray in the rarer medium (n2) makes 90° with the normal; applying Snell's law: n1 sin c = n2 sin 90° = n2; therefore sin c = n2/n1. • **n2 < n1** — since medium 1 is denser, n1 > n2, so n2/n1 < 1, ensuring sin c < 1 and a valid critical angle exists; for air (n2 = 1), sin c = 1/n1. • 💡 Wrong-option analysis: sin c = n1/n2: this is greater than 1 (since n1 > n2), which is impossible for a sine — no valid critical angle would exist; sin c = n1 - n2: subtracting refractive indices gives a dimensionless number but has no basis in Snell's law — this is not the correct relation; sin c = n1 + n2: adding refractive indices gives a sum > 2, making sin c > 1 — impossible.

Correct Answer: A. 39 degrees

• **39 degrees** = sin c = 1/n = 1/1.6 = 0.625; c = arcsin(0.625) ≈ 38.7° ≈ 39°. • **Dense glass** — n = 1.6 is typical of dense flint glass; its critical angle of ~39° is smaller than for common glass (n = 1.5, c ≈ 42°); a smaller critical angle means TIR begins at shallower angles. • 💡 Wrong-option analysis: 45 degrees: sin 45° = 0.707 would require n = 1/0.707 ≈ 1.41 — less dense than this glass (n = 1.6); 24 degrees: sin 24° ≈ 0.407 would require n = 1/0.407 ≈ 2.46 — that is diamond, not glass with n = 1.6; 30 degrees: sin 30° = 0.5 would require n = 1/0.5 = 2.0 — much denser than this glass.

Correct Answer: A. Higher refractive index means higher optical density

• **Higher refractive index means higher optical density** = optical density is defined by how much a medium slows light; a higher n means more slowing, which is the definition of being optically denser. • **Common examples** — glass (n ≈ 1.5) is optically denser than water (n ≈ 1.33) which is optically denser than air (n ≈ 1.0); this ordering matches everyday experience of refraction. • 💡 Wrong-option analysis: Optical density has no relation to refraction: optical density is defined precisely through refractive index and the slowing of light — it is directly related to refraction; Higher refractive index means lower optical density: this reverses the definition; higher n means light is slowed more, meaning higher optical density, not lower; Refractive index depends only on mass: refractive index is an electromagnetic property of the material depending on its electronic structure, not simply its mass density — lead glass is dense but not as optically dense as diamond.

Correct Answer: C. 70 degrees

• **70 degrees** = apply Snell's law: n_water sin i = n_air sin r; 1.33 × sin 45° = sin r; 1.33 × 0.707 = 0.940; r = arcsin(0.940) ≈ 70°. • **Denser to rarer** — going from water (denser) to air (rarer), the refracted angle r > i = 45°; the result 70° > 45° is consistent; note that the critical angle of water is ~49°, so 45° < 49° and TIR does not occur. • 💡 Wrong-option analysis: 45 degrees: r = i = 45° only when both media are the same (n1 = n2); since n_water ≠ n_air, refraction occurs and r ≠ 45°; 30 degrees: r = 30° < i = 45° would mean the ray bends towards the normal — that happens going into a denser medium, not from water to air; 90 degrees: r = 90° is the critical angle condition (grazing refraction), which occurs only when i equals the critical angle (~49°); since i = 45° < 49°, r < 90°.

Correct Answer: B. frequency stays same and wavelength decreases

• **frequency stays same and wavelength decreases** = frequency is fixed by the source and cannot change at a boundary; speed decreases in glass; since λ = v/f and f is constant, λ must decrease in proportion to the speed decrease. • **λ_glass = λ_air / n** — for glass with n = 1.5, the wavelength in glass is 2/3 of the wavelength in air; colour (perceived frequency) does not change, but the wavelength shrinks. • 💡 Wrong-option analysis: frequency stays same and wavelength stays same: if both stayed the same, v = fλ would also be the same, meaning no refraction (no speed change) — but glass does slow light; frequency increases and wavelength decreases: frequency cannot increase or decrease at a simple boundary — only amplitude may change; a frequency change would require a moving medium (Doppler); frequency decreases and wavelength increases: frequency decreasing is impossible at a stationary boundary; wavelength increasing would occur if the light were entering a rarer medium (higher speed), opposite to air-to-glass.

Correct Answer: D. 0.77 cm

• **0.77 cm** = step 1: find r using sin r = sin 30°/1.5 = 0.333, r ≈ 19.5°; step 2: lateral displacement d = t sin(i − r)/cos r = 4 × sin(30° − 19.5°)/cos 19.5° = 4 × sin 10.5°/cos 19.5° = 4 × 0.182/0.943 ≈ 0.77 cm. • **Formula d = t sin(i−r)/cos r** — thickness t = 4 cm, angle difference (i − r) = 10.5°; the displacement grows with t and with angle of incidence. • 💡 Wrong-option analysis: 0.50 cm: d = 0.50 would require sin(i−r)/cos r = 0.125; with r ≈ 19.5° this gives sin(i−r) ≈ 0.118, i−r ≈ 6.8°, so i ≈ 26.3° — not 30°; the calculated value for i = 30° is 0.77 cm; 1.50 cm: d = 1.50/4 = 0.375, meaning sin(i−r)/cos r = 0.375; for t = 4 this is too large for i = 30° with n = 1.5; 0.20 cm: this would be the result for a much thinner slab or smaller angle of incidence — not consistent with t = 4 cm and i = 30°.