In physics, a lens is defined as a device that either focuses or disperses the light beam falling on it using refraction. Based on this concept, the lenses are classified into two types - converging lenses that concentrate parallel rays of light falling on them and diverging lenses that cause parallel rays of light to spread out. No matter whether converging or diverging, both types of lenses are marvels of optical physics and used to create a sharp, magnified, and clear image of the object placed on one side of them. Although the principal purpose of all the lenses is to magnify or we can say make images appear larger than their actual size, still there is a remarkable difference in the images formed by them. For instance, an image formed by a converging lens differs from one formed by a diverging lens. There are several aspects like shapes, physical dimensions, etc., of a lens that impact the behaviour of a light beam falling on it, and also the characteristics of the image formed. To understand the physics of the concept of lenses and images formed by them, we need to know about lens formula. It is a key term around which our optical physics often revolves.
What is Lens Formula?
Based on the physics concept stating that lenses are formed by coupling two spherical surfaces together, lenses are of two types:
Convex lenses formed by binding two spherical surfaces that are curved outward
Concave lenses formed by binding two spherical surfaces that are curved inward.
Characteristics of images created by these lenses differ depending on the aspects of lenses and object's distance from these lenses. It is where the lens formula comes into the action. As per optical physics, lens formula relates the distance of an object (u), the distance of an image (v), and the focal length (f) of the lens. Applicable for both the convex and concave lenses, the lens formula is given as:
1/v - 1/u = 1/f
Where,
v = Distance of image formed from the optical center of the lens.
u = Distance of object from the optical center of the lens.
f = focal length of the lens.
Lens Formula Derivation
Convex Lens
Consider a convex lens with O be the optical centre, and F be the principal focus with focal length f. Now, let AB be the object kept perpendicular to the principal axis and at a distance beyond the focal length.
As the object is perpendicular to the principal axis, the image will also be perpendicular to the principal axis. To find out the location of the image formed, draw a perpendicular from point A' to point B' on the principal axis. On doing this, we can see that the ΔABO and ΔA’B’O are similar as shown in the figure.
So, A′B′/AB = OB'/OB (as ΔABO and ΔA’B’O are similar) ... (1)
Also, ΔA'B'F and ΔOCF are similar
So, A'B'/OC = FB'/OF
But, OC = AB
Therefore, A'B'/AB = FB'/OF
From the above equations, we get:
OB'/OB = FB'/OF = (OB' - OF)/ OF
Now, by using the sign convention, OB = -u, OB' = v, and OF = f, we can say:
v/-u = (v - f) /f
=> vf = - uv + uf or uv = uf - vf
Diving both sides by uvf, we will have:
uv / uvf = (uf / uvf) - (vf/ uvf)
Hence, 1/f = 1/v - 1/u
This is the required lens formula.
Concave Lens
Let AB be the object perpendicular to the principal axis and at a distance more than the focal length (f) of the convex lens. We will see that the image A'B' is erect, virtual, and formed on the same side as the object.
Now, from the figure, we can consider that:
OF1 is the focal length (f),
OA is the object distance (u),
OA is the image distance (v),
and ΔOAB and ΔOA'B' are similar
∵ Angle BAO = Angle B’A’O = 900, vertex O is common for both the triangles
So, Angle AOB = Angle A’OB’
Therefore, Angle ABO = Angle A’B’O
And, A’B’ / AB = OA’ / OA … (1)
Again, ΔOCF1 and ΔF1A’B’ are similar
So, A’B’/ OC = A’F1/ OF1
But from the diagram, we can see that OC = AB
A’B’ / AB = A’F1/ OF1 = (OF1 - OA’)/ OF1
A’B’ / AB = (OF1 - OA’)/ OF1 … (2)
From equation (1) and equation (2), we get
OA’ / OA = (OF1 - OA’)/ OF1
- v / -u = (-f - - v)/ -f
v/ u = (-f + v)/ -f
- vf = - uf + uv … (3)
Dividing equation (3) by uvf
- 1/u = - 1/v + 1/f
Hence, 1/f = 1/v - 1/u
This is the required lens formula.
