Lens Maker’s Formula
1.0Master Lens Maker's Formula in Minutes
Have you ever wondered how opticians determine the exact curvature of glass needed to fix someone's vision? Deconstruct the core mathematical relation that lens manufacturers use. Learn how a lens's focal length depends on its material properties and physical geometry through step-by-step derivations and sign-convention techniques.
2.0Learning Outcomes
After completing this lesson, you will be able to:
- Define Lens Maker's Formula and state its physical variables.
- Apply correct Cartesian sign conventions for radii of curvature ($R_1$ and$R_2$).
- Trace the two-stage refraction process across spherical boundaries to understand its derivation.
- Predict focal length modifications when a lens is immersed in different fluid media.
- Solve numerical problems combining refractive indexes, radii of curvature, and focal lengths.
Have you ever wondered how opticians determine the exact curvature of glass needed to fix someone's vision? Or how camera manufacturers design lenses to capture perfectly focused images? They rely on a fundamental equation in optics called the Lens Maker’s Formula.
For Class 10 students, this topic serves as a brilliant bridge between geometry and physics, helping you understand how a lens's physical shape directly influences its ability to focus light.
3.0What is the Lens Maker’s Formula?
The Lens Maker’s Formula is a mathematical equation that relates the focal length (f) of a lens to the refractive index of its material (n) and the radii of curvature (R1 and R2) of its two spherical surfaces.
The Universal Equation:
f1=(nmediumnlens−1)(R11−R21)
If the lens is placed in air, the refractive index of the medium (nmedium) is equal to 1. The formula simplifies to:
f1=(n−1)(R11−R21)
- f = Focal length of the lens
- n = Refractive index of the lens material (usually glass)
- R1 = Radius of curvature of the first spherical surface (where light enters)
- R2 = Radius of curvature of the second spherical surface (where light exits)
Assumptions
The following assumptions are taken for the derivation of lens maker formula.
- Let us consider the thin lens shown in the image above with 2 refracting surfaces having the radii of curvatures R1 and R2, respectively.
- Let the refractive indices of the surrounding medium and the lens material be n1 and n2, respectively.
4.0Derivation of Lens Maker’s Formula
Spherical Refraction from 1st surface,
vμ2−uμ1=Rμ2−μ1μ2=μl,μ1=μs,u=u,v=v1,R=R1v1μl−uμs=R1μl−μs………..(1)
Spherical Refraction from 2nd surface,
vμ2−uμ1=Rμ2−μ1μ2=μs,μ1=μl,u=v1,v=v,R=R2vμs−v1μl=R2μs−μl………(2)
Add equations (1) and (2)
vμs−uμs=R1μl−μs+R2μs−μlvμs−uμs=(μl−μs)(R11−R21)
v1−u1=(μsμl−1)(R11−R21)
If an object is placed at infinity, i.e. u = −∞ , then v = f
f1−−∞1=(μsμl−1)(R11−R21)f1=(μsμl−1)(R11−R21)⇒ Lens Maker’s Formula
5.0Sign Convention for Convex Lens
For a convex lens:
- Focal length is positive
- Radius of curvature of first surface is positive
- Radius of curvature of second surface is negative
Convex lenses are converging lenses.
6.0Sign Convention for Concave Lens
For a concave lens:
- Focal length is negative
- Radius of curvature follows sign convention
- Concave lenses are diverging lenses.
7.0EUREKA by ALLEN – The Future of Class 10 Learning
EUREKA by ALLEN is designed to simplify, enrich, and enhance your experience in Class 10. Through the use of fun and engaging video lessons, regular practice tests, and immediate help for any doubts you may have regarding the material; students have a firm understanding of the concepts they are studying and feel confident in their preparation for their board exams. No matter if you are attempting to receive a higher mark or develop a better understanding of your studies, EUREKA will support you as you continue to grow as a learner.
8.0Supporting Study Materials
This study material, including CBSE Notes and NCERT Solutions for the chapters focusing on Spherical Refraction and Lens Design, is designed according to the latest syllabus guidelines. It delivers precise multi-surface ray diagrams, index transition formulas, and high-yield subjective numerical questions to guarantee complete confidence in board examinations.
9.0Previous Year Question on Lens Maker’s Formula
Question: A double convex lens is made of glass of refractive index 1.5. The radii of curvature of its two surfaces are +20 cm and –20 cm respectively. Calculate the focal length of the lens.
Solution: Using Lens Maker's Formula: 1/f = (μ – 1) [ (1/R₁) – (1/R₂) ]
Given:
- μ = 1.5
- R₁ = +20 cm
- R₂ = –20 cm
Substituting values:
1/f = (1.5 – 1) [ (1/20) – (1/–20) ]
= 0.5 [ (1/20) + (1/20) ]
= 0.5 × (2/20)
= 1/20
Therefore,
f = 20 cm
Answer: The focal length of the lens is 20 cm.
10.0Lens Maker's Formula – 30 Second Quick Revision
- Used to determine the focal length of a lens.
- Depends on the refractive index of the lens material.
- Depends on the radii of curvature of the two lens surfaces.
- Formula: 1/f = (μ − 1)[(1/R₁) − (1/R₂)]
- f = Focal length of the lens.
- μ = Refractive index of the lens material.
- R₁ = Radius of curvature of the first surface.
- R₂ = Radius of curvature of the second surface.
- Convex lens → Positive focal length.
- Concave lens → Negative focal length.
- Greater curvature ⇒ Smaller focal length.
- Used in cameras, microscopes, telescopes, and spectacles.
11.0Recommended Next Topics