An Introduction to Spherical Lens
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. The choices made may differ between authors. Disagreement about sign conventions is a frequent source of confusion, misunderstandings, and even outright errors in scientific work.
In general, a sign convention is a special case of a choice of coordinate system for the case of one dimension. Sometimes, the term "sign convention" is used more broadly to include factors of i and 2π, rather than just choices of sign.
The Spherical Lens
Sign Convention for Concave and Convex Lens
Sign Convention for Lens
The picture of the convex and concave lens shows that the lens has an optical centre at O, two foci F1 and F2, and two centres of curvature of the lens 2F1 and 2F2. The primary axis is a line that runs through the centre of curvature, foci, and optical centre. The optical centre is the place where the primary axis meets the lens. Some more important points to note about Sign Convention of spherical lenses:
Since the object is always in front of the lens, object distance is always negative.
If a real image is formed, the image is formed on the right side, so image distance is positive. If a virtual image is formed, the image is formed on the left side, so the image distance is negative.
Since the object is always above the principal axis, the object height will be positive. If the image is above the principal axis, the image height will be positive. It means that the image is erect . If the image is below the principal axis, image height will be negative. It means that image is inverted
Sign Convention and Lens Formula
The same sign convention is used for lenses as it is for spherical mirrors. We use the same criteria for distance signs, except that all measurements are measured from the optical centre of the lens. According to tradition, the focal length of a convex lens is always positive, while that of a concave lens is always negative. You must use suitable signs for the values \[u,\text{ }v,\text{ }f,\]object height \[h,\]and image height \[h'.\]
Sign Convention
The Sign Conventions
Cartesian sign conventions are used to quantify the different distances in the ray diagrams of spherical lenses.
The object is always positioned on the lens's left side, according to the Cartesian sign convention.
All distances are measured from the optical centre of the lens.
Positive distances are measured in the direction of the incident light.
Negative distances are those measured in the direction opposite the incident light.
Positive distances are those that are measured upward and perpendicular to the primary axis.
Distances measured downhill and perpendicular to the primary axis are awarded negative values.
The Lens Formula
We have a formula for spherical lenses, just as we do for spherical mirrors. This formula gives the relationship between object distance\[(u)\], image distance, \[(v)\] and focal length\[(f)\]. The lens formula is expressed as:
\[\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\]
The lens formula given above is general and is valid in all situations for any spherical lens. Take proper care of the signs of different quantities while putting numerical values for solving problems relating to lenses.
Power of Lens and Magnification
The power of a lens is the measure of the degree of convergence or divergence of the light rays falling on it. The degree of convergence or divergence depends upon the focal length of the lens. Thus, we define the power of the lens as the reciprocal of the focal length of the lens used. It is given as
${\frac{{1}}{{f}}}$
The magnification of a lens is defined as the ratio of the height of an image to the height of an object. It is also given in terms of image distance and object distance. It is equal to the ratio of image distance to that of object distance.
$m = {\frac{{h}_{‘}}{{h}}} ={\frac{{u}}{{v}}}$
Where
m= magnification
h’= height of the image
h = height of an object
v is the image distance
u is the object distance
Convention Table for Lens Signs
Solved Examples
1. A plano-convex lens with a focal length of 30 cm has had its flat surface silvered. An object is placed 40 cm from the lens on the convex side. How far away is the image?
Ans: We know that, from lens formula:
${\frac{{1}}{{v}}}- {\frac{{1}}{{u}}} = {\frac{{1}}{{f}}}$
${\frac{{1}}{{v}}} = {\frac{{1}}{{30}}} - {\frac{{1}}{{40}}}$
${\frac{{1}}{{v}}} = 4 - {\frac{{3}}{{120}}}$
v = +120 cm
So now, this image will act as an object for the lens.
As a result, u = +120 cm and f = 30 cm
${\frac{{1}}{{v}}} - {\frac{{1}}{{u}}} = 1 - f$
${\frac{{1}}{{v}} {=} \frac{{1}}{{30}} {+} \frac{{1}}{{120}}}$ cm
v = +24 cm
2. A point object O is 30 cm away from a convex lens (focal length 20 cm) that has been split into two pieces, each displaced by 0.05 cm perpendicular to the main axis. What is the distance between the two images?
Ans: We will assume that each part as a separate lens with:
u = -30 cm and f = 20 cm.
