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Chapter 23: Problem 23

(a) Calculate the critical angle for a diamond surrounded by air. (b) Calculate the critical angle for a diamond under water. (c) Explain why a diamond sparkles less under water than in air.

Short Answer

Expert verified
Answer: A diamond sparkles less under water than in air because the critical angle for total internal reflection is higher under water (approx. 32.2掳) compared to air (approx. 24.4掳). This means that light rays need to strike the diamond facets at a steeper angle for total internal reflection to occur under water, reducing the number of light rays undergoing total internal reflection inside the diamond and causing it to sparkle less.

Step by step solution

01

(Step 1 - Snell's Law and Critical Angle Formula)

Snell's law relates the angle of incidence of light at the interface of two media to the angle of refraction in the second medium. It is given by: n1 * sin(胃1) = n2 * sin(胃2) where n1 and n2 are the refractive indices of the first and second media, respectively, and 胃1 and 胃2 are the angles of incidence and refraction, respectively. The critical angle is the angle of incidence at which refracted light is parallel to the interface (胃2 = 90掳). We can find it using Snell's law by setting 胃2 = 90掳 and solving for 胃1: 胃c = arcsin(n2/n1)
02

(Step 2 - Refractive Indices of Diamond, Air, and Water)

To find the critical angles, we need the refractive indices of diamond, air, and water. The refractive index of diamond is approximately 2.42, the refractive index of air is approximately 1, and the refractive index of water is approximately 1.33.
03

(Step 3 - Calculate the Critical Angle for a Diamond in Air)

Find the critical angle for a diamond surrounded by air by using the formula wederived and the refractive indices of diamond and air: 胃c_air = arcsin(n_air/n_diamond) 胃c_air = arcsin(1/2.42) 胃c_air 鈮� 24.4掳 The critical angle for a diamond surrounded by air is approximately 24.4掳.
04

(Step 4 - Calculate the Critical Angle for a Diamond in Water)

Find the critical angle for a diamond under water by using the formula we derived and the refractive indices of diamond and water: 胃c_water = arcsin(n_water/n_diamond) 胃c_water = arcsin(1.33/2.42) 胃c_water 鈮� 32.2掳 The critical angle for a diamond under water is approximately 32.2掳.
05

(Step 5 - Explain Why a Diamond Sparkles Less Under Water)

The sparkle of a diamond is due to the total internal reflection of light within the diamond. This can only occur when the angle of incidence is greater than the critical angle. In air, the critical angle is approximately 24.4掳, while under water, it's approximately 32.2掳. A higher critical angle under water means that light rays need to strike the diamond facets at a steeper angle for total internal reflection to occur. This reduces the number of light rays that undergo total internal reflection inside the diamond, and therefore, the diamond sparkles less under water than in air.

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Most popular questions from this chapter

The angle of deviation through a triangular prism is defined as the angle between the incident ray and the emerging ray (angle \(\delta\) ). It can be shown that when the angle of incidence \(i\) is equal to the angle of refraction \(r^{\prime}\) for the emerging ray, the angle of deviation is at a minimum. Show that the minimum deviation angle \(\left(\delta_{\min }=D\right)\) is related to the prism angle \(A\) and the index of refraction \(n,\) by $$ n=\frac{\sin \frac{1}{2}(A+D)}{\sin \frac{1}{2} A} $$ [Hint: For an isosceles triangular prism, the minimum angle of deviation occurs when the ray inside the prism is parallel to the base, as shown in the figure.]
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