Derivation of Lens Formula - Definition, Formula, FAQs

Derivation of Lens Formula - Definition, FAQs

Edited By Vishal kumar | Updated on Jul 19, 2025 03:20 PM IST

Lenses are pieces of curved glass or some transparent material, and they bend light to create images. Lenses are commonly found in cameras, eyeglasses, microscopes and telescopes. The formation of an image by lenses is understood with the help of the lens formula, a mathematical expression that relates the object distance (u), image distance (v), and focal length (f) of a lens. In this article, we shall derive the lens formula step by step using simple ray diagrams and basic geometry, all for the benefit of a simple-to-understand concept for students and for application to numerical problems.

This Story also Contains

What Is A Lens?
What is the Lens Formula?
State Lens Makers' Formula
What Is a Thin Lens?
Derive Lens Maker’s Equation
Lens Formula Derivation
Image Formation With A Thin Lens: Characteristics

Derivation of Lens Formula - Definition, FAQs

What Is A Lens?

The lens is a transparent object that may be made of glass or plastic, and it bends (or refracts) light rays to create images. These are lenses with curved surfaces that primarily use the principle of refraction. Lenses are found in eyeglasses, cameras, microscopes, and magnifying glasses.

Lenses are based on two basic types:

Convex lens (converging lens): Thicker in the middle, bending light rays to bring them together, and can create real or virtual images.
Concave lens (diverging lens): Thinner in the middle, diverging light rays, and always forming a virtual image.

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.

types of lens

What is the 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 centre of the lens, u is the distance of the object from the optical centre of the lens and f is the focal length of the lens.

State Lens Makers' Formula

The relationship between a lens's 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's 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, and 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 diopters (D). Power is positive for converging lenses and negative for diverging lenses.

What Is a 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.

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.

Converging (convex) Lens	Diverging (concave) Lens
• The focal length is positive (+ve)	• The focal length is negative (-ve)
• A convex lens is thicker at the center and thinner at the edges	• A concave lens is thicker at the edges and thinner at the center
• Use for correction of long-sightedness	• Use for correction of short-sightedness
• On passing the light through the lens, it bends the light rays towards each other	• On passing the light through the lens, it bends the light rays away from each other
• The image formed can be real, virtual, enlarged, or diminished	• The image formed is always virtual and diminished
• The principal focus is real	The principal focus is virtual
• It is also called a positive lens	• It is also called a negative lens
• Human eye, camera	• Lights, flashlights

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Derive Lens Maker’s Equation

lens maker equation

The lens maker's 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 na and nb, 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 = nb, n2 = na

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.

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.

lens formula derivation

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

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

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

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.

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

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

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

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

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.

Frequently Asked Questions (FAQs)

1. What is the lens formula?

1f=1v−1u

2. When is a Concave Lens' Image Virtual?

Only when the object and image are on the same side of the lens is the picture generated by a concave lens virtual.

3. When Do Convex Lenses Act Like Combined Lenses?

The combined lens works as a convex lens if the focal length of the second lens is greater than the focal length of the first lens.

4. What is the answer to the lens formula ?

The lens formula is the relationship between the object's distance u, the image's distance v, and the lens's focal length f. With the right sign conventions, this law can be applied to both concave and convex lenses.

5. Concave lenses come in a variety of shapes and sizes.

Concave lenses can be found in a variety of real-world applications.

Telescopes and binoculars

Nearsightedness can be corrected with eyeglasses.

Cameras.

Flashlights.

6. Which lens is used to correct myopia?

A concave lens is used to correct myopia.

7. Who discovered the lens maker formula?

The lens maker formula was discovered by Rene Descartes.

8. What type of lens is the human eye?

The human eye has a convex (biconvex) lens.

9. How can the lens formula be used to explain the concept of a lens's back focal length?

The back focal length (BFL) is the distance from the rear vertex of a lens to its focal point. While the simple lens formula assumes a thin lens, it can be extended to thick lenses where the BFL becomes relevant. In such cases, the formula helps in understanding how the effective focal length differs from the BFL due to the lens's thickness and shape.

10. How does the lens formula help in understanding chromatic aberration?

The lens formula itself doesn't account for chromatic aberration, but it forms the basis for understanding it. Different wavelengths of light have slightly different refractive indices, leading to different effective focal lengths (f) for each color. By applying the lens formula separately for different wavelengths, we can understand how images of different colors form at slightly different positions, causing chromatic aberration.

