The Thin-Lens Equation

This essay will tell about The Thin-Lens Equation but before we begin with the equation. We will have to learn what is the meaning of Thin lens.
The word lens comes from the Latin word for lentil, a seed whose shape is similar to that of a common lens. An optical lens is made from some transparent material (most commonly glass but sometimes plastics or crystals). One or both surfaces usually have a spherical contour. Biconvex spherical lenses (both surfaces convex) and biconcave spherical lenses (both surfaces concave).
The properties of lenses are due to the refraction of light passing through them. When light rays pass through a lens, they are bent , or deviated from their original paths, according to the law of refraction. To analyze lens refraction, we can approximate a biconvex lens by two prisms placed base to base.
A biconvex lens is a converging lens: Incident light rays parallel to the lens axis converge at a focal point on the opposite side of the lens. You may have focused the Sun’s rays with a magnifying glass (a biconvex, or converging, lens) and have witnessed the concentration of radiant energy that results. The parallel rays coming from the Sun or some other distant object converge at the focal point. This fact provides a way for experimentally determining the focal length of a converging lens.
Conversely, a biconcave lens can be approximated by two prisms placed point to point. A biconcave lens is a diverging lens: Incident parallel rays emerge from the lens as though they emanated from a focal point on the incident side of the lens.

As was the case for mirrors, it is convenient to define a point called the focal point  for a lens, For example a group of rays parallel to the axis pass through the focal point, F, after being converged by the lens. The distance from the focal point to the lens is called the focal length, f. The focal length is the image distance that corresponds to an infinite object distance.

Consider a ray of light that passes through the center of a lens, shown as in the picture. If we followed this ray through the lens by applying Snell’s law at both surface, we would find that this ray is deflected from its original direction of travel by a distance ?. We shall make here what is called the thin-lens approximation, that is, the thickness of the lens is assumed to be negligible, as a result the distance ? becomes vanishingly small. Thus, this ray will pass through the lens undeflected. Ray is parallel to the principle axis of lens, and as a result it passes through the focal point, F, after refraction. The point at which these rays intersect is the point which the image is formed.

Now we learn about the Thin lens that what is the meaning was it. Next we will learn how to calculate to find the distance of the object of the image, the focus length and also the magnification of the lens.
We first note that the tangent of the angle theta can be found by using the triangle from the picture below.

From which

Thus, the equation for magnification by a lens is the same as the equation for magnification by a mirror. We also note from the picture that the tangent of theta is

However, the height PO used in the first of these equations is the same as h, the height of the object. There for we get.

Using this in combination with Equation 1 gives

Which reduces to

This equation called The thin lens equation can be use either converging or diverging lenses if we adhere to a set of sign conventions.

Sign Convention for Thin Lenses

s is + if the object is in front of lens.
s is – if the object is in back of lens.

s’ is + if the image is in back of lens.
s’ is – if the image is in front of lens.

f is + for a converging lens
f is – for a diverging lens