Problem 92 - 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 light rays hit a glass surface at an angle of \(30^{\circ}\) from the perpendicular and are refracted to an angle of \(18^{\circ}\) from the perpendicular, then what is the refraction index for that glass? Round the answer to two significant digits.

Step-by-Step Solution

Answer

The refractive index for the glass is approximately 1.6.

Step 1: Understand Snell's Law

Snell's Law relates the angles and indices of refraction between two different media. It is given by the equation: \(n_{1} \sin \theta_{1} = n_{2} \sin \theta_{2}\). Here, \(n_{1}\) and \(n_{2}\) are the indices of refraction, and \(\theta_{1}\) and \(\theta_{2}\) are the angles of incidence and refraction, respectively.

Step 2: Identify Known Values

From the problem statement, we know that light is moving from air (\(n_{1} = 1\)) to glass (\(n_{2}\) is unknown). The angle of incidence \(\theta_{1}\) is \(30^{\circ}\) and the angle of refraction \(\theta_{2}\) is \(18^{\circ}\).

Step 3: Apply Snell's Law

Substitute the known values into Snell's Law: \(n_{1} \sin \theta_{1} = n_{2} \sin \theta_{2}\).This becomes \(1 \cdot \sin(30^{\circ}) = n_{2} \cdot \sin(18^{\circ})\).

Step 4: Simplify the Equation

Calculate the sine values: \(\sin(30^{\circ}) = 0.5\) and \(\sin(18^{\circ})\) can be found using a calculator (approximately \(0.309\)). Thus, the equation becomes \(0.5 = n_{2} \cdot 0.309\).

Step 5: Solve for the Index of Refraction

Rearrange the equation to solve for \(n_{2}: n_{2} = \frac{0.5}{0.309}\). Using a calculator, calculate \(n_{2} \approx 1.62\).

Step 6: Round to Significant Figures

The problem asks to round the answer to two significant digits. Thus, the refractive index for glass is approximately \(1.6\).

Key Concepts

Refraction

Refraction happens when light passes from one medium to another and changes direction—it bends. This bending occurs because light travels at different speeds in different materials. For example, light travels faster in air than in water or glass, leading to a change in direction at the boundary of the two materials. This explains why a straw appears bent when placed in a glass of water. Every Bending action in lenses, like in cameras or eyeglasses, uses the principle of refraction to focus light effectively for clearer images. Understanding this concept helps us explain phenomena like rainbows, which occur due to the refraction and dispersion of light.

Indices of Refraction

The index of refraction, represented as "n" in formulas, is a measure that describes how much a specific medium decreases the speed of light. It helps quantify the bending effect of light as it enters a new medium. Each material has its unique index of refraction; for instance:

• Air has an index of refraction of approximately 1.
• Water has an index of around 1.33.
• Glass generally ranges from about 1.5 to 1.9, depending on its composition.

The higher the index, the slower light travels through the material, leading to greater bending or refraction. Snell's Law, which uses these indices, enables us to calculate how light will behave at the boundary between two media.

Angle of Incidence

The angle of incidence is the angle between the incoming light ray and an imaginary line perpendicular to the surface at the point of contact, known as the "normal." To visualize this, imagine a light ray striking a surface at an angle. The angle between this ray and the normal is called the angle of incidence. It's crucial in determining how light refracts because according to Snell's Law, both the angle and the index of refraction are needed to understand and predict light's path as it moves from one medium to another. In our given problem, the angle of incidence is given as 30°.

Angle of Refraction

After light enters a new medium, it changes direction, and the resulting angle compared to the normal is known as the angle of refraction. Much like the angle of incidence, it is measured between the refracted ray and the perpendicular line, or normal, at the point of entry into the new medium. Understanding this angle is essential in comprehending the path of light and requires us to know both the indices of refraction for the materials in question and the incident angle. In the example provided, the refracted angle is 18°, showcasing how significantly the path of light has shifted due to the change in medium from air to glass.