Step-by-Step Derivation of Lens Maker’s Formula with Diagram
The Lens Maker's Formula is a fundamental concept in optics which helps in understanding how a lens forms images. It describes the relationship between the focal length of a thin lens, its refractive index, and the radii of curvature of its two surfaces.
This formula is essential for both conceptual clarity and solving numerical problems related to lens design, focal length calculation, and understanding image formation.
What is the Lens Maker’s Formula?
The lens maker's formula provides a direct method to calculate the focal length of a thin lens in air using the physical properties of the lens. The formula is:
1/f = (μ - 1) [1/R1 – 1/R2]
Where:
f = focal length of the lens
μ (or n) = refractive index of the lens material
R1 = radius of curvature of the first surface (object side)
R2 = radius of curvature of the second surface (image side)
Derivation of Lens Maker’s Formula
To understand the derivation, consider refraction at both spherical surfaces of a thin lens.
- For the first surface (with medium μ1 outside and μ2 inside):
(n2/v1) – (n1/u) = (n2–n1)/R1 - For the second surface (from lens to air):
(n1/v) – (n2/v1) = (n1 – n2)/R2 - Add both equations to eliminate v1 (the intermediate image):
(n1/v) – (n1/u) = (n2 – n1) [1/R1 – 1/R2] - For lens in air (n1 = 1, n2 = μ), the formula simplifies to:
1/f = (μ - 1) [1/R1 – 1/R2]
Sign Conventions and Assumptions
Correct sign conventions are critical:
| Physical Quantity | Convex Lens | Concave Lens |
|---|---|---|
| Object distance (u) | -u | -u |
| Image distance (v) | +v | -v |
| Radius R1 | +R1 | -R1 |
| Radius R2 | –R2 | +R2 |
| Focal length (f) | Positive | Negative |
Assumptions:
- The lens is thin.
- The aperture is small.
- Object is a point source on the principal axis.
- Angles of incidence and refraction are small.
These conditions ensure the derived formula accurately predicts focal length.
Step-by-Step Approach to Solving Problems
| Step | Description | Tips |
|---|---|---|
| 1 | Identify the type of lens and write down all given values (μ, R1, R2). | Check if the lens is convex or concave for proper signs. |
| 2 | Assign the correct signs using the sign convention rules. | For convex lens: R1 is positive, R2 is negative. |
| 3 | Substitute values into the lens maker’s formula. | Convert cm to m if SI units are needed. |
| 4 | Solve for the focal length (f). | Express your answer clearly. Check for positive (convex) or negative (concave) f. |
Example: Numerical Application
Example: A biconvex lens has both radii of curvature as 20 cm. The refractive index μ of the material is 1.5. Find the focal length.
- R1 = +20 cm (convex surface facing object)
- R2 = –20 cm (convex surface opposite object)
- μ = 1.5
- Use: 1/f = (1.5 – 1) [1/20 – 1/(–20)]
- 1/f = 0.5 [1/20 + 1/20] = 0.5 × (2/20) = 0.05
- f = 1/0.05 = 20 cm
Result: The focal length is 20 cm.
Limitations of Lens Maker’s Formula
- The lens must be thin.
- The surrounding medium should be the same on both sides.
Violating these assumptions may introduce errors in calculated focal length.
Key Formulas Table
| Formula | Usage |
|---|---|
| 1/f = (μ - 1) [1/R1 – 1/R2] | To compute focal length from lens properties |
| 1/v – 1/u = 1/f | To relate object distance, image distance, and focal length |
Related Links and Resources
- Lens Formula and Magnification
- Power of a Lens
- Concave and Convex Lens
- Difference Between Concave and Convex Lens
- Sign Convention for Spherical Lens
Next Steps for Practice
- Revise assumptions and conventions for each lens type.
- Solve more problems using the Lens Maker’s Formula to build speed and confidence.
- Explore related concepts like aberrations in lenses and advanced applications in optical instruments.
Consistent practice and clear understanding of formulas will help you excel in optics-based questions and applications.
