Question

Two lenses of powers \[ - 1.5\,{\text{D}}\] and \[ + 2.75\,{\text{D}}\] are kept in contact. Find the focal length of the combination.

Answer 1

Hint:The power of the combined lens is the sum of powers of the two lenses. The power of the lens is the reciprocal of the focal length. If the focal length is positive, the lens must be the convex lens.

Formula used:
\[f = \dfrac{1}{P}\]
Here, f is the focal length and P is the power of the lens.

Complete step by step answer:
 Let the first lens have power \[{P_1} = - 1.5\,{\text{D}}\] and focal length \[{f_1}\]. Also, the second lens has the power \[{P_2} = + 2.75\,{\text{D}}\] and focal length \[{f_2}\]. We have the formula for the power of the combined lens when the two lenses are kept next to each other.
\[P = {P_1} + {P_2}\]
Here, P is the power of the combined lens, \[{P_1}\] is the power of the first lens and \[{P_2}\] is the power of the second lens.

Substituting \[{P_1} = - 1.5\,{\text{D}}\] and \[{P_2} = + 2.75\,{\text{D}}\] in the above equation, we get,
\[P = - 1.5 + 2.75\]
\[ \Rightarrow P = + 1.25\,{\text{D}}\]
We have the relation between focal length and power of the lens,
\[f = \dfrac{1}{P}\]
Here, f is the focal length of the combined lens and P is the power of the combined lens.
Substituting \[P = + 1.25\,{\text{D}}\] in the above equation, we get,
\[f = \dfrac{1}{{1.25}}\]
\[ \Rightarrow f = 0.8\,{\text{m}}\]
\[ \therefore f = 80\,{\text{cm}}\]

Thus, the focal length of the combination of the lenses is 80 cm. Since the focal length is positive, we can say that the lens is a convex lens.

Note:The power of the lens is always expressed in dioptres or in meters. If the focal length is given in cm, you should convert it into meters before using it in the formula for power of the lens. Students must be able to recognize the nature of the lens using the sign of the focal length. Note that the concave lens has a negative focal length while a convex lens has a positive focal length.