What is the correct sign convention for the lens and mirror formula?

[rishch]

So we are learning the lens formula and I have two textbooks, my school textbook and another one that is much more detailed. My school textbook gives the lens formula as:-

1/f=1/v-1/u

while the other one gives

1/f=1/v+1/u (v is image distance and u is object distance)

There both different! I looked them some more and the problem is that my school textbook follows some "New Cartesian" sign convention. Imagine the optical center as the origin and the principal axis as the x-axis and the vertical line down the mirror as the y axis. So left is negative, right is positive, up is positive and down is negative. Just like a graph. The other one does the same except that even if the object is on the left they take it as positive.

I went to Khan Academy and he had a proof and he came up with the second one, different from the one in my school textbook. I tried doing the proofs on my own but I don't know how to include negative distances in geometry. Do you take the lengths as negative? Then you can't use similarity of triangles because some sides have negative length and others, positive. How do you work with negative distances? Getting really confused :/

 

[Simon Bridge]

The differences are due to different sign conventions all right.
I've never been able to keep them straight and I always rely on a quick sketch of the situation as a reality check.

Note: Triangles are similar if all the angles are the same ... with negative distances, you get a negative scale factor.

 

[Simon Bridge]

The differences are due to different sign conventions all right.
I've never been able to keep them straight and I always rely on a quick sketch of the situation as a reality check.

Note: Triangles are similar if all the angles are the same ... with negative distances, you get a negative scale factor.

 

[rishch]

Actually I find the cartesian plane sign convention really good as it's easy to remember if you just thing of it like a graph or number line. Left is negative and right is positive. Yes but during proofs one pair of corresponding sides may have a negative scale factor and one may have a positive scale factor. I managed to come up with an alternative neat solution to the problem after a bit of thinking though.

 

[sardar]

The correct sign convention for the lens and mirror formula is based on the Cartesian coordinate system, where the optical center is the origin and the principal axis is the x-axis. In this convention, distances to the left of the origin are considered negative, while distances to the right are positive. Similarly, distances above the x-axis are positive and distances below are negative.

In terms of the lens and mirror formula, this means that the object distance (u) is negative if the object is located to the left of the origin, and positive if it is to the right. The image distance (v) is also negative if the image is formed to the left of the origin, and positive if it is formed to the right.

Therefore, the correct lens formula should be 1/f = 1/v + 1/u, where the negative signs are taken into account. This is the same formula that is used in the other textbook that you mentioned and the one used by Khan Academy.

To understand how negative distances work in geometry, you can think of them as being just like positive distances, but in the opposite direction. So a negative distance of -5 units would be the same as a positive distance of 5 units, but in the opposite direction on the number line.

In terms of working with negative distances in the lens and mirror formula, you can use the absolute value of the distances to calculate the magnification and other properties of the image. The negative sign is important in determining the direction of the image, but the absolute value is used to calculate its size and position.

In conclusion, the correct sign convention for the lens and mirror formula is to take distances to the left and below the origin as negative, and distances to the right and above as positive. This convention is used to accurately calculate the properties of images formed by lenses and mirrors.