Plano-Convex lens cut in two halves

[zorro]

Homework Statement

A thin plano-convex lens of focal length f is split into two equal halves. One of the halves is shifted along the optical axis as shown. The separation etween the object and the image planes is 1.8m and the magnification of the image formed by one of the half lens is 2. The separation between the two halves is d. Find out f and d.

The Attempt at a Solution

Let the distance between the first half and the object be x.

This half will form an image at a distance given by (by using lens equation)
v= fx/f+x
This will act as an object for the second lens with distance v+d and the image is formed at a distance v1 = f(v+d)/f+v+d
we have x+d+v1 = 1.8

3 equations and 5 variables.
I am stuck.

 

[hikaru1221]

Geometry Optics! Eek, you ask about the field that I suck at

The question is not clear enough; I guess all the final images are at the same position, at a distance 1.8m from the object. There are 2 cases:
1/ Light from object --> first half --> straight to the image plane.
2/ Light from object --> first half --> second half --> image plane.
So that is one image. There is another image: light from object --> second half --> image plane.
From here, you can eliminate one case by noticing a hidden symmetry Hint: The two halves have the same focal length and there is symmetry between x and x' in the lens equation 1/x + 1/x' = 1/f. Also notice that the final images are at the same position.

Look for that symmetry and you will solve this problem in seconds

 

[zorro]

Haha...you don't suck at it. I can see that from your thought process

I did not get what is that symmetry.
For the first half, 1/x -1/x' = 1/f ( you missed a - sign in the equation )
For the second half, 1/(x'') - 1/(x+d) = 1/f

What is symmetric in this ?
I can observe from your cases that since the images are formed by both the lenses with similar focal length at the same point/plane, the object & its image can be interchanged. So this implies that the two lenses are equidistant from the object and the screen respectively.

 

[hikaru1221]

I did not get what is that symmetry.
For the first half, 1/x -1/x' = 1/f ( you missed a - sign in the equation )
For the second half, 1/(x'') - 1/(x+d) = 1/f

I didn't mean to apply 1/x + 1/x' = 1/f directly to this problem Let's interpret the formula in this way: if you place the object at position A and then obtain its image at position B, you should also obtain an image at position A if you place the object at position A.

A more general statement: if there is a path for light to go from point A to point B, light can also go from B to A through the same path. This is true for geometry optics.

And besides, x and x' in my formula can take negative value. This is why the formula only has + sign.

I can observe from your cases that since the images are formed by both the lenses with similar focal length at the same point/plane, the object & its image can be interchanged. So this implies that the two lenses are equidistant from the object and the screen respectively.

Yep, exactly
Now some picture The first one is my second case, the second picture is my first case. See something weird with M1 (the image of the object O through the first half)?

 

[zorro]

See something weird with M1 (the image of the object O through the first half)?

For the same object distance, the path of the ray from first half has to be either of those given in your figure.
I think that if the source and screen are interchanged, the path of the light from N (which will now become the source) should be the same as earlier with direction reversed but there will be one more possibility for the path of the light (directly through first half). In other words there is not only a change in the direction of the rays but also in the position of the rays. But this will violate laws of symmetry. So the first case if not possible.

Am I correct in my reasoning?

 

[hikaru1221]

Yep
Now that you eliminate the first picture. Proceed with the second one. See some symmetry?

 

[zorro]

I solved it and got f=0.4m and d=0.6m. Thank you very much for your help.