How Does Bessel's Method Determine Lens Focal Length?

[Perses]

I'm having a bit of trouble finding a place to start in one of my optics questions. I have to find Bessel's method for focal length of a lens. Any pointers on where to begin or a general direction, that would be greatly appreciated! Here's the question:

A thin lens is moved along the optical axis between a fixed object and a fixed image screen. The object and image positions are separated by a distance L that is more than four times the focal length of the lens. Two positions of the lens are found for which an image is in focus on the screen, magnified in one case and reduced in the other. If the two lens positions differ by distance D, show that the focal length of the lens is given by;

f = (L^2 - D^2)/4L.

 

[Perses]

any help anyone could offer would be appreciated, thnx

 

[quicknote]

Bessel's method is a technique used to determine the focal length of a lens by measuring the positions of the object and image when the lens is moved along the optical axis. In this case, the lens is placed between a fixed object and a fixed image screen, and the object and image positions are separated by a distance L that is more than four times the focal length of the lens.

To start, we can use the thin lens equation, 1/f = 1/o + 1/i, where f is the focal length, o is the object distance, and i is the image distance. We know that in the two positions of the lens, the image is in focus on the screen, and it is either magnified or reduced.

Let's call the two positions of the lens x1 and x2, where x1 is the position where the image is magnified and x2 is the position where the image is reduced. In these positions, the object and image distances are related by the following equations:

o = x1 + L and i = L - x1 (for x1)

o = L - x2 and i = x2 + L (for x2)

By substituting these values into the thin lens equation, we can get two equations:

1/f = 1/(x1 + L) + 1/(L - x1) and 1/f = 1/(L - x2) + 1/(x2 + L)

We can solve these equations for x1 and x2, respectively, and then subtract them to get:

x1 - x2 = (L^2 - D^2)/4L

Since we know that x1 - x2 = D, we can substitute this value and rearrange the equation to get:

f = (L^2 - D^2)/4L

This shows that the focal length of the lens can be determined using Bessel's method by measuring the distance between the two positions of the lens where the image is in focus, and using the known distance between the object and image on the screen. This method is useful for determining the focal length of a lens when other methods, such as the thin lens equation, cannot be used. I hope this helps in understanding Bessel's method for finding the focal length of a lens.