Problem 2881 - JEE Physics

Question

A convex lens of focal length \(f\) produces a real image \(x\) times the size of an object, Then what is the distance of the object from the lens? (A) \((\mathrm{x}+1) \mathrm{f}\) (B) \((\mathrm{x}-1) \mathrm{f}\) (C) \([(\mathrm{x}+1) / \mathrm{x}] \mathrm{f}\) (D) \([(x-1) / x] f\)

Step-by-Step Solution

Answer

The distance of the object from the lens is \( u = \frac{(x-1)f}{x} \).

Step 1: Write down the lens formula and the magnification formula.

First, write down the lens formula and magnification formula, which will be used together to determine the object's distance from the lens. Lens formula: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \) Magnification formula: \( m = -\frac{v}{u} \)

Step 2: Replace m with x in the magnification formula

Since m = x, we can write the magnification formula using x: \( x = -\frac{v}{u} \)

Step 3: Solve for v in the magnification formula.

Now, solve for v in the magnification formula: \( v = -xu \)

Step 4: Substitute v in the lens formula.

Replace v with the expression found in Step 3 in the lens formula: \( \frac{1}{f} = \frac{1}{-xu} - \frac{1}{u} \)

Step 5: Simplify and solve for u.

Simplify the equation and solve for u: \( \frac{1}{f} = \frac{1-x}{xu} \) Multiplying both sides by f and xu, we get: \( xu = (-x+1)f \) Now, divide both sides by x: \( u = \frac{(-x+1)f}{x} \) Comparing our result with the given options, we find that it matches Option (D). Thus, the distance of the object from the lens is: \( u = \frac{(x-1)f}{x} \).

Key Concepts

Lens Formula

A lens is an optical tool used to bend light and can produce images by refraction. The lens formula is a fundamental equation that helps us understand how lenses form images. It is expressed as:\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]Where:

• \( f \) is the focal length of the lens.
• \( v \) is the image distance from the lens.
• \( u \) is the object distance from the lens.

For convex lenses, the focal length \( f \) is positive, while for concave lenses, it is negative. This formula is essential as it relates the distances of the object and the image from the lens's optical center. When applying this formula, ensure that all distances are considered with respect to the lens's principal axis.
Understanding the lens formula provides a foundation for exploring other concepts such as image formation, magnification, and optical corrections. It serves as a critical tool in calculations related to lens-based optical devices.

Magnification Formula

In optics, magnification describes how much larger or smaller an image is compared to the object. The magnification formula for a lens quantifies this change in size and is given by:\[ m = -\frac{v}{u} \]Here:

• \( m \) represents the magnification factor. It can tell us whether an image is enlarged or reduced compared to the original object.
• \( v \) is the distance from the lens to the image.
• \( u \) is the distance from the lens to the object.

The negative sign indicates that for real images formed by a convex lens, the image is inverted compared to the object. If \( |m| \) is greater than 1, the image is magnified. If \( |m| \) is less than 1, the image is reduced in size.
Magnification helps in understanding the size relationship between objects and their images, which is essential in applications ranging from simple magnifying glasses to complex imaging systems like microscopes and telescopes.

Real Image

A real image is one of the types of images formed by lenses and mirrors, characterized by its ability to be projected onto a screen. It occurs when light rays actually converge at a point after passing through a lens.
Real images are always formed by the convergence of light rays that pass through the lens and meet on the opposite side from the object. For a real image:

• The image is inverted compared to the object.
• It is formed on the opposite side of the lens from where the object is placed.
• Can be captured on a screen.

Convex lenses are particularly known for their ability to produce real images when the object is placed outside the focal length. This property is leveraged in many optical devices like cameras, projectors, and the human eye itself. Understanding the characteristics of a real image aids in identifying the specifications and positioning required for image-focused applications.

Focal Length

The focal length of a lens is a fundamental property that indicates how strongly the lens converges or diverges light. It is the distance from the lens to the focus, where light rays either converge (in convex lenses) or appear to diverge from (in concave lenses).
The focal length \( f \) is crucial in determining the lens's power and usage in various applications:

• For convex lenses, a positive focal length signifies convergence of light rays to a point.
• For concave lenses, a negative focal length indicates divergence of light rays.

The focal length affects both the size and quality of the image produced. Shorter focal lengths yield wider fields of view and are used in wide-angle lenses, while longer focal lengths give narrower fields and are used in zoom or telephoto lenses.
In the context of the lens formula, the focal length relates the object and image distances, playing a pivotal role in optical calculations. Its understanding is crucial for designing and utilizing any lens-based system effectively.