Critical Angle of a Semicircular Perspex Block
A student uses a semicircular Perspex block on a ray board to find the critical angle. A narrow ray is directed towards the centre of the straight face, and the block is rotated until the emergent ray along the straight face just disappears. The angle between the incident ray and the normal is then 41.0°. The student calculates n = 1/sin 41.0° = 1.52 and concludes that the Perspex is a new type with n = 1.52, not the expected 1.49. The ray board's protractor reads to 0.5°. The student notices that the bright spot on the straight face is slightly broad and that the ray does not quite pass through the marked centre; it is about 2 mm off. The block's radius is 50 mm. A second measurement with the ray on the other side gives 42.0°.
1. Calculate the refractive index from each measurement and the percentage difference between them. Comment on whether the difference is consistent with the stated protractor resolution.
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n₁ = 1/sin 41.0° = 1/0.6561 = 1.524. n₂ = 1/sin 42.0° = 1/0.6691 = 1.494. The difference is 0.030, and the percentage difference relative to the mean (1.509) is 0.030/1.509 × 100% = 2.0%. The protractor resolution is 0.5°, which gives an uncertainty in n of about |dn/dθ| × 0.5°. With n = 1/sin θ, dn/dθ = −cos θ/sin² θ. At 41°, cos 41° = 0.755, sin² 41° = 0.431, so dn/dθ = −1.75 per radian = −0.0306 per degree. A 0.5° reading uncertainty therefore gives about ±0.015 in n, and the two values differ by about twice this. The difference is therefore only just outside what the protractor alone would allow, and it is not strong evidence of a real material difference. The broad spot and the 2 mm offset add a systematic uncertainty that is likely larger than 0.015, so the two measurements are consistent within the realistic error.
2. Evaluate the student's conclusion that the Perspex has n = 1.52 rather than 1.49. Consider the effect of the ray missing the centre by 2 mm.
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The conclusion that the Perspex has n = 1.52 rather than 1.49 is not justified. The accepted value 1.49 corresponds to a critical angle of sin⁻¹(1/1.49) = 42.2°, which is close to the second measurement (42.0°) and within the stated 0.5° resolution. The first measurement (41.0°) is 1.2° lower, which would require a real change in n of about 0.03. A 2 mm offset from the centre of a 50 mm radius block means the ray enters the curved surface at an angle of about sin⁻¹(2/50) = 2.3° to the normal. Refraction at the curved surface bends the ray towards the normal by about 0.8° (using sin i = n sin r), so the angle of incidence at the flat face is not the angle read on the board. This can easily shift the apparent critical angle by about 1°, producing an apparent n of 1.52. The broad spot also makes the 'just disappears' judgement subjective. The correct conclusion is that the measurements are consistent with ordinary Perspex within experimental error; a new material would require a much larger, reproducible difference.
3. Describe an improved procedure to determine n more reliably from critical-angle measurements.
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Align the ray so that it passes through the marked centre of the straight face; check this by observing that the undeviated ray at normal incidence passes through the centre mark and is not displaced. Use a narrower slit and a darker room so the disappearance of the emergent ray is sharper. Rotate the block in small steps (e.g. 0.2°) and record the angle at which the emergent ray just vanishes; repeat on both sides and average. Measure the critical angle several times, resetting the block each time, and calculate n for each; use the mean and the spread to estimate uncertainty. A better method is to use a rectangular block and measure the critical angle at a plane face with a goniometer, or to use a spectrometer, which gives a more precise angle. If possible, compare the value with a known Perspex sample under the same conditions, and quote n with an uncertainty rather than claiming a new material from a single 1° difference.