Lens Formula
It is a relation between focal length of a lens and distances of object and image from optical centre of the lens.
To derive this formula, we use the same new Cartesian sign conventions.
(a) Convex lens: the image formed may be virtual or real.
Real image: let C be the optical centre and F be the principle focus of a convex lens of focal length CF = ƒ.
AB is an object held perpendicular to the principle axis of the lens at a distance beyond focal length of the lens. A real, inverted and magnified image A’B’ is formed.
As ?s A’B’C and ABC are similar,
Therefore, A’B’/AB = CB’/CB (1)
Again as ?s A’B’F and CDF are similar
Therefore, A’B’/CD = B’F/CF (2)
From (1) and (2),
CB’/CB = B’F/CF = (CB’ + CF)/CF
Using new Cartesian sign conventions, let
CB = -u, CB’ = -v, CF= +ƒ
Therefore, -v/-u = (-v + ƒ)/ƒ
uv – uƒ = - vƒ
Divide both sides by uvƒ
uv/ uvƒ = (uƒ/ uvƒ) – (vƒ/uvƒ)
Or, 1/ƒ = (1/v – 1/u)
This is the required lens formula.
Concave lens: in this case, the image formed is always virtual. Let C be the optical centre and F be the principle focus of a concave lens of focal length ƒ.
AB is an object held perpendicular to the principle axis of the lens. A virtual, erect and smaller image A’B’ is formed due to refraction through concave lens as shown in fig. 3
As ?s A’B’C and ABC are similar,
Therefore, A’B’/AB = B’F/CF (3)
Again as ?s A’B’F and CDF are similar,
Therefore, A’B’/CD = B’F/CF
But CD = AB,
Therefore, A’B’/AB = B’F/CF (4)
From (3) and (4),
CB’/CB = B’F/CF = (CF – CB’)/CF (5)
Using new Cartesian sign conventions, let
CB = -u, CB’ = - v,
CF = - ƒ
-v/-u = (-ƒ + v)/-ƒ
Vƒ = uƒ – uv
Uv = uƒ – vƒ
Divide both sides by uvƒ
uv/ uvƒ = (uƒ/uvƒ – vƒ/uvƒ)
1/ƒ = 1/v = 1/u, this is the required formula.
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