

PHYSICS11
Chapter 26  Lenses
Definition:
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/(vf) [where, CA= u is object distance,
CA^{'} is image distance and CF= f is focal length] uvuf = 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
L_{1} and L^{2} 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 L_{2},
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 L_{2} is placed in contact with L_{1}, 'I' acts as object for
L_{2} 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 L_{2}, we get, 1/f
_{2} = 1/(v_{1}) + 1/v
< (ii)
Adding equation (i) and equation (ii) we get, 1/u + 1/v = 1/f_{1} +
1/f_{2}
< (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/f_{1} +
1/f_{2}
This is the required expression for the equivalent focal length of two thin
convex lenses having focal length f
_{1} and f_{2} 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(μ1)
< (ii)
From equation (i) and equation (ii) we get, h/f = A(μ1) 1/f =
A(μ1)/h
< (iii)
The normal at P and Q pass through the centre of curvature C_{1}
and C_{2} of lens surfaces as shown in figure. From
the figure,
< PMC_{2} =
< QMC_{1} = A Also,
< QMC_{1} =
< MC_{2}C +
< MC_{1}C = α+β [Since α and β are small
angles made by PC_{1} and PC_{2}
with principal axis.] A= α+β = tanα + tanβ
h/r
_{1} + h/r_{2} [r_{1} and
r_{2} be radii of curvatures which are
positive for convex lens.] therefore, A/h =
1/r_{1} + 1/r_{2}
< (iv)
Using equation (iv) in equation (iii) we get,
1/f = (μ1)(1/r_{1} +
1/r_{2}) which is the required lens
maker formula.
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