FAQs on Lens Maker’s Formula Explained: Derivation, Equation & Uses
1. What is the lens maker's formula?
The lens maker's formula relates the focal length (f) of a thin lens in air to its refractive index (μ) and the radii of curvature of its two surfaces (R1 and R2). The formula is:
1/f = (μ - 1) [1/R1 - 1/R2]
This equation is essential for designing lenses and is frequently used in optics numerical problems and exam questions.
2. How is the lens maker's formula derived?
The lens maker’s formula is derived by applying the refraction formula at each spherical surface of a thin lens.
Steps:
- Apply the refraction formula for the first surface: (μ2/v1) – (μ1/u) = (μ2 – μ1)/R1
- Apply for the second surface: (μ1/v) – (μ2/v1) = (μ1 – μ2)/R2
- Eliminate the intermediate image distance, assume object at infinity (for focal length), and combine using proper sign conventions.
The final result is:
1/f = (μ – 1) [1/R1 – 1/R2]
3. What is the significance of R1 and R2 in the lens maker’s formula?
R1 and R2 are the radii of curvature of the lens's two surfaces.
- R1: Radius of curvature of the first surface (from which light enters)
- R2: Radius of curvature of the second surface (from which light exits)
- Signs of R1 and R2 are determined by the sign convention (convex: R1 positive, R2 negative if light travels left to right).
4. What are the main assumptions and limitations of the lens maker's formula?
Key assumptions for lens maker’s formula:
- The lens is thin (negligible thickness).
- The aperture of the lens is small.
- Light rays make small angles (paraxial approximation).
- The surrounding medium is the same on both sides of the lens, usually air.
- Not accurate for thick lenses or when lens thickness cannot be ignored.
- Inconsistent results if used for different surrounding media without modification.
5. What is the difference between the lens formula and the lens maker's formula?
The lens formula is used to locate images formed by the lens and is written as:
1/v – 1/u = 1/f,
where u = object distance, v = image distance.
The lens maker’s formula is used to calculate the focal length of the lens based on its physical properties:
1/f = (μ – 1)[1/R1 – 1/R2].
In summary, lens formula deals with image formation, while lens maker’s formula deals with lens design and material properties.
6. How do you use sign conventions in the lens maker's formula?
Sign conventions are crucial for correct results:
- Object distances (u) are negative for objects placed to the left of the lens.
- For convex lenses: R1 is positive, R2 is negative (light enters left to right).
- For concave lenses: R1 is negative, R2 is positive.
- Focal length (f): Positive for convex, negative for concave lenses.
7. For which lenses is the lens maker's formula applicable?
The lens maker's formula applies to both convex and concave lenses, as long as the lens is thin and placed in the same medium (usually air) on both sides.
- For biconvex and plano-convex lenses: Use R1 positive, R2 negative.
- For biconcave and plano-concave lenses: Use R1 negative, R2 positive.
8. What are some practical applications of the lens maker's formula?
Lens maker’s formula is essential in:
- Designing eyeglasses and contact lenses
- Manufacturing microscope and camera lenses
- Calculating required lens curvature for desired focal lengths
- Understanding image formation in optics-based instruments
9. How do you calculate the power of a lens using the lens maker’s formula?
The power (P) of a lens is the reciprocal of its focal length (in meters):
P = 1/f (in dioptres, D)
To calculate power:
- First, use the lens maker’s formula to find f (in meters)
- Then, find P (dioptres) by dividing 1 by the focal length in meters
For example, if f = 0.25 m, then P = 4 D.
10. What happens to the focal length of a lens if it is placed in a medium other than air?
The focal length changes if the lens is placed in a medium other than air.
In this case, use the formula:
1/f = (nlens/nmedium – 1)[1/R1 – 1/R2]
where nlens is the refractive index of the lens material and nmedium is that of the surrounding medium.
So, focal length increases when the lens is in a denser medium compared to air.
11. What are common mistakes students make when applying the lens maker's formula?
Common mistakes include:
- Incorrect sign conventions for R1 and R2
- Mixing up lens formula with lens maker’s formula
- Using wrong units (not converting cm to m when needed)
- Not using the medium's refractive index correctly
12. Can the lens maker's formula be used for thick lenses?
No, the standard lens maker's formula assumes a thin lens.
For thick lenses, corrections are needed using the lens thickness and the more general thick lens equation, which takes into account the distance between the lens's two surfaces.
