Q.Two thin convex lenses of focal lengths $20\ \text{cm}$ and $25\ \text{cm}$ are placed in contact. Find the focal length and power of the combination.

For two thin convex lenses in contact, $\dfrac1F=\dfrac1{f_1}+\dfrac1{f_2}=\dfrac1{20}+\dfrac1{25}=\dfrac{5+4}{100}=\dfrac9{100}$, so $F=\dfrac{100}{9}\approx11.1\ \text{cm}$. Equivalently, working directly in powers: $P_1=\dfrac{100}{20}=5\ \text{D}$, $P_2=\dfrac{100}{25}=4\ \text{D}$, so $P=P_1+P_2=9\ \text{D}$, and $F=100/P=100/9\approx11.1\ \text{cm}$, the same answer either way. Since both individual lenses converge and their powers simply add, the combination is more strongly converging than either lens alone (its focal length, $11.1\ \text{cm}$, is shorter than either $20\ \text{cm}$ or $25\ \text{cm}$).

Add the reciprocals of the two focal lengths (or, equivalently, add the two individual powers in dioptres) to get the combination's focal length/power.

• Averaging the two focal lengths instead of adding reciprocals.
• Forgetting the combination's focal length must always be shorter than the shorter of the two individual focal lengths, for two same-sign lenses.