How to Derive the Formula for Combination of Thin Lenses in Contact Stepwise
Combination of Thin Lenses in Contact is a key topic in JEE Main optics. This concept explains how multiple thin lenses, aligned and touching each other, act as a single optical system. Such combinations are used in microscopes, cameras, and spectacle lenses. Understanding their combined focal length and power is critical for both numerical problems and designing optical instruments.
When two or more thin lenses are placed so their centers coincide and the distance between them is negligible, their effect on light rays can be treated as one equivalent lens. This is the principle behind the combination of thin lenses in contact—using multiple lenses together as a composite system.
Key examples include high-powered spectacle lenses, camera zoom systems, and compound microscopes, where one lens alone cannot provide the needed effect. The same analysis also supports advanced topics like lens makers’ formula for designing complex optics.
Physical Principle Behind Combination of Thin Lenses in Contact
A thin lens is one whose thickness is much smaller than the radius of its surfaces. When such lenses are placed "in contact," light passes through each sequentially without significant displacement between lenses. The combined effect can be described using the lens formula for each and treating the emerging image from one lens as the object for the next.
- All lenses must share a common optical axis.
- The distance between lenses is nearly zero compared to their focal lengths.
- Applicable for both convex and concave combinations.
- Image formation can be predicted using a single effective focal length.
Formula for Combination of Thin Lenses in Contact
If f₁, f₂,..., fn are the focal lengths of n thin lenses in contact, the equivalent or combined focal length F is:
| Type | Formula | Units |
|---|---|---|
| Combined focal length | 1/F = 1/f₁ + 1/f₂ + ... + 1/fn | metre (m) |
| Combined power (P) | P = P₁ + P₂ + ... + Pn | dioptre (D) |
Here, P is the total power and Pi = 1/fi (in metres) for each lens. This formula only works if the lenses are in contact and the system is in the same medium throughout.
Stepwise Derivation: Combination of Thin Lenses in Contact
- Let the object be at a distance u from the first lens (L₁).
- L₁ forms an image at v₁ (using 1/v₁ – 1/u = 1/f₁).
- This image acts as object for L₂; apply 1/v – 1/v₁ = 1/f₂ for the next lens.
- Rearrange: 1/v – 1/u = 1/f₁ + 1/f₂ for two lenses.
- Generalize to n lenses: 1/v – 1/u = 1/f₁ + 1/f₂ + ... + 1/fn.
- Define equivalent focal length F as 1/v – 1/u = 1/F.
- Thus, 1/F = 1/f₁ + 1/f₂ + ... + 1/fn.
The sign convention is crucial. Use the standard Cartesian sign setup: focal length of convex lens is positive, concave is negative, all distances measured from the lens’ optical center.
Power of Combination: Quick Calculation
For thin lenses in contact, powers add algebraically: total power P = P₁ + P₂ + .... This enables quick calculations, especially in numericals or in designing spectacle lenses. Remember that P = 1/f with f in metres and P in dioptres.
- Convex lens: positive power (converging).
- Concave lens: negative power (diverging).
- Powers must be added with correct signs.
Numerical Example: Combination of Two Thin Lenses in Contact
Two thin lenses of focal lengths f₁ = +20 cm (convex) and f₂ = -40 cm (concave) are placed in contact. What is their combined focal length and power?
- Convert both to metres: f₁ = +0.20 m, f₂ = -0.40 m.
- 1/F = 1/0.20 + 1/(-0.40) = 5 - 2.5 = 2.5
- So, F = 1/2.5 = 0.40 m.
- P = 1/f₁ + 1/f₂ = 5 D - 2.5 D = 2.5 D.
Thus, the system behaves like a 0.40 m focal length converging lens of 2.5 dioptres power.
Common Mistakes and Best Practices
- Forgetting to convert focal length to metres when calculating power.
- Incorrect sign convention: convex is positive, concave is negative always!
- Using this formula when lenses are not in contact—does not apply if separated by distance.
- Not accounting for the medium; formula assumes air/glass interface without change.
- Overlooking object–image relations in sequentially placed lenses.
- Assuming powers multiply—they always add algebraically, not multiply!
For details on the relevant sign conventions and more examples, explore linked revision topics at Vedantu.
Applications & Further Practice: Combination of Thin Lenses in Contact
- Spectacle “high power” lenses—layering lenses to reach required correction.
- Camera lens systems for zoom and image focusing.
- Physics laboratory experiments involving refraction through lenses.
- Numerical problems in JEE Main—typical short calculation and application questions.
The combination of thin lenses in contact lets you predict image position, system power, and design complex optics—foundational for JEE optics. Mastering the formula, derivation, and sign convention is essential for both MCQs and reasoning-based questions.
