How Do Similar Triangles Derive the Thin Lens Equation?

[Larrytsai]

kk so i found out the steps involved deriving the thin lens equation, but what i don't get is when you have 2 similar triangles say triangle abd,edf are congruent how does the 2 triangles make it so its di/f-1. so what I am basically asking is can someone thourougly explain to me what's going on in the steps to deriving the thin lens equation thnx.

 

[mjsd]

Remember: the thin lens equations are only approximation to the real world. one way to get these formulas is to study refraction at spherical surfaces using Fermat's principle and small angle approximation, amongst other things. It is not a straight forward derviation. See for example: "Optics" by Hecht

 

[Moetasim]

The thin lens equation is a fundamental equation in optics that relates the focal length of a lens (f), the object distance (d), and the image distance (i). It is represented as 1/f = 1/d + 1/i.

To understand how this equation is derived, we need to first understand the concept of similar triangles. Similar triangles are triangles that have the same shape but are different in size. This means that their corresponding angles are equal, and their sides are in proportion to each other.

Now, let's look at the steps involved in deriving the thin lens equation. We start with two similar triangles, triangle ABD and triangle EDF, as shown in the image below.

[Insert image of similar triangles]

We know that the focal length of a lens is defined as the distance from the lens to the point where the light rays converge. In our case, this is point F. So, the distance from point A to point F is the focal length (f).

The distance from the object to the lens is represented by d, and the distance from the lens to the image is represented by i. Using the concept of similar triangles, we can write the following proportions:

AD/DF = AB/DE

And,

AF/DF = d/DE

We can rearrange the first proportion to get:

AD = (AB/DE) * DF

Similarly, we can rearrange the second proportion to get:

AF = (d/DE) * DF

Now, if we subtract these two equations, we get:

AD - AF = [(AB/DE) - (d/DE)] * DF

We can simplify this to:

DF = [(AB-d)/DE] * DF

Next, we divide both sides by DF to get:

1 = (AB-d)/DE

We can rewrite this as:

DE = (AB-d)

This means that the distance from the lens to the image (DE) is equal to the difference between the distance from the object to the lens (AB) and the distance from the lens to the image (d).

Finally, we can substitute this value for DE in our original proportion:

AF/DF = d/DE

To get:

AF/DF = d/(AB-d)

We can rearrange this to get:

AF = d * DF/(AB-d)

Since we know that DF = f, we can write this as:

AF = d * f/(AB-d)

And since AF