We know there are two types of lens: concave lens, and convex lens. These lenses are used as per the requirement and play an important role in the study of optics. Lens formula is a well-designed formula that is applicable for concave as well as convex lenses. The lens formula is used to find image distance, type of image formed, and the focal length (f). Let us know the derivation of the lens formula.
What is Lens Formula?
In optics, the relationship between the distance of an image (v), the distance of an object (u), and the focal length (f) of the lens is given by the formula known as Lens formula. Lens formula is applicable for convex as well as concave lenses. These lenses have negligible thickness. The formula is as follows:
1/ v − 1/ u = 1/ f
Lens Formula Derivation
Consider a convex lens with an optical center O. Let F be the principle focus and f be the focal length. An object AB is held perpendicular to the principal axis at a distance beyond the focal length of the lens. A real, inverted magnified image A’B’ is formed as shown in the figure.
From the given figure, we notice that △ABO and △A’B’O are similar.
Therefore,
A ′ B ′ /A B = O B ′ /O B
Similarly, △A’B’F and △OCF are similar, hence
A ′ B ′ /O C = F B ′ /O F
But,
O C = A B
Hence,
A ′ B ′ A B = F B ′ O F
(2)
Equating eq (1) and (2), we get
O B ′ /O B = F B ′ /O F
=O B ′ − O F /O F
Substituting the sign convention, we get
OB=-u, OB’=v and OF=f
v /− u = v − f/ f
v f = − u v + u f o r u v = u f − v f
Dividing both the sides by uvf, we get
u v /u v f = u f /u v f − v f /u v f
⇒ 1 f = 1 v − 1 u
The above equation is known as the Lens formula.
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