Focal Length Equation
The complete reference to the focal length equation in all its forms. Learn the lens maker equation, thin lens equation, every algebraic rearrangement, simplified forms for special cases, and solve problems with our free calculator.
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1. The Standard Focal Length Equation
The fundamental focal length equation, also called the lens maker equation or lensmaker formula, relates the focal length of a thin lens to its geometry and material:
1/f = (n − 1)(1/R₁ − 1/R₂)
Variables
- f = focal length
- n = refractive index of lens material
- R₁ = radius of curvature of first surface
- R₂ = radius of curvature of second surface
Sign Convention
R > 0 for convex (center of curvature to the right); R < 0 for concave. All lengths in consistent units (meters).
2. All Rearranged Forms
Solve for any variable by algebraically rearranging the focal length equation:
Solve for f (focal length)
f = 1 / [(n − 1)(1/R₁ − 1/R₂)]
Invert both sides of 1/f = (n−1)(1/R₁ − 1/R₂)
Solve for n (refractive index)
n = 1 + 1 / [f × (1/R₁ − 1/R₂)]
Divide both sides by (1/R₁ − 1/R₂), then add 1
Solve for R₁
1/R₁ = 1/R₂ + 1/[f(n − 1)]
R₁ = 1 / (1/R₂ + 1/[f(n − 1)])
Isolate 1/R₁, then take reciprocal
Solve for R₂
1/R₂ = 1/R₁ − 1/[f(n − 1)]
R₂ = 1 / (1/R₁ − 1/[f(n − 1)])
Isolate 1/R₂, then take reciprocal
3. The Thin Lens Equation
The thin lens equation relates focal length to object and image distances. It is different from the focal length equation (lens maker equation) but uses the same f:
1/f = 1/v − 1/u
or equivalently: 1/f = 1/dᵢ + 1/dₒ (depending on sign convention)
How They Relate
The focal length equation gives f from lens geometry. The thin lens equation uses that f to relate object distance u and image distance v. Use the first to design a lens; use the second to predict image position.
4. Simplified Equations for Special Cases
Biconvex Lens (R₁ = R₂)
When both surfaces have equal curvature: R₁ = R, R₂ = −R (opposite signs by convention)
1/f = (n − 1)(2/R) → f = R / [2(n − 1)]
Plano-Convex (R₂ = ∞)
One flat surface: 1/R₂ = 0
1/f = (n − 1)(1/R₁) → f = R₁ / (n − 1)
Plano-Concave (R₁ = ∞)
Flat first surface: 1/R₁ = 0
1/f = (n − 1)(−1/R₂) → f = −R₂ / (n − 1)
Symmetric Biconvex (R₁ = −R₂)
Equal radii, opposite signs (typical biconvex)
1/f = (n − 1)(2/|R|) → f = |R| / [2(n − 1)]
5. Comparison Table
| Equation | Formula | Relates | Use When |
|---|---|---|---|
| Focal Length Equation | 1/f = (n−1)(1/R₁ − 1/R₂) | f to n, R₁, R₂ | Designing lenses, known geometry |
| Thin Lens Equation | 1/f = 1/v − 1/u | f to u, v (object/image distance) | Image formation, ray tracing |
| Thick Lens Equation | 1/f with thickness term | f to n, R₁, R₂, thickness t | Thick lenses (t non-negligible) |
6. Focal Length Equation Calculator
Use the interactive calculator below to solve the focal length equation. Enter n, R₁, and R₂ to compute f.
Lens Maker Formula
Calculate focal length from lens parameters
Formula
1/f = (n-1)(1/R₁ - 1/R₂)
Typical: 1.5 (glass), 1.33 (water), 1.52 (crown glass)
Positive for convex, negative for concave
Positive for convex, negative for concave
7. Worked Examples
Example 1: Solve for f (Biconvex Lens)
n = 1.5, R₁ = 0.1 m, R₂ = −0.1 m
1/f = (1.5 − 1)(1/0.1 − 1/(−0.1))
1/f = 0.5 × (10 + 10) = 10
f = 0.1 m
Example 2: Solve for n
f = 0.2 m, R₁ = 0.15 m, R₂ = −0.15 m
1/R₁ − 1/R₂ = 1/0.15 + 1/0.15 = 13.33
n = 1 + 1/(0.2 × 13.33) = 1 + 0.375
n ≈ 1.38
Example 3: Plano-Convex (Special Case)
n = 1.6, R₁ = 0.25 m, R₂ = ∞
1/f = (1.6 − 1)(1/0.25) = 0.6 × 4 = 2.4
f = 0.417 m ≈ 42 cm
Example 4: Thin Lens Equation (Image Distance)
f = 0.1 m, u = −0.2 m (object 20 cm left of lens)
1/v = 1/f + 1/u = 10 + 1/(−0.2) = 10 − 5 = 5
v = 0.2 m (image 20 cm right of lens)
8. Frequently Asked Questions
What is the focal length equation?
The focal length equation (lens maker equation) is 1/f = (n−1)(1/R₁ − 1/R₂), where f is focal length, n is the refractive index of the lens material, R₁ is the radius of curvature of the first surface, and R₂ is the radius of the second surface.
How do you rearrange the focal length equation to solve for n?
To solve for the refractive index n: n = 1 + 1/[f × (1/R₁ − 1/R₂)]. First calculate the curvature term (1/R₁ − 1/R₂), then divide 1/f by this term, and add 1.
What is the difference between the focal length equation and the thin lens equation?
The focal length equation (lens maker equation) relates focal length to lens geometry: 1/f = (n−1)(1/R₁ − 1/R₂). The thin lens equation relates focal length to object and image distances: 1/f = 1/v − 1/u. They describe different aspects of lens behavior.
What happens when R₁ = R₂ in the focal length equation?
If R₁ and R₂ have the same sign and magnitude (e.g. symmetric biconvex with R₁ = R, R₂ = −R), the curvature term simplifies to 2/R, giving f = R/[2(n−1)]. If R₁ = R₂ with the same sign, 1/R₁ − 1/R₂ = 0 and the lens has no net power (f → ∞).