Understanding the Combination of Thin Lenses in Contact

The combination of thin lenses in contact involves placing two or more lenses with negligible separation along a common optical axis. This system acts as a single lens with a new effective focal length and power, allowing for simplified analysis of image formation and optical design in advanced physics problems.

Physical Principles of Lenses in Contact

When thin lenses are placed in contact, they share a common optical axis and the distance between them is negligible compared to their focal lengths. This configuration is valid for combinations of both convex and concave lenses.

Each lens refracts light sequentially such that the image formed by one lens acts as the object for the next. The final position of the image can then be determined using a combined or effective focal length.

Analysis of the optical system utilizes the thin lens formula and superposition of the individual lens powers for accurate predictions. For a related topic, see Sign Convention In Lenses.

Formula for the Combination of Thin Lenses in Contact

For $n$ thin lenses with focal lengths $f_1, f_2, \ldots, f_n$ placed coaxially in contact, the combined or effective focal length $F$ is given by the following relation:

$\dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2} + \ldots + \dfrac{1}{f_n}$

Here, each focal length is taken with its proper sign: positive for a convex (converging) lens and negative for a concave (diverging) lens.

The combined optical power $P$ (in dioptres, D) of the lens system is

$P = P_1 + P_2 + \ldots + P_n$, where $P_i = \dfrac{1}{f_i}$ with $f_i$ in metres.

Derivation: Combination of Thin Lenses in Contact

Consider two thin lenses $L_1$ and $L_2$ with focal lengths $f_1$ and $f_2$ in contact. Let an object be placed at distance $u$ from the combination. The final image forms at distance $v$ from the lenses.

First, $L_1$ forms an intermediate image at $v_1$, given by the lens formula: $\dfrac{1}{f_1} = \dfrac{1}{v_1} - \dfrac{1}{u}$.

This image acts as the object for $L_2$ such that: $\dfrac{1}{f_2} = \dfrac{1}{v} - \dfrac{1}{v_1}$.

Adding both equations: $\dfrac{1}{f_1} + \dfrac{1}{f_2} = \left(\dfrac{1}{v_1} - \dfrac{1}{u}\right) + \left(\dfrac{1}{v} - \dfrac{1}{v_1}\right) = \dfrac{1}{v} - \dfrac{1}{u}$.

By the definition of a system with equivalent focal length $F$, $\dfrac{1}{F} = \dfrac{1}{v} - \dfrac{1}{u}$. Therefore, $\dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2}$, which generalizes for $n$ lenses.

This derivation assumes the separation between lenses is negligible and all are placed on a common axis for accurate image alignment.

Sign Conventions for Thin Lenses in Contact

Convex (converging) lenses have positive focal length and power, while concave (diverging) lenses have negative values. All values must use the Cartesian sign convention for consistency.

In calculations, convert focal lengths to metres when determining power in dioptres. See Mirror Formula And Magnification for further understanding of optical sign convention.

Numerical Example: Combination of Two Thin Lenses in Contact

Suppose a convex lens with $f_1 = +20\ \mathrm{cm} = +0.20\ \mathrm{m}$ and a concave lens with $f_2 = -40\ \mathrm{cm} = -0.40\ \mathrm{m}$ are placed in contact. Calculate the combined focal length and power.

• $\dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2}$
• $\dfrac{1}{F} = \dfrac{1}{0.20} + \dfrac{1}{-0.40} = 5 - 2.5 = 2.5$
• $F = \dfrac{1}{2.5} = 0.40\ \mathrm{m}$
• $P = \dfrac{1}{0.20} + \dfrac{1}{-0.40} = 5 - 2.5 = 2.5\,\text{D}$

Therefore, the combination acts as a single converging lens with focal length $0.40$ m and power $2.5$ D.

Key Points and Common Mistakes

• Always convert focal lengths to metres when calculating power
• Use proper sign convention for lens type
• Combination formula applies only for lenses in contact
• The medium between lenses must remain the same
• Power values are always added algebraically, not multiplied

Incorrect use of sign for focal length or failure to convert units may lead to errors in calculation. The combination rule is not valid if lenses are separated by a finite distance; in that case, a more complex formula involving separation is required.

Applications of Thin Lens Combinations in Contact

The principle of combining thin lenses in contact is widely applied in designing optical instruments such as cameras, microscopes, and high-power spectacles.

Layering multiple lenses allows for precise control of focal length and system power, enabling correction of vision defects and generation of required magnification. For additional concepts, refer to Difference Between Lens And Mirror.

Skill in using these formulas is essential for JEE Main and similar exams, particularly in both calculation-based and reasoning questions. Related topics include Concave Mirror Image Formation and Thin Film Interference.

Laboratory experiments on refraction and lens combinations reinforce the practical understanding of these formulas, ensuring accurate placement and alignment of optical elements for desired results.

Questions on lens combinations also assess logical application of sequential image-object relations. Learning systematic substitution and correct use of measurement units is critical for avoiding mistakes, as discussed in Application Of Echo.