Understanding the Object-Image Distance in a Concave Lens

Introduction

Concave lenses are a fascinating topic in optics, playing a crucial role in various applications from photography to corrective lenses. In this article, we will delve into the properties and mathematical relationships of a concave lens with a specific focal length to understand the object-image distance when the image is one-quarter the size of the object.

Understanding Concave Lenses: Key Terms

To fully grasp the concept, let's define some key terms: Concave Lens: A lens that is curved outward, causing light rays to diverge as they pass through it. Focal Length: The distance from the lens to its focal point, denoted as f. Magnification: The factor by which an object's size is increased or decreased in the final image, given by M S'/S. Object Distance: The distance of the object from the lens, denoted as S. Image Distance: The distance of the image from the lens, denoted as S'.

Mathematical Relationships in Concave Lenses

The behavior of light rays through a lens can be described by the lens equation and the magnification formula:

Lens Equation

The lens equation is given by:

1/f 1/S 1/S'

Where: f: Focal length S: Object distance S': Image distance

Magnification Formula

The magnification M is given by:

M S'/S

In this specific problem, the image is one-quarter the size of the object, meaning:

S' 0.25S

This implies:

S 4S'

Solving the Problem: Object and Image Distances

Given the focal length f 40 cm, and the magnification M 0.25, we can solve for the object and image distances.

First, we use the magnification formula:

M S'/S 0.25

Therefore:

S 4S'

Substituting this into the lens equation:

1/f 1/S 1/S'

becomes:

1/f 1/4S' 1/S'

To combine the fractions:

1/f (1 4)/4S' 5/4S'

Therefore:

1/f 5/4S'

Rearranging to solve for S':

S' 5f/4

Substituting f 40 cm:

S' 5 * 40 cm / 4 200 cm / 4 50 cm

Now, using S 4S' to find S:

S 4 * 50 cm 200 cm

Verification

To verify the solution, we can substitute back into the lens equation and magnification formula:

Lens equation:

1/40 1/200 1/50

Which simplifies to:

0.025 0.005 0.02

Indeed:

0.025 0.025

This confirms the solution is correct.

Conclusion

In this article, we explored the mathematical relationships in a concave lens with a focal length of 40 cm, where the image is one-quarter the size of the object. We found that the object distance S is 200 cm and the image distance S' is 50 cm. This understanding is crucial for anyone working with optics and lens systems.

Related Keywords

Keywords: Concave lens, focal length, magnification, image distance, object distance