Problem 71 - Arihant AIEEE Physics

Question

A lens forms a sharp image of a real object on a screen. On inserting a parallel slide between the lens and the screen with its thickness along the principal axis of the lens, it is found necessary to shift the screen parallel to itself distance \(d\) away from the lens for getting image sharply focussed on it. If the refractive index of the glass relative to air is \(\mu\), the thickness of the slab is: (a) \(\frac{d}{\mu}\) (b) \(\mu d\) (c) \(\frac{r d}{\mu-1}\) (d) \((\mu-1) \frac{d}{\mu}\)

Step-by-Step Solution

Answer

The correct answer is (c): \( \frac{d \mu}{\mu - 1} \).

Step 1: Understanding the Problem

We have a lens that forms an image on a screen. Inserting a glass slab shifts the image, requiring the screen to be moved a distance \( d \) to refocus. We need to calculate the slab's thickness \( t \) using the given refractive index \( \mu \) of the slab.

Step 2: Recall the Principle of Shift Due to a Slab

When a parallel slab of thickness \( t \) and refractive index \( \mu \) is introduced, the position of the light path is shifted by \( \frac{t}{\mu} \). The effective shift in image position due to the slab is \( (\mu - 1)\frac{t}{\mu} \). This is equal to the distance \( d \) that the screen is moved.

Step 3: Set up the Equation

By the above principle, equate the effective shift caused by the slab to the screen displacement: \[(\mu - 1)\frac{t}{\mu} = d\] This equation relates the slab's thickness \( t \) to the displacement \( d \) and refractive index \( \mu \).

Step 4: Solve for Thickness \( t \)

Rearrange the equation to solve for \( t \):\[(\mu - 1)\frac{t}{\mu} = d\]\[t = \frac{d \cdot \mu}{\mu - 1}\]

Step 5: Select the Correct Choice

Comparing our expression \( t = \frac{d \cdot \mu}{\mu - 1} \) to the given choices, we see that it matches choice (c): \( \frac{d \cdot \mu}{\mu - 1} \). Thus, the correct answer is (c).

Key Concepts

Lens Equations

Lens equations are essential in optics for predicting how lenses focus light. These equations help us determine the position and size of the image formed by a lens from a given object.
One commonly used lens equation is the lens maker's formula:

• \ \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)

Here, n

\( f \) is the focal length of the lens,

\( v \) is the image distance,

\( u \) is the object distance.

The formula is essential to finding out where the image will form when an object is placed at a certain distance from a lens. Using these equations ensures precision and helps in devising clear process paths in optical setups, like cameras and eyeglasses.

Refractive Index

The refractive index, often denoted by \( \mu \), is crucial in understanding how light behaves as it passes through different mediums. This concept explains why light bends when it moves between materials like air and glass.
The refractive index is defined by the ratio:

• \( \mu = \frac{\text{Speed of light in vacuum}}{\text{Speed of light in the medium}} \)

A higher refractive index indicates that light travels slower in the medium compared to a vacuum.
For a glass slab, this bending or refraction causes a shift in the light path, which can affect the position of the image formed by a lens system.

Image Formation

Image formation is a pivotal concept in optics and refers to how lenses capture and project images of objects. This process is governed by the way lenses bend light rays.

• A real image is formed when light rays converge at a point. This can project onto a screen.
• A virtual image appears when light rays diverge and cannot be caught on a screen but can be seen through a lens.

In this exercise, when the glass slab is placed, image formation is affected due to the change in light path caused by the slab. The need to adjust the screen demonstrates how sensitive image formation is to changes in the environment.

Glass Slab Effect

The glass slab effect refers to how a parallel-sided piece of glass shifts the path of light passing through it. This optical phenomenon is closely tied to the refractive index of the glass.
When light enters a glass slab:

• It bends due to a change in speed.
• The shift depends on the slab's thickness and the refractive index.

In the exercise, this shift forced the screen to be moved to a new location to refocus the image. The formula

• \( (\mu - 1)\frac{t}{\mu} \)

represents the actual shift in the image position. Understanding this effect helps in correcting displacement in optical instruments, ensuring accurate image framing and focus.