Problem 94 - Precalculus

Question

When light passes from one substance to another, such as from air to water, its path bends. This is called refraction and is what is seen in eyeglass lenses, camera lenses, and gems. The rule governing the change in the path is called Snell's law, named after a Dutch astronomer: \(n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2},\) where \(n_{1}\) and \(n_{2}\) are the indices of refraction of the different substances and \(\theta_{1}\) and \(\theta_{2}\) are the respective angles that light makes with a line perpendicular to the surface at the boundary between substances. The figure shows the path of light rays going from air to water. Assume that the index of refraction in air is \(1 .\) (GRAPH CANNOT COPY) If the refraction index for a rhinestone is \(1.9,\) then to what angle is light refracted if it enters the rhinestone at an angle of \(30^{\circ} ?\) Round the answer to two significant digits.

Step-by-Step Solution

Answer

The light is refracted to an angle of approximately \( 15^{\circ} \).

Step 1: Identify Known Values

We know that the index of refraction of air, \( n_1 \), is 1, and the index of refraction of the rhinestone, \( n_2 \), is 1.9. The angle of incidence, \( \theta_1 \), is \( 30^{\circ} \). Our goal is to find \( \theta_2 \), the angle of refraction.

Step 2: Write Snell's Law

Snell's Law is given as \[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \] where \( n_1 = 1 \), \( \theta_1 = 30^{\circ} \), and \( n_2 = 1.9 \). We need to solve for \( \theta_2 \).

Step 3: Substitute Known Values into Snell's Law

Substitute the known values into the equation: \[ 1 \cdot \sin(30^{\circ}) = 1.9 \cdot \sin \theta_2 \] Since \( \sin(30^{\circ}) = 0.5 \), the equation becomes: \[ 0.5 = 1.9 \cdot \sin \theta_2 \]

Step 4: Solve for \( \sin \theta_2 \)

To find \( \sin \theta_2 \), divide both sides of the equation by 1.9: \[ \sin \theta_2 = \frac{0.5}{1.9} \] Calculate the value: \( \sin \theta_2 \approx 0.2632 \).

Step 5: Find \( \theta_2 \) Using the Inverse Sine Function

Use the inverse sine function to find \( \theta_2 \): \( \theta_2 = \sin^{-1}(0.2632) \). Calculate \( \theta_2 \) which is approximately \( 15.3^{\circ} \).

Step 6: Round the Angle of Refraction to Two Significant Digits

Round \( \theta_2 \) to two significant digits to get the final answer. So, \( \theta_2 \approx 15^{\circ} \).

Key Concepts

Refraction

When light travels from one medium to another, it bends, a phenomenon called **refraction**. This bending occurs because light travels at different speeds in different materials. For example, light moves faster in air than it does in water or glass. Refraction is what causes a straw in a glass of water to appear bent at the surface.
Snell's Law governs this change in direction. It tells us how much light bends when it enters a new medium. This bending of light through different materials is what allows lenses to focus images, whether in cameras, glasses, or even telescopes. Thanks to refraction, we can correct vision problems, capture sharp photographs, and observe distant celestial objects.

• Light bends because it changes speed when moving from one substance to another.
• The degree of bending depends on the **index of refraction** of the materials involved.

Index of Refraction

The **index of refraction** ( italicized words are terminology ) is a measure of how much a medium can bend light. Every transparent material has its own unique index of refraction, denoted by the symbol **n**. This value is crucial in determining how light behaves when entering or exiting a substance.
For example, the index of refraction for air is approximately **1**, meaning it minimally affects the speed and direction of light. In contrast, the index of refraction for rhinestones is **1.9**, indicating that light slows down significantly more upon entry, as demonstrated in the exercise.
Materials with higher indices of refraction bend light more sharply compared to those with lower indices. This property is utilized in designing optics like lenses and prisms to control and manipulate light paths.

• The index of refraction determines how much light "bends" when transitioning between different materials.
• Higher indices signify greater bending of light rays.

Angle of Incidence

The **angle of incidence** refers to the angle at which a ray of light strikes the surface of a material. It is measured relative to an imaginary line called the normal. The normal is perpendicular to the surface at the point where the light hits.
In our exercise example, the light enters the rhinestone at an angle of **30 degrees**. This is the angle of incidence. Understanding this angle is essential for calculating how much the light will bend or refract due to Snell's Law.
The angle of incidence, along with the index of refraction, are pivotal in applying Snell's Law, which accurately predicts the light's path as it moves from one medium to another.

• The angle of incidence is measured from the incoming light ray to the normal.
• This concept is fundamental in optics, influencing how lenses and other transparent materials are designed and used.