Image Formation in Galilei Binoculars: Solving for the Image Position

[Mulz]

Homework Statement

A galileibinocular consists of two lenses with ƒ_objective = +30 cm and ƒ_okular = -10 cm. The distance between the lenses is 30 cm and the binocular is aimed at an object 90 cm away from the objective lens.

Where is the image of the object seen through the binoculars? (the binoculars are not in their normal position)

Homework Equations

1/a₁ + 1/b₁ = 1/ƒ_objective
1/a₂ + 1/b₂ = 1/ƒ_okular
L = ƒ_objective + ƒ_okular

The Attempt at a Solution

I started by:
[/B]
1/(90 cm) + 1/b₁ = 1/(30 cm) which gave me b₁ = 45 cm that is the image that the objective generates. Now I need to do the same for the okular but I don't know how to get a₂.

 

[stevendaryl]

It's easiest to understand when all the distances are positive. Then the situation looks like this:

The object is on the right, a distance $a_1$ from the the right lens. It produces an image on the left side of the right lens at a distance of $b_1$ from the lens. Then this image is a distance $a_2$ from the left lens. it produces an image on the left side of the left lens at a distance $b_2$ from the lens. The distance between the lenses is $a_2 + b_1$.

The same general idea applies even when some of the parameters $(a_1, b_1, f_1, a_2, b_2, f_2)$ are negative. It's just that

1. If $b_2$ is negative, that means that the first image is a virtual image, on the right side of the right lens.
2. If $a_2$ is negative, that means that the first image serves as a virtual object of the left lens.
3. If $b_2$ is negative, that means that the final image is a virtual image, on the right side of the left lens.

In a telescope or microscope or binoculars, you actually want $b_2$ to be negative, because you want to put your eye close to the left lens and see the image through the lens, on the right side of the lens.

 

[Merlin3189]

I agree with Steve*, but would draw it with the correct lenses and relative positions.

If b1 is 45 cm and the separation of the lenses is 30 cm, then you know the size of a2. The diagram will show what sort of object it is for the eyepiece lens.

* Edit: though IMO his ray direction is unconventional.

 

[Mulz]

I solved it, thank you both.