Problem 93 - 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 diamond is \(2.4,\) then to what angle is light refracted if it enters the diamond at an angle of \(30^{\circ} ?\) Round the answer to two significant digits.

Step-by-Step Solution

Answer

The angle of refraction is approximately 12°.

Step 1: Identify Given Values

From the problem, we know that the light is entering from air where the index of refraction \( n_1 = 1 \) and the angle of incidence \( \theta_1 = 30^{\circ} \). In the diamond, the index of refraction \( n_2 = 2.4 \). We need to determine the angle of refraction \( \theta_2 \).

Step 2: Use Snell's Law

Snell's law is given by the equation: \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \). Plugging in the known values: \( 1 \times \sin 30^{\circ} = 2.4 \times \sin \theta_2 \).

Step 3: Calculate the Sine of the Angle of Incidence

The sine of 30 degrees is common knowledge and is \( \sin 30^{\circ} = 0.5 \). Substitute into Snell's law: \( 1 \times 0.5 = 2.4 \times \sin \theta_2 \).

Step 4: Solve for Sine of the Angle of Refraction

Rearrange the equation for \( \sin \theta_2 \): \( \sin \theta_2 = \frac{0.5}{2.4} \). Calculate: \( \sin \theta_2 = 0.2083 \).

Step 5: Determine the Angle of Refraction

Find \( \theta_2 \) by taking the inverse sine (arcsin) of \( 0.2083 \): \( \theta_2 = \arcsin(0.2083) \). Using a calculator, we find \( \theta_2 \approx 12.04^{\circ} \).

Step 6: Round the Result

Round the angle of refraction to two significant digits: \( \theta_2 \approx 12^{\circ} \).

Key Concepts

Snell's Law

Snell's Law is a fundamental principle used to describe how light behaves when it travels from one medium to another. Named after Willebrord Snell, a Dutch astronomer, it explains why light bends, or refracts, as it passes across the boundary between two substances. The formula for Snell's Law is:

• \[n_{1} \sin \theta_{1} = n_{2} \sin \theta_{2}\]

Here:

• \(n_{1}\) is the index of refraction of the first medium.
• \(n_{2}\) is the index of refraction of the second medium.
• \(\theta_{1}\) is the angle of incidence, which is the angle between the incoming ray and the normal (a line perpendicular to the surface at the point of incidence).
• \(\theta_{2}\) is the angle of refraction, which is the angle between the refracted ray and the normal.

This law helps in predicting how much the light will bend and is essential in designing various optical devices such as glasses and cameras. It is important to know the indices of refraction of the mediums involved and the initial angle of incidence, as these values allow you to calculate the angle of refraction using this straightforward mathematical relationship. Remember that Snell's Law assumes that the mediums are uniform and the light interacts with them in a predictable manner.

Angle of Incidence

The angle of incidence is a key concept in understanding how light behaves when it encounters different surfaces. It's defined as the angle between the ray of light that is striking a surface and the normal to that surface at the point of incidence.
This angle is crucial in determining how light will refract or reflect.

• A larger angle of incidence generally results in a larger angle of refraction or reflection, following Snell’s Law for refraction and the law of reflection for reflective surfaces.
• The angle of incidence can vary depending on the initial conditions, such as the source of the light and the medium it travels through before reaching the boundary.

The concept of the angle of incidence is applicable in various scenarios, including optical lenses, mirrors, and even natural phenomena like the shimmering effect of sunlight on water surfaces.
In the exercise, we have the light entering a diamond at an angle of \(30^{\circ}\), which is a typical scenario for observing refraction. Understanding this angle is essential because it directly influences the amount of bending the light experiences.

Index of Refraction

The index of refraction, often shown as \(n\), is a number that describes how light propagates through a specific medium. It is an intrinsic property of materials and depends on the light's wavelength and the medium through which it travels.

• The index of refraction measures the reduction in the speed of light within a medium compared to the speed of light in a vacuum (the speed of light in a vacuum is the fastest speed at which light can travel).
• An index of refraction greater than 1 indicates that light moves slower in the medium than in a vacuum, as seen with air \(n \approx 1.0003\) or glass \(n \approx 1.5\).
• A higher index of refraction means that light bends more as it enters the medium.

In practical terms, knowing the index of refraction is vital for calculating how much light will bend when entering or exiting a material.
For instance, in the exercise provided, air has an index of refraction of 1, while the diamond has an index of refraction of 2.4. This large difference causes a significant bending of the light, which is what makes diamonds sparkle and their cuts appealing. Understanding the index of refraction helps in analyzing various optical effects and is indispensable for designing lenses and other optical instruments.