11. What does the lens formula tell us about the relationship between a lens's focal length and its light-gathering ability?

While the lens formula doesn't directly address light-gathering ability, it's related to the concept. Lenses with shorter focal lengths (larger 1/f) can have larger apertures relative to their focal length (smaller f-number). This allows more light to be gathered, which is crucial in low-light photography and astronomy. The formula helps in understanding the trade-offs between focal length, aperture, and light-gathering power.

12. How can the lens formula be used to explain the concept of diopters in vision correction?

The lens formula relates directly to diopters, the unit of measurement for lens power. Since diopters are defined as the reciprocal of the focal length in meters (D = 1/f), the lens formula (1/f = 1/u + 1/v) can be rewritten in terms of diopters. This helps in calculating the required lens power for vision correction based on a person's near and far points.

13. How does the lens formula help in understanding the concept of principal planes in complex lens systems?

While the simple lens formula assumes a thin lens with coincident principal planes, it forms the basis for understanding more complex systems. In thick lenses or multi-lens systems, principal planes are the theoretical planes where refraction is considered to occur. The lens formula can be adapted to use distances measured from these planes, helping to analyze more complex optical setups.

14. What role does the lens formula play in understanding the working of the human eye?

The lens formula is fundamental in understanding how the human eye forms images. It helps explain how the eye's lens changes shape (accommodation) to focus on objects at different distances, effectively changing its focal length to maintain a clear image on the retina. This application of the formula is crucial in optometry and ophthalmology.

15. How does the lens formula help in understanding the concept of a lens's nodal points?

The lens formula, while not directly involving nodal points, helps in understanding their significance. Nodal points are locations where light rays entering and exiting the lens appear to have no angular deviation. The formula's assumption of a thin lens approximates the two nodal points as coinciding at the lens's center, simplifying calculations for image formation.

16. Can the lens formula be used to explain why concave lenses always produce virtual images for real objects?

Yes, the lens formula can explain this phenomenon. For a concave lens, f is negative. When dealing with a real object (positive u), the formula 1/v = 1/f - 1/u always results in a negative v, indicating a virtual image. This mathematical result aligns with the physical reality that concave lenses diverge light rays, forming virtual images for real objects.

17. How does the lens formula relate to the concept of focal planes in optical systems?

The lens formula helps define focal planes in optical systems. The focal plane is where objects at infinity form their images, which occurs when 1/u approaches zero in the formula. This simplifies to v = f, defining the focal plane's location. Understanding this relationship is crucial in designing complex optical systems like cameras and telescopes.

18. How does the lens formula relate to the concept of depth of field in photography?

While the lens formula doesn't directly calculate depth of field, it's fundamental to understanding it. The formula shows how slight changes in object distance (u) affect image distance (v). In photography, depth of field refers to the range of object distances that appear acceptably sharp. The lens formula helps explain why larger apertures (shorter effective f) result in a shallower depth of field.

19. Can the lens formula be applied to any type of lens?

The lens formula can be applied to thin lenses, including both convex and concave types. However, it becomes less accurate for thick lenses or complex lens systems, where more advanced optical formulas are needed to account for additional factors like lens thickness and multiple refracting surfaces.

20. Why is the focal length considered positive for convex lenses and negative for concave lenses?

The sign convention for focal length (positive for convex, negative for concave) is based on whether the lens converges or diverges light. Convex lenses converge parallel rays to a real focal point, while concave lenses diverge them, creating a virtual focal point. This convention simplifies calculations and helps predict image characteristics.

21. What assumptions are made when deriving the lens formula?

The main assumptions in deriving the lens formula are: 1) The lens is thin (negligible thickness compared to focal length), 2) Rays are paraxial (close to and nearly parallel to the principal axis), 3) The medium on both sides of the lens is the same (usually air), and 4) The lens material is homogeneous.

22. Can the lens formula be used to find the focal length of an unknown lens?

Yes, the lens formula can be used to find the focal length of an unknown lens. By measuring the object distance (u) and the corresponding image distance (v), you can plug these values into the formula 1/f = 1/u + 1/v and solve for f. This method is often used in practical optics experiments.

23. What happens to the image when the object is placed at the focal point of a convex lens?

When an object is placed at the focal point of a convex lens, the image forms at infinity. Mathematically, if u = f in the lens formula, then 1/v becomes zero, meaning v is infinite. Physically, this results in parallel rays emerging from the lens, forming no real image.

