Problem 18 - Master Resource Book in JEE Main Physics

Question

A thin lens has focal length, \(f_{1}\) and its aparture has diameter \(d\). It forms an image of intensity \(I .\) Now the central part of the aparture upto diameter \(\frac{d}{2}\) is blocked by an opaque paper. The focal length and image intensity will be change to (a) \(f\) and \(\frac{1}{4}\) (b) \(f\) and \(\frac{3 l}{4}\) (c) \(\frac{f}{2} \operatorname{and} \frac{l}{2}\) (d) \(\frac{3 f}{4}\) and \(\frac{1}{2}\)

Step-by-Step Solution

Answer

(b) Focal length remains \(f\) and intensity is \(\frac{3}{4}I\).

Step 1: Understanding Lens Characteristics

The focal length, \(f\), of a thin lens is determined by its material and curvature, as outlined by the lens maker's formula. Closing off part of the lens's aperture does not change these factors, so the focal length \(f\) remains unchanged.

Step 2: Analyzing Intensity Impact

The original aperture diameter is \(d\), and its area is \(A_1 = \frac{\pi d^2}{4}\). Blocking the central part of the aperture up to diameter \(\frac{d}{2}\) leaves just the peripheral ring open. The blocked area \(A_2 = \frac{\pi (\frac{d}{2})^2}{4} = \frac{\pi d^2}{16}\). The open area of the aperture becomes \(A = A_1 - A_2 = \frac{\pi d^2}{4} - \frac{\pi d^2}{16} = \frac{3\pi d^2}{16}\).

Step 3: Calculating the New Intensity

The intensity of the image is proportional to the area of the aperture that remains open, since a larger area allows more light to pass through. Initially, the intensity was \(I\). Now, with the new open area being \(\frac{3\pi d^2}{16}\), compared to the original \(\frac{\pi d^2}{4}\), the new intensity \(I'\) is given by \(I' = I \times \frac{\frac{3}{16} \pi d^2}{\frac{1}{4} \pi d^2} = I \times 3/4\).

Key Concepts

Thin Lens

Thin lenses are optical devices characterized by their small thickness compared to their focal length and aperture size. They are used extensively in applications ranging from cameras to eyeglasses due to their ability to converge or diverge light efficiently.
A thin lens has two principal focuses on either side, where light converging through it will meet or appear to diverge from if going the opposite direction. The key property of a thin lens is its ability to form clear images by bending light rays, and this functioning is governed by the lens-maker's formula which defines the focal length based on the lens's curvature and material refractive index.

Focal Length

The focal length of a lens, denoted as \(f\), is a measure of how strongly it converges or diverges light rays. It is the distance between the lens and its focal point, and is pivotal in determining the lens's optical power.
For thin lenses, the focal length remains unchanged even if part of the aperture is obstructed. This is because closure of parts of the aperture does not affect the shape or material of the lens considered in the lens-maker's formula.
The focal length is intrinsic to the lens's design, which means blocking the central part of an aperture won't alter the focal length. It continues to provide the same convergence or divergence of light as before.

Image Intensity

Image intensity relates closely to the amount of light passing through the lens's aperture onto the image plane. Intensity \(I\) is proportional to the open area of the aperture.
When a portion of the aperture is blocked, less light enters, reducing the intensity. In the case of the exercise, when the central portion up to diameter \(\frac{d}{2}\) is blocked, the open area becomes a peripheral ring only. This reduces the intensity to \(\frac{3}{4}\) of its initial value, since the remaining open area is only a fraction of the total lens area.
This is crucial for applications needing precise illumination levels, such as photography, telescopes, or microscopes, where image brightness significantly affects overall image quality.

Lens Aperture

The lens aperture is an opening through which light travels before it is focused by the lens system. The diameter of this opening significantly affects both the depth of field and the amount of light that reaches the image sensor.
A larger aperture permits more light, increasing image brightness and reducing depth of field for artistic focus effects. Conversely, a smaller aperture reduces light intake, often requiring longer exposure times or higher sensitivity settings to achieve the same image brightness.
In the given exercise, when the central portion of the aperture is blocked, the aperture effectively decreases in area, resembling a smaller aperture ring pattern. This not only affects image intensity by cutting out central ray paths but also impacts how the lens resolves image details. Understanding apertures help photographers and optical designers optimize performance for specific scenarios.