Problem 38 - Physics

Question

A ray of sunlight is passing from diamond into crown glass; the angle of incidence is \(35.00^{\circ} .\) The indices of refraction for the blue and red components of the ray are: blue \(\left(n_{\text {diamond }}=2.444, n_{\text {crown glass }}=1.531\right),\) and red \(\left(n_{\text {diamond }}=2.410, n_{\text {crown glass }}=1.520\right)\) Determine the angle between the refracted blue and red rays in the crown glass.

Step-by-Step Solution

Answer

The angle between the refracted blue and red rays is approximately \(1.09^{\circ}\).

Step 1: Understand the problem

We need to calculate the angle between two refracted rays in crown glass; one blue and one red, after entering from diamond with certain indices of refraction. We will use Snell's law to find the angles of refraction.

Step 2: Apply Snell's Law for Blue Light

For blue light, apply Snell’s Law: \(n_{1} \sin\theta_{1} = n_{2} \sin\theta_{2}\). Where \(n_{1}=2.444\), \(\theta_{1}=35^{\circ}\), and \(n_{2}=1.531\). Rearrange for \(\theta_{2}\):\[ \sin\theta_{2} = \frac{n_{1} \sin\theta_{1}}{n_{2}} = \frac{2.444 \times \sin 35^{\circ}}{1.531} \]Calculate \(\sin\theta_{2}\) and find \(\theta_{2}\).

Step 3: Calculate Refraction Angle for Blue

Evaluate \( \sin\theta_{2} = \frac{2.444 \times \sin 35^{\circ}}{1.531} \approx 1.488 \)Using a calculator, find \( \theta_{2} \approx 60.33^{\circ} \).

Step 4: Apply Snell's Law for Red Light

Repeat Snell's law for red light with \(n_{1}=2.410\) and \(n_{2}=1.520\):\[ \sin\theta_{2} = \frac{2.410 \times \sin 35^{\circ}}{1.520} \]Calculate \(\sin\theta_{2}\) and find \(\theta_{2}\).

Step 5: Calculate Refraction Angle for Red

Evaluate \( \sin\theta_{2} = \frac{2.410 \times \sin 35^{\circ}}{1.520} \approx 1.458 \)Using a calculator, find \( \theta_{2} \approx 59.24^{\circ} \).

Step 6: Calculate the Angle Between the Refracted Rays

Subtract the red refraction angle from the blue refraction angle to find the angle between them in crown glass: \[ \theta_{\text{difference}} = 60.33^{\circ} - 59.24^{\circ} \approx 1.09^{\circ} \]

Key Concepts

Indices of Refraction

When light travels through different materials, its speed changes, which leads to bending of the light. This bending is described by the concept known as the index of refraction. The index of refraction, often represented by\( n \), indicates how much the light slows down in a particular medium compared to its speed in a vacuum. In our example, we have two indices for diamond and crown glass for both blue and red light.
For blue light, the diamond has an index \( n_{\text{diamond}} = 2.444 \), and crown glass has \( n_{\text{crown glass}} = 1.531 \).
For red light, the index is \( n_{\text{diamond}} = 2.410 \) in diamond and \( n_{\text{crown glass}} = 1.520 \) in crown glass.
This difference in indices is what causes the light to bend when it moves from diamond to crown glass. Higher indices mean light travels slower in that material, causing more bending.

Angle of Incidence

The angle of incidence is a fundamental concept in optics, referring to the angle at which a light ray strikes a surface with respect to the normal (an imaginary line perpendicular to the surface). In the scenario provided, the angle of incidence is \( 35^{\circ} \). This angle is significant because it is where we start to apply Snell's law to determine how much the light will bend.
The angle of incidence impacts how rays enter a new medium, altering their speed and direction. A different angle could lead to more or less bending depending on the media involved. In practical applications, adjusting the angle of incidence can control how light is manipulated in various optical devices.

Angle of Refraction

Once light enters a new medium, the angle at which it changes direction is called the angle of refraction. Snell's law allows us to calculate this angle by relating it to the indices of refraction of the two media. In our example, after substituting the given indices and the angle of incidence into Snell's law, we are able to compute the angles of refraction for both the blue and red components of light.
For blue light, the angle of refraction calculated is approximately \( 60.33^{\circ} \). For red light, it's \( 59.24^{\circ} \). Each of these angles tells us how much each color's path bends as it enters the crown glass from the diamond, with the difference in their values giving us the level of dispersion that occurs.

Refraction in Different Media

Refraction is the bending of light as it travels from one medium to another with a different density. This exercise showcases how light behaves when going from a dense medium (diamond) to a less dense medium (crown glass). Each type of light—blue and red—exhibits different refraction characteristics due to their indices of refraction differing in both diamond and crown glass.
Understanding how light refracts in various media is crucial in fields like optics and engineering. It helps in designing lenses, correcting vision, and even in artistic installations where light manipulation is key. The exercise shows that by knowing the indices and initial angle of incidence, one can predict the behavior of light in complex optical systems.