Now, we know that,
${\frac{{1}}{{v}}}- {\frac{{1}}{{u}}} ={\frac{{1}}{{f}}}$
${\frac{{1}}{{v}}}- (-{\frac{{1}}{{30}}}) = -120$
v = 60 cm
Therefore, each part will form a real image of the point object O at 60 cm from the lens.
Now, from magnification formula we have:
$m = {\frac{{v}}{{u}}} = {\frac{{h}_{i}}{{h}_{o}}}$
${\frac{{60}}{{30}}} = {\frac{{h}_{i}}{{0.05}}}$
${{h}_{i}}$ = 0.1 cm
Thus, the first image will be 0.1 cm above the first half of the lens (say L1) optic axis and the second image will be 0.1 cm above the second half of the lens (say L2) optic axis. Also, separation between the two lenses = 0.05+0.05= 0.1 cm
We can say that
Distance between the two images = 0.1 + 0.1+0.1 =0.3 cm
Summary
The sign convention in lenses is vital for picture quality and formation. This article has provided all of the pertinent information on the subject. The sign convention is required to determine the orientation of the picture. In addition, the article informs us about the working principles and properties of various lenses. The behaviour of light as it passes through the convex and concave lenses differs.
Convex lenses behave like converging lenses because they converge the light and form images, whereas concave lenses behave like diverging lenses because they diverge the incoming light and produce images. We also learnt about the lens sign convention table and had a firm grasp on the magnification sign convention.
FAQs on Sign Convention for Spherical Lens
1. What is the official sign convention for measuring distances in spherical lenses as per the CBSE 2025–26 syllabus?
For spherical lenses, all distances are measured from the optical centre of the lens.
- Distances measured in the direction of incident light (to the right) are considered positive.
- Distances measured against the direction of incident light (to the left) are negative.
- The object distance (u) is always negative as the object is usually placed on the left of the lens.
- The focal length (f) is positive for a convex lens and negative for a concave lens.
2. How do you determine the sign of the image distance and magnification for images formed by spherical lenses?
The sign of image distance (v) depends on the image location:
- If the image is real, it forms on the right side of the lens; image distance is positive.
- If the image is virtual, it forms on the left side of the lens; image distance is negative.
- Magnification (m) is positive for upright (virtual) images and negative for inverted (real) images.
3. Why is the object distance always taken as negative in lens sign conventions?
The object distance is always negative because, by convention, the object is placed to the left of the lens, opposite to the direction of incident light. According to the CBSE sign convention, distances to the left of the optical centre are considered negative.
4. What is the difference in sign convention between concave and convex lenses?
The key difference is in the focal length:
- For a convex (converging) lens, focal length (f) is positive.
- For a concave (diverging) lens, focal length (f) is negative.
- The sign convention for object and image distances applies equally to both types; only the focal length sign changes based on lens type.
5. Can a convex lens form a real and inverted image of the same size as the object? If yes, under what condition?
Yes, a convex lens forms a real, inverted image of the same size as the object when the object is placed at a distance equal to twice the focal length (u = –2f) from the optical centre of the lens. The image forms at 2f on the other side.
6. How does the lens formula incorporate sign conventions when solving problems?
The lens formula is 1/v – 1/u = 1/f. While applying this formula:
- All values of u, v, and f must be substituted with the correct signs based on the lens sign convention.
- Failing to use the proper signs leads to incorrect results for image position and nature.
7. What are common errors students make with sign conventions in spherical lens questions?
Common mistakes include:
- Forgetting to assign a negative sign to object distance (u).
- Assigning a wrong sign to the focal length (mixing up convex and concave).
- Calculating image distance (v) without considering the image’s side relative to the optical centre.
- Not applying sign convention when determining magnification.
8. Why is understanding sign convention important for solving ray diagrams and calculation-based questions?
Proper use of sign convention ensures accuracy in locating images, calculating magnification, and interpreting the nature of images (real or virtual, erect or inverted). Errors in sign convention can lead to incorrect answers, which is why exam questions often test this conceptual understanding.
9. What real-world applications depend on accurate use of lens sign conventions?
Applications such as microscopes, telescopes, cameras, eyeglasses, and magnifying glasses rely on precise calculations involving lens sign conventions to work effectively. Incorrect sign usage would result in faulty image formation in such devices.
10. If a virtual image is formed by a spherical lens, what would be the sign of image distance and why?
A virtual image is always formed on the same side as the object (left of the optical centre). Therefore, the image distance (v) is negative, reflecting its position according to the sign convention.
