Lens makers formula
Lens maker’s formula
Lens makers formula :- Lens maker’s formula is the relation between the focal length of a lens, the refractive index of its material and the radii of curvature of its two surfaces. It is used by lens manufacturers to make the lenses of particular power from the glass of a given refractive index.
Let us suppose that a double convex lens made of glass of refractive index nL is placed in a rarer medium of refractive index nM. Below figure shows the geometry of image formation by the lens.
The image formation can be seen in terms of two steps :-
(i) Formation of image I1 of the object O by the first refracting surface XP1Y
If the lens material were continuous and there were no boundary/second surface XP2Y of the lens, then the refraction would take place only at surface XP1Y and the refracted ray would go straight meeting the ray going along principle axis at point I1. Therefore I1 would have been a real image of O formed after refraction at XP1Y as shown in figure below :-
Now using the formula of refraction at spherical surface, i.e.,
Here nmaterial of refracted ray = nL and nmaterial of incident ray = nM
Object distance P1O = u and image distance P1I1 = v1
So we get …..(1)
(ii) Formation of image I of the virtual object I1 by the second refracting surface XP2Y
Actually the lens material is not continuous. Therefore the refracted ray again suffers refraction at surface XP2Y of the lens, meeting the principle axis actually at I. Therefore I is the final real image of O, formed after refraction through the convex lens as shown in figure below :-
For refraction at the second surface XP2Y, we can consider I1 as a virtual object, whose real image is formed at I.
Now nmaterial of refracted ray = nM and nmaterial of incident ray = nL
Object distance P2I1 ≅ P1I1 = v1 and image distance P2I = v
So we get …..(2)
Adding equation (1) and (2), we get
Now put nL/nM = n = refractive index of material of the lens w.r.t. surrounding medium.
When object on the left of lens is at infinity, image is formed at the principle focus of the lens.
∴ when u = ∞, v = f = focal length of the lens.
From equation (3),
This is the Lens Makers Formula.
(1). For a convex lens R1 is positive and R2 is negative.
⇒ focal length of convex lens is positive.
(2). For a concave lens R1 is negative and R2 is positive.
⇒ focal length of concave lens is negative
3. Lens immersed in a liquid
Suppose that focal length of lens in air is fa and in liquid is fl, then
(A) If a lens is immersed in a liquid whose refractive index is less than refractive index of substance of lens, then
Then fl > fa
So the focal length increases but its nature does not change.
(B) If a lens is immersed in a liquid whose refractive index is equal to the refractive index of substance of lens, then
⇒ fl = ∞ and Pl = 0
It means focal length of lens will become infinite and it will act like a flat transparent plate.
(C) If a lens is immersed in a liquid whose refractive index is greater than refractive index of substance of lens, then
then the value of fl will be negative
It means the convex lens in the air will behave like a concave lens in liquid. The nature of the lens will change that is why, the air bubble (convex) in water behaves like a concave lens.
(D) The focal length of lens (n =1.5) is f then it’s focal length will become 4f, when immersed in water(refractive index of water is 4/3).
(E) From , we get
⇒ focal length of lens is the highest for red color and lowest for violet color.
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