An erect image \[2.0\,cm\] high is formed \[12\,cm\]from a lens, the object being \[0.5\,cm\] high. Find the focal length of the lens.

Hint: To solve this question we have to know about focal length. We know that, focal length is a notable word in optical material science. Central length is the distance over which the equal beams either merge or veer. For convex lenses the central length is positive. Then again, the central length of the inward focal point is negative.

Complete step by step answer:
Lens manufacturers use the lens maker’s formula to manufacture lenses of the desired focal length. Lenses of different focal lengths are used for different optical instruments. The focal length of a lens depends on the refractive index of the lens and the radii of curvature. The lens maker’s equation is another formula used for lenses that give us a relationship between the focal length, refractive index, and radii of curvature of the two spheres used in lenses.

In this question the given elements are,
Height of the image- \[h' = 2cm\]
Image distance- \[v = - 12cm\]
Object height- \[h = 0.5cm\]
We know that, \[m = \dfrac{v}{u} = \dfrac{{h'}}{h}\]
After putting the values we will get,
\[\dfrac{v}{u} = \dfrac{2}{{0.5}}\]
\[\Rightarrow u = \dfrac{{0.5 \times v}}{2} \\
\Rightarrow u= \dfrac{{ - 12}}{4} \\
\Rightarrow u= - 3\,cm\]
Len’s formula,
\[\dfrac{1}{f} = - \dfrac{1}{v} - \dfrac{1}{u} \]
Here, f= focal length of the lens, v is the distance between the image and the centre of the lens and u is the object distance.
After putting the values we will get,
\[f = \dfrac{{vu}}{{u - v}} \\
\Rightarrow f= \dfrac{{( - 12)( - 3)}}{{ - 3 + 12}} \\
\therefore f= - 4\,cm\]
So the focal length of the lens is \[4\] cm.

Note: We can say, for convex lenses, the focal length is a distance over which all the parallel rays will converge. A certain point over that distance all the parallel rays will converge, that point is known as focal point. Similarly, for concave lenses, the focal length is a distance where the parallel rays will diverge.We know that the refractive index depends on the material of the lens and focal length of the lens. Refractive index is a dimensionless number through which we can get to know how fast light will pass through that lens or any material.