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What is the Sign Convention of Lens Formula?
Sign Convention
i) Distances along the principal axis are to be measured from the optical centre of the lens.
ii) Distances (from the optical centre) to be measured opposite to the direction of the incident ray are taken as negative and those to be measured in the direction of the incident ray are taken as positive. According to the above convention the focal length of a convex lens is positive and that of a concave lens is negative.
iii) If the principal axis of the lens be taken as x -axis, distances along y -axis above the principal axis are taken as positive and distances along y-axis below the principal axis are taken as negative.
Assumptions are made during discussion of refraction through lens:
- The lens will be thin and its aperture will be small.
- Direction of incident ray will be shown from left to right i.e., the object should be considered to be placed on the left side of the lens.
- The optical centre O of the lens will he the origin of the cartesian frame of reference and the principal axis of the lens will be x-axis.
General Formula of Lens
The relation among object distance, image distance and focal length of a lens is known as general formula of lens.
Convex lens and real image: in Fig., LL’ is a convex lens. An object FQ is placed perpendicular to the principal axis of the lens. A ray from P travelling parallel to the principal axis after refraction through the lens passes through the second principal focus F. Another ray from P moves straight through the optical centre O. These two refracted rays meet at the point p which is the image of P. From p, pq is drawn per-pendicular to the principal axis. So, pq is the image of PQ. This image is real and inverted.
As the triangles POQ and pOq are similar,
∴ \(\frac{P Q}{p q}\) = \(\frac{O Q}{O q}\) …. (1)
On the other hand, as the triangles AFO and pFq are similar,
∴ \(\frac{A O}{p q}\) = \(\frac{F O}{F q}\)
or \(\frac{P Q}{p q}\) = \(\frac{F O}{F q}\) [∵ AO = PQ] …… (2)
From equations (1) and (2) we get,
\(\frac{O Q}{O q}\) = \(\frac{F O}{F q}\)
or \(\frac{O Q}{O q}\) = \(\frac{F O}{O q-O F}\) …. (3)
Now, according to sign convention, object distance = OQ = -u, image distance = Oq = + v, focal length = OF = +f
Putting these values in equation (3) we get,
\(\frac{-u}{v}\) = \(\frac{f}{v-f}\) or, -uv+uf = vf
or, uf- vf = uv or, \(\frac{u f}{u v f}-\frac{v f}{u v f}\) = \(\frac{u v}{u v f}\)
or, \(\frac{1}{v}\) – \(\frac{1}{u}\) = \(\frac{1}{f}\) …. (4)
Convex lens and virtual image: The object PQ is placed perpendicular to the principal axis of the convex lens LL’ and is placed between the focus and the lens. So, the virtual image pq has been formed [Fig.].
As the triangles POQ and pOq are similar,
∴ \(\frac{P Q}{p q}\) = \(\frac{O Q}{O q}\) …. (5)
On the other hand, as the triangles AFO and pFq are similar,
∴ \(\frac{A O}{p q}\) = \(\frac{O F}{q F}\)
or \(\frac{P Q}{p q}\) = \(\frac{O F}{q F}\) [∵ AO = PQ] …. (6)
From equations (5) and (6) we get,
\(\frac{O Q}{O q}\) = \(\frac{O F}{q F}\) or, \(\frac{O Q}{O q}\) = \(\frac{O F}{O q+O F}\) ….. (7)
Now, according to sign convention, object distance = OQ = -u, image distance = Oq = -v, focal length = OF = +f
Putting these values in equation (7) we get,
\(\frac{-u}{-v}\) = \(\frac{f}{-v+f}\) or, uv – uf = -vf or, uf – vf = uv
or, \(\frac{1}{v}\) – \(\frac{1}{u}\) = \(\frac{1}{f}\) …. (8)
The focal length of a convex lens is taken as positive. In formation of a real image of a real object, u is negative but v is positive. So, for formation of a real image of a real obj ect by a convex lens the modified form of the general formula is as follows-
\(\frac{1}{v}\) – \(\frac{1}{-u}\) = \(\frac{1}{f}\) or, \(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\)
Concave lens and virtual image: in Fig., LL’ is a concave lens, An object PQ is placed perpendicular to the principal axis of the lens. A ray from P travelling parallel to the principal aids after refraction through the lens appears to diverge from the focus F. Another ray from P moves straight through the optical centre O. The two refracted rays virtually meet at p. The point p from where the emergent rays appear to diverge after refraction through the lens is the image of P. From p, pq is drawn perpendicular to the principal axis. So, pq is the image of PQ. The image is virtual and erect.
As the triangles POQ and pOq are similar,
∴ \(\frac{P Q}{p q}\) = \(\frac{O Q}{O q}\) …. (9)
On the other hand, as the triangles AFO and pFq are similar,
∴ \(\frac{A O}{p q}\) = \(\frac{O F}{q F}\)
or \(\frac{P Q}{p q}\) = \(\frac{O F}{q F}\) [∵ AO = PQ] ….. (10)
From equations (9) and (10) we get,
\(\frac{O Q}{o q}\) = \(\frac{O F}{q F}\)
or, \(\frac{O Q}{O q}\) = \(\frac{O F}{O F-O q}\) …. (11)
Now, according to sign convention, object distance = OQ image distance = Oq = -v, focal length = OF = -f
Putting these values in equation (11) we get,
\(\frac{-u}{-v}\) = \(\frac{-f}{-f+v}\) or, uf – uv = vf or, uf – vf = uv
or, \(\frac{u f}{u v f}-\frac{v f}{u v f}\) = \(\frac{u v}{u v f}\)
or, \(\frac{1}{v}-\frac{1}{u}\) = \(\frac{1}{f}\) …. (12)
This is the conjugate foci relation of the lens, also known as general formula of lens.
Definition: Two points on the axis of a lens are said to be conjugate foci when an object on being placed at any one of the two, forms its image at the other.
The term conjugate means that the two points are interchangeable. This follows from the principle of reversibility of light path. For these lenses the distances of conjugate foci i.e., u and v are given by the relation \(\frac{1}{v}\) – \(\frac{1}{u}\) = \(\frac{1}{f}\). So this relation is often called conjugate foci relation.