Lens Maker Formula Tutorial
A comprehensive guide to understanding, deriving, and applying the lens maker formula. Perfect for students, educators, and optical engineers.
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1. What is the Lens Maker Formula?
The lens maker formula is a fundamental equation in optics that relates the focal length of a lens to its physical properties: the refractive index of the material and the radii of curvature of its two surfaces.
The Formula
1/f = (n-1)(1/R₁ - 1/R₂)
This formula is essential for designing optical systems, from simple magnifying glasses to complex camera lenses and microscope objectives.
Key Variables
- f = Focal length (meters)
- n = Refractive index of lens material
- R₁ = Radius of first surface (meters)
- R₂ = Radius of second surface (meters)
Prerequisites
- • Basic understanding of refraction
- • Knowledge of Snell's Law
- • Familiarity with geometry
- • Understanding of thin lens approximation
2. Formula Derivation
The lens maker formula is derived by applying Snell's Law at each surface of the lens and using the thin lens approximation.
Step 1: Refraction at First Surface
Using the refraction formula for a single spherical surface:
n₁/s + n₂/s' = (n₂ - n₁)/R₁
Where n₁ = 1 (air), n₂ = n (lens material), s is object distance, s' is image distance.
Step 2: Refraction at Second Surface
The image from the first surface becomes the object for the second surface:
n/s' + 1/s'' = (1 - n)/R₂
Step 3: Combine and Simplify
Adding the equations and using the thin lens approximation (lens thickness ≈ 0):
1/f = (n - 1)(1/R₁ - 1/R₂)
📚 Important Note
This derivation assumes the lens is thin (thickness much smaller than radii of curvature) and surrounded by air (n = 1). For lenses in other media, the formula must be modified.
3. Sign Convention
Understanding the sign convention is crucial for correctly applying the lens maker formula. The most common convention (Cartesian sign convention) is:
✓ Positive (+)
- R > 0: Center of curvature is to the right of the surface (convex surface facing left)
- f > 0: Converging lens (real focus)
✗ Negative (−)
- R < 0: Center of curvature is to the left of the surface (concave surface facing left)
- f < 0: Diverging lens (virtual focus)
🎯 Quick Reference
Biconvex lens:
R₁ > 0, R₂ < 0
Biconcave lens:
R₁ < 0, R₂ > 0
Plano-convex:
R₁ > 0, R₂ = ∞
Plano-concave:
R₁ < 0, R₂ = ∞
4. Step-by-Step Calculation Guide
Identify the lens type
Determine if the lens is convex, concave, or a combination.
Measure or identify radii
Find R₁ (first surface) and R₂ (second surface) in meters.
Apply sign convention
Assign positive or negative signs based on surface curvature.
Find refractive index
Look up or measure the refractive index (n) of the lens material.
Calculate (n-1)
Subtract 1 from the refractive index.
Calculate (1/R₁ - 1/R₂)
Compute the difference of the reciprocals of the radii.
Multiply and invert
Multiply the results from steps 5 and 6, then take the reciprocal to find f.
5. Worked Examples
Example 1: Biconvex Lens
Given:
- • Refractive index n = 1.5 (crown glass)
- • First surface radius R₁ = +10 cm = 0.1 m (convex)
- • Second surface radius R₂ = −10 cm = −0.1 m (convex on right side)
1/f = (n - 1)(1/R₁ - 1/R₂)
1/f = (1.5 - 1)(1/0.1 - 1/(−0.1))
1/f = 0.5 × (10 − (−10))
1/f = 0.5 × 20 = 10
f = 0.1 m = 10 cm (converging)
Example 2: Biconcave Lens
Given:
- • Refractive index n = 1.6 (flint glass)
- • First surface radius R₁ = −15 cm = −0.15 m (concave)
- • Second surface radius R₂ = +15 cm = 0.15 m (concave on right side)
1/f = (n - 1)(1/R₁ - 1/R₂)
1/f = (1.6 - 1)(1/(−0.15) - 1/0.15)
1/f = 0.6 × (−6.67 − 6.67)
1/f = 0.6 × (−13.34) = −8.0
f = −0.125 m = −12.5 cm (diverging)
6. Common Mistakes to Avoid
Incorrect sign for radii
✓ Fix: Always apply the sign convention consistently. Remember: convex surfaces facing the light source are positive.
Using diameter instead of radius
✓ Fix: The formula requires radius of curvature, not diameter. Divide diameter by 2.
Forgetting to subtract 1 from n
✓ Fix: The formula uses (n-1), not n. This is because air has n=1.
Mixing units
✓ Fix: Keep all lengths in the same unit (preferably meters) throughout the calculation.
Applying to thick lenses
✓ Fix: This formula is for thin lenses only. Use the thick lens formula for lenses where thickness matters.
7. Practice Problems
Test your understanding with these practice problems. Use our calculator to verify your answers.
Problem 1: A plano-convex lens (R₁ = 20 cm, R₂ = ∞) is made of glass with n = 1.5. Find the focal length.
Show Answer
40 cm
Problem 2: Calculate n for a biconvex lens (R₁ = 10 cm, R₂ = −15 cm) with f = 12 cm.
Show Answer
n ≈ 1.5
Problem 3: A meniscus lens has R₁ = 10 cm and R₂ = 15 cm with n = 1.6. Is it converging or diverging?
Show Answer
Converging (f > 0)