24. What is the lens formula and why is it important in optics?

The lens formula is a mathematical equation that relates the object distance (u), image distance (v), and focal length (f) of a lens: 1/f = 1/u + 1/v. It's important because it allows us to predict where an image will form and its characteristics for any given lens and object position, which is crucial for designing optical systems and understanding image formation.

25. Why does the lens formula use reciprocals of distances?

The lens formula uses reciprocals of distances (1/u and 1/v) because it simplifies the mathematical relationship between object distance, image distance, and focal length. This form allows for easier calculations and a more intuitive understanding of how changes in one variable affect the others.

26. How does the lens formula differ for convex and concave lenses?

The lens formula itself (1/f = 1/u + 1/v) remains the same for both convex and concave lenses. However, the convention for signs changes: for convex lenses, f is positive, while for concave lenses, f is negative. This sign convention helps account for the different image-forming properties of these lens types.

27. How does changing the object distance affect the image distance according to the lens formula?

According to the lens formula, as the object distance (u) increases, the image distance (v) decreases, assuming the focal length remains constant. This inverse relationship means that moving an object farther from a lens will bring the image closer to the lens, and vice versa.

28. How does the lens formula help in understanding the concept of power of a lens?

The lens formula relates directly to the power of a lens. The power (P) is defined as the reciprocal of the focal length (P = 1/f). Since the lens formula uses 1/f, it naturally incorporates the concept of lens power, helping us understand how a lens's ability to converge or diverge light affects image formation.

29. How is the lens formula derived using ray diagrams?

The lens formula is derived using ray diagrams by considering two key rays: one passing through the optical center and another parallel to the principal axis. By analyzing the similar triangles formed by these rays, we can establish relationships between object distance, image distance, and focal length, ultimately leading to the lens formula.

30. How does the lens formula relate to magnification?

The lens formula is closely related to magnification. The magnification (m) is given by m = -v/u, where v is the image distance and u is the object distance. By combining this with the lens formula, we can derive relationships between magnification, focal length, and object position.

31. How does the lens formula help in understanding the concept of infinite conjugates?

The lens formula helps explain infinite conjugates by showing what happens when an object is very far from the lens (u approaches infinity). In this case, 1/u approaches zero, and the formula simplifies to 1/f ≈ 1/v, indicating that the image forms at the focal point. This concept is crucial in understanding telescopes and other optical systems dealing with distant objects.

32. What happens mathematically in the lens formula when an object is placed beyond twice the focal length of a convex lens?

When an object is placed beyond twice the focal length of a convex lens (u > 2f), the lens formula shows that the image distance (v) will be between f and 2f. Mathematically, this means 1/v will be smaller than 1/f but larger than 1/(2f). This scenario results in a real, inverted, and diminished image.

33. How can the lens formula be used to understand the concept of afocal systems?

Afocal systems, like Galilean telescopes, have no net focal power (1/f = 0). The lens formula helps explain this by showing how the diverging power of one element (1/f negative) is exactly balanced by the converging power of another (1/f positive). This results in parallel input rays remaining parallel after passing through the system, a key characteristic of afocal systems.

34. How does the lens formula account for virtual images?

The lens formula accounts for virtual images through sign conventions. For virtual images, the image distance (v) is considered negative. This allows the formula to work consistently for both real and virtual images, maintaining its mathematical validity across different image-forming scenarios.

35. What is the significance of the '1/f' term in the lens formula?

The '1/f' term in the lens formula represents the optical power of the lens. It quantifies the lens's ability to converge or diverge light. A larger 1/f value indicates a stronger lens (shorter focal length), while a smaller value indicates a weaker lens (longer focal length).

36. How does the lens formula relate to the concept of conjugate points?

The lens formula directly relates to conjugate points - pairs of points where an object at one point forms an image at the other. For any given focal length, the formula shows how object and image distances are related, effectively describing all possible pairs of conjugate points for that lens.

37. Can the lens formula be applied to mirrors? If so, how?

Yes, the lens formula can be applied to mirrors with a slight modification. For mirrors, the formula becomes 1/f = 1/u + 1/v, where f is the focal length of the mirror. The main difference is in sign conventions: for concave mirrors, f is positive, while for convex mirrors, f is negative.

