Derivation of Lens Formula - Definition, FAQs

The Lens Formula is a fundamental equation in optics that establishes a relationship between the object distance, image distance, and focal length of a lens. Whether dealing with a convex or concave lens, this formula provides a precise mathematical way to calculate the position of the image formed by a lens. Understanding the derivation of lens formula in physics helps in gaining deeper insights into how lenses manipulate light to form images. In this article, we will explore the derivation of the lens formula, highlighting the underlying principles of ray optics and the geometry involved in image formation by thin lenses.

This Story also Contains

1. What Is A Lens?
2. What is Lens Formula?
3. State Lens Makers' Formula
4. What Is Thin Lens?
5. Derive Lens Maker’s Equation
6. Lens Formula Derivation
7. Image Formation With A Thin Lens: Characteristics

What Is A Lens?

The lens is a piece of transparent material with two refracting surfaces such that at least one surface is curved and the refractive index of its material is different from that of the surroundings.

Different Types of Lens

The lens is divided into two types depending on how the light rays behave when they travel through the lens. These are further divided into the following types.

What is Lens Formula?

The lens equation or lens formula is an equation that links the focal length, image distance, and object distance.

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

where,

v is the distance of the image from the optical center of the lens

u is the distance of the object from the optical center of the lens

f is the focal length of the lens

State Lens Makers' Formula

The relationship between a lens' focal length, the refractive index of its material, and the radii of curvature of its two surfaces is known as the lens maker's formula. Lens manufacturers use it to build lenses with a specific power from glass with a specific refractive index. The focal length, f, is described by the lens maker formula:

$\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

where

R₁ and R₂ are the radii of curvature

n is the index of refraction

Focal length

The focal length of an optical system is the inverse of the system's optical power; it measures how strongly the system converges or diverges light. A system with a positive focal length converges light, while one with a negative focal length diverges light.

Power

It is the reciprocal of the focal length and is measured in dioptre(D). Power is positive for converging lenses and negative for diverging lenses.

What Is Thin Lens?

A thin lens is defined as one whose thickness is insignificant in comparison to its curvature radii. The thickness (t) is significantly lower than the two curvature radii R₁ and R₂.

The focal length, image distance, and object distance are all connected in the lens formula for concave and convex lenses. The given formula can be used to establish this link.

$$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$$

where,

f is the focal length of the lens

v is the distance of the generated image from the lens' optical center

u is the distance between an item and the optical center of the lens.

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Different Types Of Thin Lens

There are two types of thin lenses- convex lens and concave lens. Read the below table to learn more about these.

Derive Lens Maker’s Equation

The lens maker formula is derived using the assumptions listed below.

• Consider the thin lens in the picture above, which has two refracting surfaces with curvature radii R₁ and R₂, respectively.
• Assume that the surrounding medium and the lens material have refractive indices of n_a and n_b, respectively.
• Consider an object O placed on the principal axis of the thin lens.
• Assume a ray from object O incident on the convex surface of the lens at point A which has a radius of curvature R₁
• _{It forms an image at Q.}
• For convex lens $\mathrm{n}_1=\mathrm{n}_{\mathrm{a}}, \mathrm{n}_2=\mathrm{n}_{\mathrm{b}}$

The whole derivation of the lens maker formula is provided further below. We can say that, using the formula for refraction at a single spherical surface,

For the first surface,

Object Distance = -u

Image Distance = v = x

Radius of Curvature= R₁

Thus the formula becomes,

$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R_1} \ldots$

By substituting we get

$\frac{n_b}{x}-\frac{n_a}{u}=\frac{n_b-n_a}{R_1} \ldots$ --------------(1)

• The ray gets refracted due to the concave surface at point B after getting refracted at point A and reaches the principal axis at point I.
• The image Q of object O due to the convex surface is taken as the object for the concave surface.
• Object distance PQ = u = x, Image distance = PI = v, radius of curvature = -R₂
  For concave surface , n₁ = n_b, n₂ = n_a

For the second surface,

$\frac{n_2}{v}-\frac{n_1}{x}=\frac{n_2-n_1}{-R_2}$ --------------- ( 2)

Now adding equation (1) and (2),

$\begin{aligned} & \frac{n_a}{v}-\frac{n_b}{u}=\left(n_b-n_a\right)\left[\frac{1}{R_1}-\frac{1}{R_2}\right] \\ \\& \Rightarrow \frac{1}{v}-\frac{1}{u}=\left(\frac{n_b}{n_a}-1\right)\left[\frac{1}{R_1}-\frac{1}{R_2}\right]\end{aligned}$

We know that,

$\frac{1}{v}-\frac{1}{u} =\frac{1}{f} $

Hence the equation becomes,

$\frac{1}{f}=\left(\frac{n_2}{n_1}-1\right)\left[\frac{1}{R_1}-\frac{1}{R_2}\right]$

Therefore, we can say that,

$\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Where μ is the material's refractive index.

This is the derivation of the lens maker formula. Examine the constraints of the lens maker's formula to have a better understanding of the lens maker's formula derivation.

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Lens Formula Derivation

Let AB represent an object at a distance greater than the focal length f of the convex lens that is located at right angles to the primary axis. The image A1B1 is generated between O and F1 on the same side as the item, and it is virtual and erect.

• Focul length = OF₁ = f
• Object distance = OA =u
• Image distance = OA¹ = v

$\triangle O A B$ and $\triangle O A^1 B^1$ are similar

$\left[\begin{array}{l}\because \angle B A O=\angle B^1 A^1 O=90^{\circ}, \text { vertex is common for both the triangles } \\ \text { so } \angle A O B=\angle A^1 O B^1, \therefore \angle A B O=\angle A^1 B^1 O\end{array}\right]$

$\frac{A^1 B^1}{A B}=\frac{O A^1}{O A}---(1)$

$\triangle O C F_1$ and $\triangle F_1 A^1 B^1$ are similar

$\frac{A^1 B^1}{O C}=\frac{A^1 F_1}{O F_1}$

But from the ray diagram, we see that OC = AB

$\begin{aligned} & \frac{A^1 B^1}{A B}=\frac{A^1 F_1}{O F_1}=\frac{O F_1-O A^1}{O F_1} \\ & \frac{A^1 B^1}{A B}=\frac{O F_1-O A^1}{O F_1}---(2)\end{aligned}$

From equation (1) and equation (2), we get

$\begin{aligned} & \frac{O A^1}{O A}=\frac{O F_1-O A^1}{O F_1} \\ & \frac{-v}{-u}=\frac{-f--v}{-f} \\ & \frac{v}{u}=\frac{-f+v}{-f} \\ & -v f=-u f+u v\end{aligned}$

Dividing throughout by uvf

$-\frac{1}{u}=-\frac{1}{v}+\frac{1}{f}$

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

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Image Formation With A Thin Lens: Characteristics

It's not enough to know the thin lens formula for convex lenses. The characteristics of a ray of light going through converging and diverging lenses must be understood.

1. On the opposite side, parallel rays going through converging lenses will intersect at point f.

2. Parallel rays seem to emerge from point f in front of diverging lenses.

3. The direction of light rays traveling through the center of converging or diverging lenses does not change.

4. Light rays that enter a converging lens through its focal point always exit parallel to the lens's axis.

5. On the other side of a diverging lens, a light ray going towards the focal point will also emerge parallel to its axis.

A concave or divergent lens has a negative focal length. When the picture is generated on the side where the object is positioned, the image distance is also negative. The image is virtual in this case. A converging or convex lens, on the other hand, has a positive focal length.