Two thin lenses are in contact and the focal length of the combination is 60 cm. If the focal length of one lens is 20 cm, the focal length of the other would be:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → - 30 cm
For thin lenses in contact, the powers add, so the reciprocals of the focal lengths add.
\( \displaystyle \frac{1}{F} = \frac{1}{f_{1}} + \frac{1}{f_{2}} \;\Rightarrow\; \frac{1}{f_{2}} = \frac{1}{F} - \frac{1}{f_{1}} \)
\( \displaystyle \frac{1}{f_{2}} = \frac{1}{60\,\mathrm{cm}} - \frac{1}{20\,\mathrm{cm}} = \frac{1 - 3}{60\,\mathrm{cm}} = -\frac{1}{30\,\mathrm{cm}} \)
\( f_{2} = -30\,\mathrm{cm} \)
The minus sign shows that the second lens is diverging (concave). It weakens the 20 cm convex lens so that the pair has the longer focal length of 60 cm.