Question

A convex lens of focal length \[40\,{\text{cm}}\]is in contact with a concave lens of focal length \[25\,{\text{cm}}\]. The power of the combination is:
A. \[ + 6.67\,{\text{D}}\]
B. \[ - 6.5\,{\text{D}}\]
C. \[ - 1.5\,{\text{D}}\]
D. \[ + 6.5\,{\text{D}}\]

Answer 1

Hint: Find the power of each lens using the given values of focal lengths for both lenses. Then use the formula for power of combined lenses to calculate the answer. Check whether the units of the quantities are the same.

Complete step by step answer:
Given, focal length of convex lens, \[{f_1} = 40\,{\text{cm}}\]
Focal length of concave lens, \[{f_2} = - 25\,{\text{cm}}\]
The focal length of a concave lens is always taken to be negative.

We have the formula for power of combined lenses as,
\[P = {P_1} + {P_2}\] (i)
Power of a lens can be written as,
\[P = \dfrac{1}{f}\] where \[f\] is the focal length of the lens.
Now, let us take out the power of each lens. Before proceeding, we observe that in the options, the unit of power is given in dioptre (D) which is \[{{\text{m}}^{ - 1}}\] but the focal length is given in cm. So, we convert the unit of focal length from cm to m.
\[{f_1} = 40\,{\text{cm}} = 0.4\,{\text{m}}\]
\[{f_2} = - 25\,{\text{cm}} = - 0.25\,{\text{m}}\]

Power of convex lens we will,
\[{P_1} = \dfrac{1}{{{f_1}}} \\
\Rightarrow {P_1} = \dfrac{1}{{0.4}}\\
\Rightarrow {P_1} = 2.5\,\,{\text{D}} \]

Power of concave lens we will,
\[{P_2} = \dfrac{1}{{{f_2}}} \\
\Rightarrow {P_2} = - \dfrac{1}{{0.25}} = - 4\,\,{\text{D}} \]
Formula for power of the combination from equation (i) we have
\[P = {P_1} + {P_2}\]
Now, putting the values of \[{P_1}\] and \[{P_2}\], we have
\[P = 2.5 + ( - 4) \\
\therefore P = - 1.5\,{\text{D}} \]
Therefore, the power of the combination is \[ - 1.5\,{\text{D}}\].

Hence, the correct answer is option C.

Note: Most of the time students make mistakes in units. So, before proceeding always check for the units. See in which units are the options given, like here options were in SI units, so we convert the given values in SI units. And if any question the options are in CGS unit then convert the given quantities in CGS unit.