FAQs on Combination of Thin Lenses in Contact: Concept, Derivation & Formulas
1. How do you derive the formula for the combination of thin lenses in contact?
To derive the formula for the combination of thin lenses in contact, consider two or more lenses placed so their centers are perfectly aligned. The combined focal length (F) is given by:
1/F = 1/f1 + 1/f2 + ... + 1/fn
The derivation follows these steps:
- Assume the image formed by the first lens acts as the object for the second lens.
- Apply the lens formula (1/f = 1/v - 1/u) sequentially for each lens.
- Add all the equations to arrive at the result: the reciprocal of the effective focal length is the sum of reciprocals of individual focal lengths.
This result is important in JEE, NEET, and board exams.
2. What happens when two thin lenses are placed in contact?
When two thin lenses are placed in contact, their optical powers add algebraically, giving a new, single lens system with an effective focal length. Effects include:
- The system behaves as a single lens with focal length F where 1/F = 1/f1 + 1/f2.
- The combined power is the sum of the individual powers.
- Image properties (position, size) change according to the combined focal length.
- This is commonly used in devices like spectacles, microscopes, and cameras.
3. What is the combined power of two thin lenses in contact?
The combined power of two thin lenses in contact is the sum of their individual powers:
- If P1, P2 are the powers of two lenses,
Combined Power, P = P1 + P2
- Power is measured in dioptres (D), and P = 1/f (with focal length f in meters).
- For more than two lenses, add all individual powers: P = P1 + P2 + ... + Pn.
4. Can you give an example of solving numericals with combined lenses?
Certainly! For numericals on lenses in contact:
Example: If two lenses of focal lengths f1 = 20 cm and f2 = 30 cm are in contact,
- Calculate effective focal length: 1/F = 1/20 + 1/30 = (3 + 2)/60 = 5/60
- So, F = 60/5 = 12 cm
For power: P = 1/f1 + 1/f2 (convert cm to meters first!)
This method is frequently asked in JEE/NEET exams.
5. Is the combination formula valid if lenses are separated by some distance?
No, the combination formula for thin lenses in contact (1/F = 1/f1 + 1/f2 + ... ) is only valid when the lenses are placed directly in contact (no separation).
- If a distance separates the lenses, additional optical path considerations are required and the effective focal length changes.
- Different formulas must then be used, involving the separation distance.
This is important for exam questions and lens system design.
6. What is a combination of lenses called?
A combination of lenses placed together is called a combined lens system or composite lens.
- When the lenses are in contact, it is also known as thin lenses in contact.
- Such systems are widely used in optical instruments like microscopes, cameras, and spectacles to achieve desired optical properties.
7. How to solve a combination of lenses?
To solve a combination of lenses in contact:
1. Write the given focal lengths (f1, f2, etc.).
2. Use the formula: 1/F = 1/f1 + 1/f2 + ...
3. Calculate combined focal length (F).
4. If asked, compute total power: P = P1 + P2 + ...
5. Substitute values carefully, always using the correct sign convention (convex is positive, concave is negative).
8. Does the medium between lenses affect the combination formula?
Yes, the medium between the lenses can affect the combination formula.
- The standard formula 1/F = 1/f1 + 1/f2 assumes all lenses are in the same medium (usually air).
- If the medium changes (e.g., lenses immersed in water), you need to consider the new focal lengths as calculated for that medium, and then apply the formula.
This is particularly relevant in advanced optics and lab experiments.
9. What sign convention should be followed for focal lengths in the combination formula?
Sign convention is crucial for correct calculation. Follow these rules:
- Convex (converging) lens: Focal length is positive (+).
- Concave (diverging) lens: Focal length is negative (−).
- Substitute these values correctly into the formula 1/F = 1/f1 + 1/f2.
Always use the Cartesian sign convention for consistency, as outlined in the CBSE/JEE syllabus.
10. What common mistakes do students make in lens combination numericals?
Students often make the following mistakes in lens combination numericals:
- Forgetting to convert focal lengths to meters when calculating power.
- Mixing up the sign convention (positive for convex, negative for concave).
- Forgetting the formula only applies when lenses are in contact.
- Not aligning the optical centers in diagrams.
- Misinterpreting the sequence when dealing with more than two lenses.
Careful reading and stepwise substitution helps avoid errors.
11. How is the combination of thin lenses in contact used in real life?
The combination of thin lenses in contact is applied in many real-life optical instruments:
- Microscopes (objective and eyepiece lenses)
- Telescope designs
- Compound lenses in cameras
- Spectacles with combined corrective power
This principle is essential for improving image formation, magnification, and correcting vision problems, as per NCERT and board syllabus.
