Problem 12 - University Physics with Modern Physics

Question

A horizontal, parallel-sided plate of glass having a refractive index of 1.52 is in contact with the surface of water in a tank. A ray coming from above in air makes an angle of incidence of \(35.0^{\circ}\) with the normal to the top surface of the glass. (a) What angle does the ray refracted into the water make with the normal to the surface? (b) What is the dependence of this angle on the refractive index of the glass?

Step-by-Step Solution

Answer

(a) The refracted angle in water is approximately \(25.8^{\circ}\). (b) Higher glass refractive index decreases refraction in glass, increasing it in water.

Step 1: Identify known values

The refractive index of air is approximately 1. The refractive index of glass is given as 1.52. We also know the refractive index of water is approximately 1.33. The angle of incidence in air is given as \(35.0^{\circ}\).

Step 2: Apply Snell's Law at Air-Glass Interface

Snell's law states \(n_1 \sin \theta_1 = n_2 \sin \theta_2\). Plug in the values for light passing from air into the glass:\(1 \times \sin(35.0^{\circ}) = 1.52 \times \sin(\theta_{glass})\).This can be rearranged to find \(\theta_{glass}\):\[\sin(\theta_{glass}) = \frac{\sin(35.0^{\circ})}{1.52}\]. Solve this to find \(\theta_{glass}\).

Step 3: Calculate Angle of Refraction in Glass

Calculate \(\sin(\theta_{glass})\) using the equation from Step 2:\(\sin(\theta_{glass}) = \frac{\sin(35.0^{\circ})}{1.52} \approx 0.375\).Next, find \(\theta_{glass}\):\(\theta_{glass} = \sin^{-1}(0.375) \approx 22.0^{\circ}\).

Step 4: Apply Snell's Law at Glass-Water Interface

Now apply Snell's Law for the transition from glass to water:\(1.52 \sin(22.0^{\circ}) = 1.33 \sin(\theta_{water})\).Rearrange to find \(\theta_{water}\):\[\sin(\theta_{water}) = \frac{1.52 \times \sin(22.0^{\circ})}{1.33}\].

Step 5: Calculate Angle of Refraction in Water

Calculate \(\sin(\theta_{water})\):\(\sin(\theta_{water}) = \frac{1.52 \times \sin(22.0^{\circ})}{1.33} \approx 0.434\).Then find \(\theta_{water}\):\(\theta_{water} = \sin^{-1}(0.434) \approx 25.8^{\circ}\).

Step 6: Discuss Dependence on Glass Refractive Index

According to Snell's Law, the angle of refraction \(\theta_{water}\) depends on the refractive index of the glass. A higher refractive index of the glass would result in a smaller \(\theta_{glass}\), leading to a larger \(\theta_{water}\), and vice versa. The refractive index determines how much the light bends at each interface.

Key Concepts

Refraction

Refraction is a fascinating phenomenon that happens when light transitions from one medium to another. This process causes the path of the light to bend. The bending occurs because light travels at different speeds in different materials. The most common example is witnessing a straw appearing bent when placed in a glass of water. But why does this happen? Here's a simple breakdown:

When light enters a new medium, its speed changes:

• If moving from a less dense to a more dense medium (like air to glass), it slows down and bends towards the normal line.
• Conversely, if it moves from a dense to a less dense medium (like water to air), it speeds up and bends away from the normal.

This change in direction has everything to do with the refractive index of the materials involved, an essential concept in optics.

Angle of Incidence

The angle of incidence is a term in optics that refers to the angle between an incoming ray of light and a line perpendicular to the surface it's hitting, known as the normal line. Understanding this is crucial for predicting how light will behave when striking a surface.

When a light beam approaches a boundary between two materials:

• The angle it makes with the normal at this boundary is the angle of incidence.
• This angle affects how much and in what manner the light refracts, based on Snell's Law.

In the problem at hand, a 35-degree angle of incidence was used to calculate how the light bends as it moves through the glass and into the water.

By knowing this angle, and the refractive indices of the involved media, you can determine the path of light through complex systems.

Refractive Index

The refractive index, often symbolized as 'n', is a dimensionless number that describes how fast light travels through a material compared to a vacuum. It is an intrinsic property of substances affecting how much light will bend when entering them.

Here's why it's essential in the refraction process:

• A material with a higher refractive index slows down the light more and bends it more towards the normal.
• Each time light crosses an interface, Snell's Law uses the refractive indices to predict the new angle.

For our problem, the refractive indices were crucial for calculating how the light refracts between air, glass, and water, affecting the final angle as shown through the calculations using Snell's Law.

Optics

Optics is a branch of physics dealing with the behavior and properties of light. It encompasses the study of things like reflection, dispersion, and refraction, involving applications from simple magnifying glasses to complex fiber optics.

Within optics,

• The principles governing light, like Snell's Law, can be applied to predict behavior in scenarios such as lenses focusing light, or fibers guiding information through refraction and total internal reflection.
• Understanding basics like the angles of incidence and refraction is fundamental to mastering more complex optical systems.

Our example problem illustrates optics concepts crucial to developing technologies like camera lenses or corrective eyewear, showcasing how manipulations of light behavior benefit everyday life.