Introduction to the Lens Maker's Equation

Relevance of the Topic

Geometric optics is a fundamental branch of Physics and the study of lenses is an integral part of this field.
The Lens Maker's Equation is a vital tool for understanding image formation and the optics of spherical lenses, hence the considerable importance of this topic.
This equation has a directly practical application in everyday life, being used in the design of glasses lenses, cameras, telescopes, and microscopes.

Within the Physics curriculum, understanding the behavior of light as it passes through material media such as lenses is a central component of broader topics.
At this point in our curriculum journey, we have already acquired solid knowledge of geometric optics and light refraction, which prepares us to explore and deepen the concepts around lenses.
After understanding the lens maker's equation, we will be more capable of predicting and explaining how light rays interact with lenses in the world around us.

Biconvex Spherical Lenses: Fundamental concept necessary for understanding the lens maker's equation. This type of lens has two curved surfaces, with the face of greater curvature being convex. They are the main object of study in geometric optics.
Optical Center of a Lens: The point in the lens where a ray of light passing through the center before refracting does not suffer deviation. This element of the lens is closely linked to the lens maker's equation.
Focal Distance and Parallel Rays: The focal distance of a lens is a measure of how "focused" the lens is, that is, the distance at which parallel rays of light converge after passing through the lens. This is important because the focal distance is used in the lens maker's equation to calculate the position of the image formed by the lens.
Object and Image: In the lens maker's equation, the object is the original light source and the image is the reproduction of the object after passing through the lens. Understanding the nature of these components is fundamental to understanding and applying the equation.

Convex Lens: A lens whose center is thicker than its edges and can be biconvex (curvature on both faces) or plano-convex (one flat face and one convex face). The convex surface deflects light rays inward, causing convergence.
Refraction: Phenomenon of the change in direction and speed of a wave when passing from one medium to another of different density. This change is what allows lenses to deflect light.
Parallel Rays: Rays of light that approach a lens from a parallel direction to each other. In the lens maker's equation, parallel rays are used to calculate the focal distance and the position of the image.
Paraxial Approximation: In geometric optics, the paraxial approximation is a method used to simplify the calculation of light rays when the aperture (or angle) of an optical system, such as a lens, is small.

Case of the biconvex lens with parallel rays: Suppose we are working with a biconvex lens with parallel rays of light incident on it. Using the lens maker's equation, we can determine the focal distance of the lens and the position of the image formed after passing through the lens.
Case where the object-lens distance is greater than the focal distance: In this case, the image will be real, inverted, and smaller than the object. This can occur, for example, when we are using a magnifying glass to read a book.
Case where the object-lens distance is less than the focal distance: In this case, the image formed by the lens will be virtual, upright, and larger than the object. An example is when we use a magnifying glass to enlarge the image of a nearby object.

Types of Lenses: Biconvex spherical lenses are the type of lens commonly studied in geometric optics, as they can be found in various practical applications, such as glasses and camera lenses.
Optical Center: The concept of the optical center is essential for understanding the lens maker's equation. This is the point in the lens where a ray of light that passes straight through it does not suffer deviation.
Focal Distance: The focal distance is an intrinsic parameter of a lens, which depends on its shape and the refractive index of the material from which it is made. This distance is essential in the lens maker's equation, as it determines how light rays are refracted as they pass through the lens.
Object and Image: In the lens maker's equation, the object is what produces the light and the image is where the light is focused after passing through the lens. Understanding these concepts is fundamental for solving problems involving the lens maker's equation.
Parallel Rays: Rays of light that approach a lens from a parallel direction to each other. In the lens maker's equation, the interaction of these rays with the lens is used to determine the position of the image formed.

The lens maker's equation is a powerful statement of geometric optics because it combines the refractory behavior of the lens with the relative positions of the object, the lens, and the image.
The equation provides a precise way to predict where an image formed by a lens will be located, whether it will be upright or inverted, larger or smaller than the original object, and whether it will be real or virtual.
We understand that converging lenses (convex) have a positive focal distance, which implies that, depending on the object-lens distance, images can be projected at different distances and sizes.

Exercise 1: Considering a biconvex lens with a focal distance of 20 cm, an object at 30 cm from the lens, use the Lens Maker's Equation to determine the position, nature (real or virtual), size, and direction of the image formed by the lens.
Exercise 2: Given a biconvex lens where the object-lens distance is 5 cm and the focal distance is 10 cm, calculate the position, nature (real or virtual), size, and direction of the image formed. Interpret the result.
Challenge: Discuss how the parameters involving the lens (radius of curvature, refractive index, focal distance) and the position of the object in relation to the lens influence the formation of the image. Give concrete examples to illustrate your answer.