Question

Find the focal length and nature of the lens which should be placed in contact with a lens of focal length $10cm$ so that the power of the combination becomes $5 \,dioptre$ .

Answer 1

Hint : The power of a lens is the reciprocal of the focal length (in meter) of the lens. The SI unit of power of a lens is dioptres represented by $D$ . Convex lenses have positive focal length, so the power of a convex lens is also positive. A concave lens has a negative focal length, so the power of a concave lens is also negative.

Formula Used:
$\Rightarrow P = \dfrac{1}{f}$
$P$ is the power of the lens in diameter and $f$ is the focal length of the lens in meters.

Complete step by step answer
Let the power of the given lens be ${P_1}$ and the power of the required lens be ${P_2}$ , and the focal length of the required lens be ${f_2}$ .
$\Rightarrow P = \dfrac{1}{f}$
$P$ is the power of the lens in diameter and $f$ is the focal length of the lens in meters.
The focal length of the given lens is $10cm$ i.e. $0.1m$ because $1m = 100cm$ .
Hence,
$\Rightarrow {P_1} = \dfrac{1}{{0.1}}$
$\Rightarrow {P_1} = 10D$
We know that,
Power of combination of two lenses of power ${P_1}$ and ${P_2}$ will be ${P_1} + {P_2}$
We want a lens of power $5D$ .
Hence,
​ $\Rightarrow 5D = \;{P_1}\; + {\text{ }}10D$
$\Rightarrow {P_2}\; = - 5D$
We know that,
$\Rightarrow {P_2} = \dfrac{1}{{{f_2}}}$
$\Rightarrow {f_2} = \dfrac{1}{{ - 5}}m$
$\Rightarrow {f_2} = - 0.2m$
Hence,
The power and focal length of the lens are negative so the given lens is a concave lens.
Therefore the answer to our question is $- 0.2m$ and concave lens.

Additional Information
A concave lens and convex lens can be detected by holding them because the lens with a bulge in middle will be the convex lens and one with a cavity in the middle will be the concave lens.

Note
A concave lens is also called a diverging lens and a convex lens is also called a converging lens. Whenever we are dealing with both power and focal length of the lens together, always be careful to convert the unit of focal length into meters.