# Deriving the lens formula for a convex lens-say

• ananthu
In summary, the sign convention used in the lens equation has an alternative where distances measured in the opposite direction of the incident ray are considered negative.

#### ananthu

I will be thankful if the following points are clarified.

1. While deriving the lens formula for a convex lens-say, 1/f= 1/v - 1/u where u and v are the object and image distances, the minus sign is obtained after applying sign conventions. But, I don't understand the logic behind taking the value of u again with a minus sign and substituting in the formula while solving problems.For example, if the object distance is given as 45 cm and the image distance as 90 cm we again apply the sign convention, take u as -45 cm and substitute in the above formula and calculate f as 30 cm. In fact, while doing so the above formula becomes 1/f = 1/v + 1/u. What is the meaning behind applying negative sign in a formula which is itself obtained by applying the negative sign already?

2. Again, take the convex lens formula, 1/f = (n-1) (1/R1 -1/R2) where n is the refractive index and R1 and R2 are radii of the two faces of the lens. For a biconvex lens, assuming that R1 = R2, and n= 1.5 roughly for glass,then substituting these values in the above formula, we get the value of f as infinity, which is really absurd. Can anyone throw light on the above points?

1. The formula most people, and most textbooks, use is
1/f = 1/v + 1/u​
with v and/or u positive for a real image or object, and negative for virtual image or object. For the typical case where object and image are both real, both are positive as is f. In your example this becomes simply
1/30 = 1/90 + 1/45​

2. For a biconvex lens, R2 would be negative as it bulges in the opposite direction as R1 -- this is the common sign convention for surface radii. So in your example,
(1.5-1) (1/R1 - 1/(-R1)) = 0.5 * (1/R1 + 1/R1) = 0.5 * 2/R1 = 1/R1​
If you wanted to change the convention and say R is positive for any convex face, then the formula would have (1/R1)+(1/R2) instead.

https://www.physicsforums.com/library.php?do=view_item&itemid=148

Scroll down to the "Extended explanation" section to read about sign conventions.

Redbelly98 said:
1. The formula most people, and most textbooks, use is
1/f = 1/v + 1/u​
with v and/or u positive for a real image or object, and negative for virtual image or object. For the typical case where object and image are both real, both are positive as is f. In your example this becomes simply
1/30 = 1/90 + 1/45​

2. For a biconvex lens, R2 would be negative as it bulges in the opposite direction as R1 -- this is the common sign convention for surface radii. So in your example,
(1.5-1) (1/R1 - 1/(-R1)) = 0.5 * (1/R1 + 1/R1) = 0.5 * 2/R1 = 1/R1​
If you wanted to change the convention and say R is positive for any convex face, then the formula would have (1/R1)+(1/R2) instead.

The above formula was obtained by considering the distances measured in the opposite direction of the incident ray as negative.
So,again my basic question remain unanswered. "Why should we apply the sign convention two times?"
If you apply the other convention - that is distances for real object and real images as positive, we don't face any problem.

If I understand you and your example from Post #1, then:

u = -45 cm
v = 90 cm
f = 30 cm​

Is it true that 1/f = 1/v - 1/u?

1/30 = 1/90 - 1/(-45) ?
1/30 = 1/90 - (-1/45) ?
0.0333... = 0.0111... - (-0.0222...) ?
0.0333... = 0.0111... + 0.0222... ?
0.0333... = 0.0333... ?​

Yes, it does.

Redbelly98 said:
If I understand you and your example from Post #1, then:

u = -45 cm
v = 90 cm
f = 30 cm​

Is it true that 1/f = 1/v - 1/u?

1/30 = 1/90 - 1/(-45) ?
1/30 = 1/90 - (-1/45) ?
0.0333... = 0.0111... - (-0.0222...) ?
0.0333... = 0.0111... + 0.0222... ?
0.0333... = 0.0333... ?​

Yes, it does.

Thank you Redbelly, for your sincere efforts. My doubt is not about solving the above problem. Any way, if I have confused you, I am sorry.

this one had me puzzled too. it appears that the sign convention used in the lens equation has an alternative to the one I learned many years ago.

http://www.practicalphysics.org/go/Guidance_122.html;jsessionid=alZLdQlAHb1?topic_id=2&guidance_id=1 [Broken] for details.

Where it might get fun is that I have no idea how widespread the change is.

====

On a personal note, as much as adding "curvature" seems like a nice approach I don't care for it as it sacrifices the symetry inherent in optics where the light path can be reversed and nothing changes beyond the trivial exchange of u and v.

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## 1. What is the lens formula for a convex lens?

The lens formula for a convex lens is 1/f = 1/v - 1/u, where f is the focal length, v is the distance of the image from the lens, and u is the distance of the object from the lens.

## 2. How is the lens formula derived for a convex lens?

The lens formula for a convex lens is derived using the thin lens equation, which states that 1/f = 1/v + 1/u. This equation is based on the principles of refraction and can be derived using the laws of refraction and the geometry of a convex lens.

## 3. Why is the lens formula important for a convex lens?

The lens formula is important because it allows us to calculate the position and size of an image formed by a convex lens, as well as the magnification of the image. It is also used in the design and construction of optical instruments such as cameras and telescopes.

## 4. Can the lens formula be used for other types of lenses?

Yes, the lens formula can be used for any type of thin lens, including concave lenses. However, the sign convention for distances (positive for real objects and images, negative for virtual objects and images) may differ depending on the type of lens being used.

## 5. What are some common applications of the lens formula for a convex lens?

The lens formula for a convex lens is used in many practical applications, including eyeglasses, microscopes, projectors, and magnifying glasses. It is also essential in the study of optics and the behavior of light in different mediums.