Problem 18 - University Physics with Modern Physics

Question

A transparent liquid fills a cylindrical tank to a depth of 3.60 m. There is air above the liquid. You look at normal incidence at a small pebble at the bottom of the tank. The apparent depth of the pebble below the liquid's surface is 2.45 m. What is the refractive index of this liquid?

Step-by-Step Solution

Answer

The refractive index of the liquid is approximately 1.469.

Step 1: Understanding the Problem

We need to find the refractive index of a liquid in a cylindrical tank. Given that the actual depth of the liquid is 3.60 m, and the apparent depth seen by an observer looking straight down is 2.45 m. This difference occurs due to light bending as it travels from the liquid to the air above it.

Step 2: Introduction to Snell's Law for Apparent Depth

The relationship between actual depth and apparent depth in optics when looking at an object in a medium is given by \[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{\text{Refractive Index}} \]Here, the apparent depth is 2.45 m and the actual depth is 3.60 m.

Step 3: Setting Up the Equation

Substitute given values into the apparent depth formula: \[ 2.45 = \frac{3.60}{n} \] where \( n \) is the refractive index of the liquid that we need to find.

Step 4: Solving for the Refractive Index

Rearrange the equation to solve for \( n \):\[ n = \frac{3.60}{2.45} \] Calculate this division to find the refractive index.

Step 5: Calculation

Perform the division:\[ n = 1.469 \] Thus, the refractive index of the liquid is approximately 1.469.

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that describes how light bends, or refracts, when it passes through different media. It helps explain why a straw looks bent when it's partially submerged in a glass of water. The law is given by the formula: \[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the two media, and
• \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction respectively.

This relation tells us that the light's path will bend towards the normal if it enters a medium with a higher refractive index, and away if it enters a medium with a lower refractive index. Although Snell's Law typically deals with angles, it also applies in scenarios involving apparent depths, explaining why objects under water appear shallower than they really are.

Apparent Depth

When you look into a pool or a tank of clear water, objects at the bottom appear closer than they really are. This is due to a phenomenon called apparent depth.

• Apparent depth is the depth at which an object appears to be, rather than its true depth.
• It occurs because light bends when moving from water to air, changing the view of the object's position.

The formula connecting true depth and apparent depth can be expressed as:\[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{\text{Refractive Index}} \]
Thus, if you know the actual depth and the apparent depth at which you observe an object, you can calculate the refractive index of the medium with this relationship. Understanding apparent depth is crucial in making accurate measurements in liquids and is a practical application of Snell's Law.

Optics

Optics is the branch of physics that studies the behavior and properties of light and its interactions with matter. It's a field that includes phenomena like reflection, refraction, diffraction, and polarization. In this context:

• Refraction is the bending of light as it passes between different transparent materials with different refractive indices.
• The refractive index is a measure of how much light is bent or slowed down in a medium.

Optics is essential not just in scientific research, but in everyday life – from using cameras and eyeglasses, to understanding how fiber optic cables transmit information. By learning about these concepts, you gain a deeper insight into the visual effects you observe and how devices utilize these principles. Understanding optics allows us to control and manipulate light for various technological applications, making it a foundational subject in physics.

Transparent Liquid

A transparent liquid, like water or oil, allows light to pass through it without significant scattering so objects can be viewed clearly through them.

• Transparency in liquids depends on the absence of impurities and the medium's molecular structure.
• Refractive index plays a crucial role: it determines how much light is bent as it travels through the liquid, affecting how we perceive objects.

When light transitions from air into a transparent liquid, it slows down, making objects appear slightly different than when viewed directly in air. This property of transparent liquids is widely utilized in various fields, from optics in camera lenses to light control in visual displays. Knowing the refractive index of a liquid helps in designing optical instruments and aids in applications such as underwater photography and aquatic research.