Question

One of the factors that cause a diamond to sparkle is its relatively small critical angle. Compare the critical angle for diamond in air with that for diamond in water.

Short answer

Answer: The critical angle for a diamond in air is approximately 24.4 degrees, while the critical angle for a diamond in water is approximately 32.3 degrees. This means that a diamond in water requires a larger incident angle for total internal reflection to occur as compared to a diamond in air. Consequently, a diamond in air produces more total internal reflection and has more sparkle than a diamond in water due to its smaller critical angle.

Step-by-step solution

Gather the necessary values and formula

Refractive indices: 1. Diamond: n_diamond = 2.42 2. Air: n_air = 1.00 3. Water: n_water = 1.33 Snell's law for the critical angle: \(sin(\theta_c) = \frac{n_{medium}}{n_{diamond}}\)

Calculate the critical angle for diamond in air

Using Snell's law, we can find the critical angle for the diamond in air: \(sin(\theta_{c,air}) = \frac{n_{air}}{n_{diamond}} = \frac{1.00}{2.42}\) Now, we will find the critical angle: \(\theta_{c,air} = arcsin(\frac{1.00}{2.42})\) \(\theta_{c,air} ≈ 24.4^\circ\)

Calculate the critical angle for diamond in water

Similarly, applying Snell's law for the diamond in water: \(sin(\theta_{c,water}) = \frac{n_{water}}{n_{diamond}} = \frac{1.33}{2.42}\) Now, we will find the critical angle: \(\theta_{c,water} = arcsin(\frac{1.33}{2.42})\) \(\theta_{c,water} ≈ 32.3^\circ\)

Compare the critical angles

Now that we have the critical angles for the diamond in both air and water, we can compare them: 1. Critical angle for diamond in air: \(\theta_{c,air} ≈ 24.4^\circ\) 2. Critical angle for diamond in water: \(\theta_{c,water} ≈ 32.3^\circ\) The critical angle for a diamond in water is larger than that in air. This means that a diamond in water will need a larger incident angle for total internal reflection to occur and produce the same level of sparkle as a diamond in air. A diamond in air will produce a smaller critical angle, allowing for more total internal reflection and more sparkle.

Key concepts

Snell's Law

Snell's Law is a fundamental principle in optics that describes how light bends, or refracts, when it passes from one medium into another. This law is expressed with the equation:
\[\begin{equation} n_1 \times \text{sin}(\theta_1) = n_2 \times \text{sin}(\theta_2) \[5pt]\end{equation}\] where

• \(n_1\) and \(n_2\) are the refractive indices of the two media,
• \(\theta_1\) is the angle of incidence in the first medium, and
• \(\theta_2\) is the angle of refraction in the second medium.

The law shows that the product of the refractive index and the sine of the angle is conserved across the interface between two different media. It's essential when calculating critical angles for understanding phenomena such as total internal reflection. By rearranging Snell's Law, we can isolate the sine of the critical angle, \( sin(\theta_c) \), and use the refractive indices of the two media to find the value of this angle.

Total Internal Reflection

Total internal reflection (TIR) is an optical phenomenon that occurs when light attempts to move from a medium with a higher refractive index to one with a lower refractive index, and it strikes the boundary at an angle greater than the critical angle specific to those materials.
Under such circumstances, all the light is reflected back into the denser medium, and none of it refracts through the boundary. This principle is why a diamond sparkles so brilliantly; when light enters a diamond, it is subjected to total internal reflection multiple times, intensifying its brightness and color.
For total internal reflection to happen, two conditions must be met:

• The light must travel from a denser to a less dense medium, meaning from higher to lower refractive index, and
• The angle of incidence must be greater than the critical angle of the medium.

TIR is harnessed in fiberoptic technology, where the light signal is contained within the fiber by repeated internal reflections, efficiently transmitting data over long distances with minimal loss.

Refractive Index

The refractive index, often denoted as \(n\), is a dimensionless number that describes how fast light travels through a material compared to its velocity in a vacuum. It's an intrinsic property of every material and varies depending on the wavelength of light. The refractive index determines how much the path of light is bent, or refracted, when entering the material.
In mathematical terms, the refractive index is calculated as:
\[\begin{equation} n = \frac{c}{v} \[5pt]\end{equation}\] l where

• \(c\) is the speed of light in a vacuum (approximately \(3 \times 10^8\) meters per second), and
• \(v\) is the speed of light in the material.

High refractive index materials, like diamonds, bend light more and have smaller critical angles. This results in a more pronounced effect of total internal reflection, making them ideal for applications where light retention and sparkle are desired. In contrast, low refractive index materials cause less light bending, resulting in larger critical angles and less potential for sparkle.