A gem must be cut to specific dimensions because:

Question

BlackTom AI · Accepted Answer

Refraction of light at an interface is governed by the material’s index of refraction, so the only way to control how much a gem bends and traps light is by cutting its facets to specific angles and proportions.

1. Relevant knowledge points and concepts
- Snell’s law: n1·sinθ1 = n2·sinθ2. For a gem (n_gem) to air (n_air ≈ 1.00) the outgoing angle is fixed by the incident angle and n_gem.
- Critical angle and total internal reflection (TIR): θc = arcsin(n2/n1) for n1>n2. If internal incidence is greater than θc, light is reflected back into the gem instead of escaping.
- Brilliance and “fire” depend on maximizing internal reflection and controlled refraction so light returns to the observer’s eye.

2. Step-by-step reasoning
- The gem’s index of refraction is an intrinsic property that sets how much light bends at the surface according to Snell’s law.
- To make light entering the gem reflect internally and then exit toward the viewer (maximizing sparkle), facet angles must be chosen so incoming rays meet facet planes at angles that produce desired refraction or TIR.
- If facets are too shallow or too steep relative to the refractive behavior, light will either leak out the pavilion (reducing brilliance) or not be directed to the eye.

3. Intermediate calculation example
- Diamond: n ≈ 2.42 → critical angle θc = arcsin(1/2.42) ≈ arcsin(0.413) ≈ 24.4°. Pavilion facets are cut so internal incidence often exceeds this, promoting TIR. This numerical example shows how n fixes the angular behavior and thus dictates facet geometry.

4. Why the correct answer is correct
- Because n_gem determines the exact bending (refraction) for any incident ray, cutter must set facet angles and proportions to work with that fixed bending to produce optimal visual effects. The statement “the index of refraction will bend light only a specific amount” captures this optical constraint.

5. Why other options are incorrect (brief)
- Non-optical reasons (e.g., for weight retention, fashion, or ease of setting) do not explain the precise geometric proportions required for brilliance; they are secondary. Structural or cosmetic reasons don’t determine the exact facet angles needed to control light as the refractive index does.

6. Conclusion
- Therefore, the gem must be cut to specific dimensions because the index of refraction will bend light only a specific amount, which matches the provided correct answer.

A gem must be cut to specific dimensions because:单项选择题

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