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Chapter 26 - Lenses


1. Convex lens: The lens having thick middle portion and thin corner (or edge) is called convex lens. It is also
known as converging lens.
2. Concave lens: The lens having thin middle portion and thick corner (or edge) is called concave lens. It is also
known as diverging lens.

Focal length for convex lens:

Consider a convex lens of focal length 'f'. Let, AB be an object placed normally on the principal axis of the lens. The rays
of light from the object AB after refracting through the convex lens meets at point 'B'. So, A'B' is the
real image of the object AB.
Since ΔABC and ΔA'B'C are similar so their corresponding sides are proportional.
AB/A 'B' = CA/CA' <--------- (i)
Similarly ΔCDF and ΔA'B'F are similar. So,
CF/FA ' = CD/A'B'
But, CD = AB
therefore, CF/FA' = AB/A'B' <---------- (ii)
From equation (i) and equation (ii)
CA/CA ' = CF/FA'
CA/CA ' = CF/(CA'-CF)
u/v = f/(v-f) [where, CA= u is object distance, CA' is image distance and CF= f is focal length]
uv-uf = fv
uv = vf+uf
dividing both sides by uv
uv/uv = (vf/uv) + (uf/uv)
1 = f/u + f/v
or 1 = f[(1/u)+(1/v)]
therefore, 1/f = 1/u + 1/v , which is len's formula.

Focal length of thin convex lens in contact.

Let us consider two thin convex lens L1 and L2 respectively.
There is a point object O placed on the principal axis at a distance 'u' from optical centre C.
In the absence of lens L2, I' be the image of O formed by lens L 1
at a distance 'v' from C shown in figure.
Using lens formula we get,
1/f 1 = 1/u + 1/v' <--------- (i)
When lens L2 is placed in contact with L1, 'I' acts as object for L2
and real image I is formed at a distance v from C. Since lenses are thin therefore point of contact can be taken
as C. Now using lens formula for L2, we get,
1/f 2 = 1/(-v1) + 1/v <-------- (ii)
Adding equation (i) and equation (ii) we get,
1/u + 1/v = 1/f1 + 1/f2 <--------- (iii)
If we replaced the combined lens by a single lens having focal length F produced image of I of object O so that,
1/F = 1/u + 1/v <-------- (iv)
From equation (iii) and equation (iv) we get,
1/F = 1/f1 + 1/f2

This is the required expression for the equivalent focal length of two thin convex lenses having focal length
f 1 and f2 respectively.

Lens maker formula:

Lens maker formula: It gives the relation between focal length, radii of curvature of surfaces and refractive index
of its material.

Let us consider a ray OP incident on a convex lens parallel to the principal axis at a height 'h' shown in figure. f be the focal
length of the lens. After refraction through the lens, the emergent ray passes through the focus F. If, 'δ' is the angle
of deviation then,
tanδ= h/f (in ΔMCF)
δ= h/f <-------- (i)
[For small angle of deviation, tanδ ≈ δ]
For a thin lens, the deviation can be considered as deviation by a small angle prism. Tangents at P and Q form a prism of
small angle A. For a small angle of incidence and point of incidence close to C.
We can write, δ= A(μ-1) <--------- (ii)
From equation (i) and equation (ii) we get,
h/f = A(μ-1)
1/f = A(μ-1)/h <---------- (iii)
The normal at P and Q pass through the centre of curvature C1 and C2 of lens surfaces as
shown in figure.
From the figure, < PMC2 = < QMC1 = A
Also, < QMC1 = < MC2C + < MC1C = α+β
[Since α and β are small angles made by PC1 and PC2 with principal axis.]
A= α+β = tanα + tanβ
h/r 1 + h/r2 [r1 and r2 be radii of curvatures which are positive for convex lens.]
therefore, A/h = 1/r1 + 1/r2 <--------- (iv)
Using equation (iv) in equation (iii) we get,
1/f = (μ-1)(1/r1 + 1/r2)
which is the required lens maker formula.

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