Problem 17 - Arihant AIEEE Physics

Question

A light ray strikes a flat glass plate, at a small angle ' \(\theta^{\prime}\). The glass plate has thickness ' \(t\) ' and refractive index ' \(\mu^{\prime}\). What is the lateral displacement ' \(d^{\prime}\) ? (a) \(\frac{t \theta(\mu+1)}{\mu}\) (b) \(\frac{t \theta(\mu-1)}{\mu}\) (c) \(\frac{t}{\theta \mu}(\mu-1)\) (d) \(\frac{\mu}{t \theta}(\mu+1)\)

Step-by-Step Solution

Answer

Option (b) is correct: \(\frac{t \theta(\mu-1)}{\mu}\).

Step 1: Understanding the Problem

We need to find the lateral displacement (\(d^\prime\)) of a light ray passing through a glass plate of thickness \(t\) and refractive index \(\mu^\prime\), given that the ray strikes the plate at a small angle \(\theta^\prime\).

Step 2: Formula for Lateral Displacement

The formula for the lateral displacement \(d\) when light passes through a medium of thickness \(t\) at an angle \(\theta\) with refractive index \(\mu\) is \(d = t \cdot \theta \cdot \frac{(\mu - 1)}{\mu}\) for small angles.

Step 3: Apply Variables to the Formula

Replace \(\theta\), \(\mu\), and \(t\) with \(\theta^\prime\), \(\mu^\prime\), and \(t\) in the formula: \(d^\prime = t \cdot \theta^\prime \cdot \frac{(\mu^\prime - 1)}{\mu^\prime}\).

Step 4: Simplify to Match Options

Rewrite and compare \(d^\prime = \frac{t \cdot \theta^\prime \cdot (\mu^{\prime} - 1)}{\mu^\prime}\) to find a match in the given options. Notice that option (b) matches: \(\frac{t \theta^\prime(\mu^\prime-1)}{\mu^\prime}\). Therefore, the correct expression is: \(\frac{t \cdot \theta^\prime \cdot (\mu^\prime - 1)}{\mu^\prime}\).

Key Concepts

Refractive Index

The refractive index, denoted as \( \mu \), is a measure of how much a light ray slows down and bends when it enters a new medium from another. In this context, we're looking at light moving from air into a glass plate. The refractive index is a ratio between the speed of light in vacuum and its speed in the material. If \( \mu \) is greater than 1, light slows down as it enters the material. For glass, \( \mu \) is typically around 1.5.

Important points to note about refractive index:

• It's dimensionless since it's a ratio.
• It depends on the wavelength of light used.
• This means different colors bend differently, a phenomenon you see in a prism.
• As the refractive index increases, light bends more towards the normal line.

Knowing the refractive index helps predict how light behaves as it passes through different materials, such as calculating lateral displacement.

Angle of Incidence

When light strikes a surface, the angle at which it hits is called the angle of incidence, denoted as \( \theta \). This angle is measured between the incoming light ray and the perpendicular line to the surface, known as the normal.

Here's why the angle of incidence is crucial:

• A larger angle means the light ray strikes the surface more slanted.
• A small angle suggests a more direct hit.
• The angle of incidence directly affects how much the light ray bends as it transitions between mediums.

Understanding the angle of incidence is essential in predicting the refraction pattern, which is significant in applications ranging from lenses to studying light paths.

Thickness of Glass Plate

The thickness of the glass plate, indicated by \( t \), plays a significant role in determining how much lateral displacement a light ray undergoes. It's the direct path along the thickness that influences how long light travels within the medium.

Why does thickness matter? Here are some points:

• Greater thickness allows more opportunity for lateral displacement.
• As light travels within a thicker plate, its path deviates more from its original trajectory.
• In our formula, \( t \) multiplies the initial angle effect, amplifying the displacement.

In practical scenarios, knowing the thickness is fundamental to understanding how light behaves in various optical devices like lenses and coatings.

Light Refraction

Light refraction is the bending of light when it passes from one material into another. This change in direction is due to a change in speed as light moves between media with different refractive indices.

Here's a simple breakdown of what happens:

• Light speeds up or slows down when it enters a different medium.
• This causes the light ray to change its direction or bend.
• The amount it bends depends on the angle of incidence and the refractive indices of the materials.

Refraction causes the light to deviate from its straight-line path, leading to effects like lens focusing, optical illusions, and lateral displacement. Mastering the concept of refraction is vital for fields ranging from optical engineering to everyday applications like eyeglasses.