38. What does the lens formula tell us about the relationship between object and image distances when they are equal?

When object and image distances are equal (u = v), the lens formula simplifies to 1/f = 2/u. This means the object and image are both located at twice the focal length (u = v = 2f). This scenario results in an image that is real, inverted, and the same size as the object, which is important in understanding 1:1 imaging systems.

39. What does the lens formula reveal about the relationship between a lens's focal length and its ability to form real images?

The lens formula shows that for real images to form (positive v), the term 1/u must be smaller than 1/f for convex lenses (positive f). This mathematically demonstrates that real images can only form when the object is beyond the focal point, revealing the crucial relationship between focal length and real image formation.

40. Can the lens formula be used to explain why moving a convex lens closer to an object eventually results in a larger image?

Yes, the lens formula explains this phenomenon. As u decreases (lens moves closer to the object), v increases more rapidly to maintain the equality in 1/f = 1/u + 1/v. Since magnification is given by m = -v/u, the increasing v/u ratio results in a larger image. This mathematical relationship aligns with the physical observation of image enlargement as the lens approaches the object.

41. What does the lens formula tell us about the maximum possible magnification for a single lens?

The lens formula, combined with the magnification equation (m = -v/u), reveals that the maximum theoretical magnification occurs as the object approaches the focal point. As u approaches f, v approaches infinity, and the magnification (v/u) becomes very large. However, practical limitations prevent achieving infinite magnification, illustrating the theoretical limits of single-lens systems.

42. What does the lens formula reveal about the relationship between a lens's focal length and its field of view?

While the lens formula doesn't directly calculate field of view, it helps explain the relationship. A shorter focal length (larger 1/f) allows objects at a given distance to form images closer to the lens. This geometrically translates to a wider field of view. Conversely, longer focal lengths result in narrower fields of view, a principle crucial in understanding lens selection in photography and telescope design.

43. Can the lens formula be used to explain why concave lenses always produce upright virtual images?

Yes, the lens formula helps explain this. For concave lenses (negative f), the formula always yields a negative v for positive u, indicating a virtual image. The magnification m = -v/u is always positive in this case, meaning the image is upright. This mathematical result aligns with the physical behavior of concave lenses diverging light rays.

44. How does the lens formula relate to the concept of working distance in microscopy?

In microscopy, the working distance is the space between the objective lens and the specimen. The lens formula helps understand how changing this distance (effectively changing u) affects the image formation. It shows that as the working distance decreases (smaller u), the image distance (v) increases, which relates to the magnification and the design of microscope objectives.

45. How can the lens formula be used to understand the concept of telecentric lenses?

Telecentric lenses are designed so that the chief rays are parallel to the optical axis in object or image space. While the basic lens formula doesn't directly describe telecentricity, it forms the foundation for understanding how these lenses work. By manipulating the object or image distance to approach infinity, we can create conditions where magnification becomes independent of object position, a key feature of telecentric systems.

46. What does the lens formula reveal about the relationship between a lens's focal length and its depth of focus?

The lens formula helps explain depth of focus, which is related to depth of field but on the image side of the lens. Lenses with longer focal lengths (smaller 1/f) have a shallower depth of focus. This is because small changes in object distance (u) result in larger changes in image distance (v) for lenses with longer focal lengths, as evident from the formula's relationships.

47. How does the lens formula help in understanding the concept of hyperfocal distance in photography?

The lens formula is fundamental in calculating hyperfocal distance - the closest distance at which a lens can be focused while keeping objects at infinity acceptably sharp. By manipulating the formula and incorporating the concept of circle of confusion, photographers can determine the optimal focus setting to maximize depth of field, a crucial technique in landscape and architectural photography.

48. Can the lens formula be used to explain why convex lenses can form both real and virtual images?

Yes, the lens formula explains this versatility of convex lenses. For objects beyond the focal point (u > f), the formula yields a positive v, indicating a real image. For objects within the focal length (u < f), v becomes negative, indicating a virtual image. This mathematical behavior aligns with the physical reality of how convex lenses interact with light rays at different object distances.

49. How does the lens formula relate to the concept of lens power addition in optometry?

The lens formula is directly related to lens power addition. Since lens power is defined as P = 1/f, the formula can be rewritten in terms of power: P = 1/u + 1/v. This additive nature of lens powers is crucial in optometry, especially when combining lenses (like in bifocals) or calculating the total power of a lens system.