Problem 91 - 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 the water's surface at an angle of \(30^{\circ}\) from the perpendicular and are refracted to an angle of \(22^{\circ}\) from the perpendicular, then what is the refraction index for water? Round the answer to two significant digits.

Step-by-Step Solution

Answer

The refraction index of water is approximately 1.3.

Step 1: Identify Given Values

Extract the given values from the problem statement. Here, the index of refraction of air is given as \( n_{1} = 1 \), the angle in air, \( \theta_{1} \), is \( 30^{\circ} \), and the angle in water, \( \theta_{2} \), is \( 22^{\circ} \). You need to find the index of refraction of water, \( n_{2} \).

Step 2: Apply Snell's Law

Use Snell's Law: \( n_{1} \sin \theta_{1} = n_{2} \sin \theta_{2} \). Substitute the known values into the equation so that it becomes \( 1 \cdot \sin 30^{\circ} = n_{2} \cdot \sin 22^{\circ} \).

Step 3: Calculate Sine Values

Calculate the sine of the angles. \(\sin 30^{\circ} = 0.5 \) and \( \sin 22^{\circ} \approx 0.3746 \). Insert these values back into the equation.

Step 4: Solve for the Unknown

Now the equation \( 0.5 = n_{2} \cdot 0.3746 \) needs to be solved for \( n_{2} \). Divide both sides by \( 0.3746 \) to isolate \( n_{2} \). Thus, \( n_{2} = \frac{0.5}{0.3746} \).

Step 5: Perform the Calculation

Calculate \( n_{2} = \frac{0.5}{0.3746} \approx 1.334 \). Round the answer to two significant digits, which gives \( n_{2} \approx 1.3 \).

Key Concepts

Refraction

Refraction occurs when light passes from one medium to another, causing it to change direction. This bending of light happens because light travels at different speeds in different substances. For example, when light moves from air to water, it slows down, causing the light ray to bend towards the normal line (a line perpendicular to the surface). This phenomenon is everyday visible in objects like eyeglass lenses and camera lenses.

• Refraction is responsible for effects like a spoon appearing bent in a glass of water.
• The extent of refraction is determined by the difference in the speed of light in the two substances.

Understanding refraction helps in designing lenses and understanding natural phenomena such as rainbows.

Optics

Optics is the branch of physics that deals with the behavior and properties of light, including its interactions with matter and instruments like lenses and mirrors. Snell's Law, which governs refraction, is a crucial concept in optics.

• Optics is central to various technologies, such as microscopes, telescopes, and optical fibers.
• It involves studying various aspects of light, including reflection, refraction, and diffraction.

Through optics, we gain insights into both natural and artificial lighting systems, improving vision and imaging systems.

Indices of Refraction

Indices of refraction measure how much a substance slows down light, relative to its speed in a vacuum. Each medium has a unique refractive index, represented as "n."

• A vacuum has a refractive index of 1, the baseline.
• Air, being close to a vacuum, has a refractive index slightly greater than 1.
• Water's refractive index is around 1.3, indicating light travels through it slower than air.

Snell's Law uses these indices to calculate how much light bends as it passes from one medium to another.

Angle of Incidence

The angle of incidence is the angle between the incoming light ray and the normal line at the surface's point of contact. It's a key variable in Snell's Law, affecting how much light bends during refraction.

• The angle of incidence is measured in the medium where the light is coming from.
• It directly influences the angle of refraction according to the media's refractive indices.

Understanding this angle is crucial for applications like adjusting glasses prescription, camera focus, and more.

Angle of Refraction

The angle of refraction is formed between the refracted light ray and the normal line in the new medium. According to Snell’s Law, this angle is determined by the relative indices of refraction of the two media.

• It's usually smaller when light moves from a rarer medium (like air) to a denser medium (like water).
• Determining this angle helps in calculating the path of light in lenses and predicts how light travels through various substances.

The angle of refraction is pivotal in designing optical instruments and understanding visual distortions.