FAQs on Derivation of Lens Formula
1. What is the Lens Formula and what do its variables represent?
The Lens Formula is a fundamental equation in optics that describes the relationship between the focal length of a lens, the distance of an object from the lens, and the distance of the image formed by the lens. The formula is expressed as: 1/f = 1/v - 1/u. Here’s what each variable represents:
- f is the focal length of the lens.
- v is the image distance from the optical centre of the lens.
- u is the object distance from the optical centre of the lens.
2. What are the essential sign conventions used in the derivation of the lens formula?
According to the Cartesian Sign Convention for spherical lenses, as per the CBSE 2025-26 syllabus, the following rules are critical for a correct derivation:
- All distances are measured from the optical centre (O) of the lens.
- Distances measured in the same direction as incident light are considered positive (+).
- Distances measured in the opposite direction to incident light are considered negative (-).
- Heights measured upwards and perpendicular to the principal axis are positive (+).
- Heights measured downwards and perpendicular to the principal axis are negative (-).
3. What are the key steps to derive the thin lens formula for a convex lens forming a real image?
The derivation involves using the properties of similar triangles formed by the object, image, and lens. The main steps are:
- Draw a ray diagram showing an object placed beyond the focal point of a thin convex lens, forming a real, inverted image.
- Identify two pairs of similar triangles. The first pair is typically formed by the object and the optical centre (e.g., ΔABO) and the image and the optical centre (ΔA'B'O).
- From the first pair, establish the ratio A'B'/AB = v/u.
- Identify the second pair of similar triangles, usually involving the focal point (e.g., ΔMFO and ΔA'B'F).
- From the second pair, establish another ratio for A'B'/AB, which will be A'B'/AB = (v-f)/f.
- Equate the two expressions for A'B'/AB and substitute the values using the correct sign convention (u is negative, v and f are positive for a real image from a convex lens).
- Simplify the resulting equation by dividing all terms by 'uvf' to arrive at the final lens formula: 1/f = 1/v - 1/u.
4. What fundamental assumptions are made for the derivation of the thin lens formula to be valid?
The derivation of the lens formula is an idealised model and relies on several important assumptions to hold true:
- The lens must be thin, meaning its thickness is negligible compared to the object distance, image distance, and radii of curvature.
- The aperture of the lens must be small to minimise spherical aberration.
- The object must be a point object situated on the principal axis.
- The rays of light striking the lens are paraxial, meaning they are close to the principal axis and make small angles with it.
5. How is the Lens Formula different from the Lens Maker's Formula?
These two formulas are related but serve different purposes. The Lens Formula (1/f = 1/v - 1/u) connects the object distance (u), image distance (v), and focal length (f) for a specific situation. In contrast, the Lens Maker's Formula (1/f = (n-1)(1/R₁ - 1/R₂)) relates the focal length (f) of a lens to its physical properties: the refractive index (n) of its material and the radii of curvature (R₁ and R₂) of its two surfaces. Essentially, the Lens Maker's Formula is used to design a lens with a desired focal length, while the Lens Formula is used to find the image position for that lens.
6. Does the derivation of the lens formula change for a concave lens?
No, the fundamental method of derivation using similar triangles remains the same for both convex and concave lenses. The key difference lies in the application of the sign convention. For a concave lens, which always forms a virtual and erect image, the focal length (f) and image distance (v) are taken as negative, while the object distance (u) is also negative. When these signs are correctly substituted into the geometric ratios derived from the similar triangles, the final equation simplifies to the exact same expression: 1/f = 1/v - 1/u. This shows the universality of the formula for thin spherical lenses.
7. Why does the lens formula use a negative sign (1/v - 1/u) while the mirror formula uses a positive sign (1/v + 1/u)?
The difference in sign arises from the way light interacts with lenses versus mirrors and the standard sign conventions. In a mirror, reflection occurs, and real images are formed on the same side as the object (in front of the mirror). In a lens, refraction occurs, and real images are formed on the opposite side of the lens from the object. This physical difference, when combined with the Cartesian sign convention (where distances are measured from the optical centre/pole), leads to the different signs in the final derived formulas to correctly relate f, v, and u in each case.
