Problem 15 - Physics

Question

Light in a vacuum is incident on a transparent glass slab. The angle of incidence is \(35.0^{\circ} .\) The slab is then immersed in a pool of liquid. When the angle of incidence for the light striking the slab is \(20.3^{\circ},\) the angle of refraction for the light entering the slab is the same as when the slab was in a vacuum. What is the index of refraction of the liquid?

Step-by-Step Solution

Answer

The index of refraction of the liquid is approximately 1.657.

Step 1: Understand Snell's Law

Snell's Law states that \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where \( n_1 \) and \( n_2 \) are the indices of refraction of the media, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively. We need to apply this law to find the unknown index of refraction of the liquid.

Step 2: Apply Snell's Law for Air to Glass

When light is incident from air (index of refraction \( n_{air} = 1.0 \)) to the glass slab, we use the angle \( 35.0^{\circ} \). According to Snell's Law, \( 1.0 \sin 35.0^{\circ} = n_{glass} \sin \theta_{glass} \).

Step 3: Apply Snell's Law for Liquid to Glass

When the slab is immersed in the liquid, the angle of incidence is \( 20.3^{\circ} \). Since the angle of refraction inside the glass is the same as in the air case, we have \( n_{liquid} \sin 20.3^{\circ} = n_{glass} \sin \theta_{glass} \).

Step 4: Equate the Two Scenarios

Since \( \sin \theta_{glass} \) is the same for both scenarios, equate them: \( 1.0 \sin 35.0^{\circ} = n_{liquid} \sin 20.3^{\circ} \).

Step 5: Solve for the Index of Refraction of the Liquid

Solve the equation \( 1.0 \sin 35.0^{\circ} = n_{liquid} \sin 20.3^{\circ} \) for \( n_{liquid} \).\[ n_{liquid} = \frac{\sin 35.0^{\circ}}{\sin 20.3^{\circ}} \]

Step 6: Calculation

Calculate \( n_{liquid} \) using the values:\[ n_{liquid} = \frac{\sin 35.0^{\circ}}{\sin 20.3^{\circ}} \approx \frac{0.5736}{0.3462} \approx 1.657 \]

Key Concepts

Index of Refraction

The index of refraction, often denoted by the symbol \( n \), is a measure of how much a medium can bend light. It is a crucial concept in optics that defines the speed of light in a given medium compared to the speed of light in a vacuum. The index of refraction is calculated using the formula:
\[ n = \frac{c}{v} \]
where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.
- **If \( n > 1\)**: The medium is denser than a vacuum, causing light to slow down.- **If \( n = 1\)**: The medium is a vacuum, where light travels fastest.
Materials like glass or water have indices of refraction greater than 1, meaning they bend light more than air (which has an \( n \) of approximately 1). This bending is what causes effects like the apparent bending of a straw when placed in a glass of water. Understanding indices of refraction allows us to determine how light will travel through different substances.

Angle of Incidence

The angle of incidence refers to the angle at which a light ray hits a surface. In optics, this angle is measured between the incoming light ray and the perpendicular (normal) to the surface at the point of incidence.
The concept of the angle of incidence is significant because it affects the bending or refraction of light as it enters a new medium. Snell's Law uses this angle to predict the behavior of light as it crosses boundaries between different media.
In our exercise, we had an angle of incidence of \(35.0^{\circ}\) when the light traveled from air to a glass slab. Then, when the slab was immersed in a liquid, the angle was \(20.3^{\circ}\). These angles show how light initially interacts with the surface, and are key to applying Snell's Law correctly.

Angle of Refraction

The angle of refraction is the angle at which light exits a medium, compared to the normal. It is directly influenced by the nature of both the incident medium and the refracting medium and is predicted using Snell's Law.
When light enters a denser medium, it slows down, causing the angle of refraction to be less than the angle of incidence.
Conversely, when moving to a less dense medium, light speeds up, increasing the angle of refraction compared to incidence. Understanding this angle is crucial for calculating how the path of light changes between different media, allowing precise predictions and applications in lenses, glasses, and various optical instruments.

Transparent Media

Transparent media are materials that allow light to pass through with minimal absorption and with some degree of refraction. When light enters a transparent medium, its speed is reduced, but it maintains its ability to transmit clear images.
Examples include materials like glass, water, and certain plastics. These media are fundamental in optics because they affect how light is bent and focused.

• **Glasses and Lenses**: Use refraction to correct vision or focus images.
• **Scientific Instruments**: Include telescopes and microscopes that rely on transparent media to magnify images and provide better clarity.

The clarity offered by transparency, combined with the bending of light, makes these materials invaluable in designing equipment that performs precise optical tasks.

Optics

Optics is the branch of physics that studies light behavior and properties. It covers phenomena related to light reflection, refraction, and diffraction. Understanding optics is essential for developing technologies and instruments that manipulate light effectively.
- **Applications**: Optics is vital for eyeglasses, cameras, microscopes, and even medical imaging technologies like MRIs. - **Light Manipulation**: Knowledge of optics helps in designing systems to control light, enhancing our visual capabilities and the performance of optical devices.
From everyday items like sunglasses to advanced lasers, optics provides the principles to harness the power of light effectively. The study of optics enhances our ability to see the world around us and develop new technologies that use light in innovative ways.