Thin Lenses in Combination

The Power of a Lens ($P$) is a measure of its ability to deviate light, defined as the reciprocal of the focal length ($P = \frac{1}{f}$).

When lenses are combined, their individual powers add together algebraically to determine the total power ($P_{Total}$) of the system.

This additive property works because the first lens changes the convergence or divergence of the light, and the second lens acts upon that already-modified wavefront.

To find the total power of a compound lens, use the summation formula: $P_{Total} = P_1 + P_2 + \dots + P_n$ where each $P$ is measured in dioptres (D).

To determine the effective focal length ($f_{Total}$) of the combination, use the reciprocal relationship: $\frac{1}{f_{Total}} = \frac{1}{f_1} + \frac{1}{f_2} + \dots + \frac{1}{f_n}$

Ensure all focal lengths are converted to meters before calculating power, as $1 \text{ D} = 1 \text{ m}^{-1}$.

When combining a convex and concave lens, the total power may be positive, negative, or zero, depending on which lens is stronger.

If $P_{Total} > 0$, the system acts as a converging lens; if $P_{Total} < 0$, it acts as a diverging lens.

Check Sign Conventions: Always assign a negative sign to the power or focal length of a diverging (concave) lens before adding it to the total.

Unit Consistency: Examiners often provide focal lengths in centimeters; you must divide by 100 to get meters before calculating power in dioptres.

The Reciprocal Step: When asked for the total focal length, students often calculate $P_{Total}$ and forget to take the reciprocal ($f = \frac{1}{P}$) for the final answer.

Alignment Assumption: Standard problems assume the lenses are touching and their principal axes are perfectly aligned; if they are separated by a distance $d$, the simple additive formula no longer applies.