Combined Effect of Multiple Lenses: Find Focus

[indigojoker]

So there are two lenses. The first lens is a converging lens and the second is a diverging lens with focal length f,-f respectively, separated by a distance D. The question asks to show the combined effect of the lenses is focusing however, I am not getting them to be always focusing.

using: $$\frac{1}{o}+\frac{1}{i}=\frac{1}{f}$$ we see that an object placed at infinity gives i=f

Moving onto the second lens, the object is located at D-f so we get:

$$\frac{1}{D-f}+\frac{1}{i}=-\frac{1}{f}$$
$$-\frac{1}{D-f}-\frac{1}{f}=\frac{1}{i}$$
$$-(\frac{1}{D-f}+\frac{1}{f})=\frac{1}{i}$$
$$-(\frac{f+D-f}{f(D-f)})=\frac{1}{i}$$
$$-(\frac{f+D-f}{f(D-f)})=\frac{1}{i}$$
$$\frac{D}{f^2-fD}=\frac{1}{i}$$
$$\frac{f^2-fD}{D}=i$$

we see that if D>f then the image is negative, telling use that there isn't convergence

any ideas?

 

[anathema]

I would approach this problem by first reviewing the basic principles of optics and lens behavior. The equation \frac{1}{o}+\frac{1}{i}=\frac{1}{f} is known as the lens equation and it describes the relationship between the object distance (o), image distance (i), and focal length (f) of a single lens. This equation assumes that the lens is thin and has a small aperture, and it is valid for both converging and diverging lenses.

In this situation, we have two lenses in close proximity to each other. When light passes through the first converging lens, it forms a real image at a distance f from the lens. This image then becomes the object for the second diverging lens. The key here is to remember that the image formed by the first lens is a real image, meaning the light rays actually converge at that point.

Now, let's consider the second lens. The object for this lens is located at D-f, which is the distance between the two lenses minus the focal length of the first lens. Using the same lens equation, we can calculate the image distance for the second lens. However, we need to be careful with the sign convention. A positive image distance indicates a real image, while a negative image distance indicates a virtual image.

If we plug in the values for the object distance and focal length of the second lens, we get:

\frac{1}{D-f}+\frac{1}{i}=\frac{1}{-f}

Simplifying this equation, we get:

\frac{1}{D-f}=\frac{1}{i}+\frac{1}{f}

This shows that the image formed by the second lens is a virtual image, as the right side of the equation is positive.

So, what does this mean for the combined effect of the two lenses? It means that the final image is formed by the second lens as a virtual image, but it is located at a distance i from the second lens. This image is then located at a distance D-i from the first lens.

To determine if this final image is a real or virtual image, we need to consider the sign convention again. The final image is real if D-i is positive, which means that D>i. This means that the distance between the two lenses must be greater than the image distance formed by the second lens. If this condition

 

[ogb p]

I would suggest that the combined effect of multiple lenses is not always focusing because it depends on the placement of the lenses and the characteristics of the objects being focused. In this specific scenario, we can see that the first lens is a converging lens, meaning it brings parallel light rays to a focus at a point, while the second lens is a diverging lens, meaning it spreads out parallel light rays.

When the object is placed at infinity, the first lens will bring the light rays to a focus at a point, but the second lens will then diverge those rays again. This means that the combined effect of the two lenses is not focusing, but rather spreading out the light rays. This is why we get a negative image when using the equation \frac{1}{D-f}+\frac{1}{i}=-\frac{1}{f}.

On the other hand, if the object is placed at a distance less than D-f, the second lens will act as a converging lens and bring the light rays to a focus, resulting in a positive image. So, the combined effect of the lenses can be focusing in certain scenarios, but it is not always the case.

It is important to consider the placement and characteristics of the lenses and objects when studying the combined effect of multiple lenses. This is why it is crucial for scientists to conduct experiments and gather data to fully understand the behavior and limitations of lenses in different situations.