homework-and-exerciseslensesoptics
I am having a problem drawing optical ray lines for optical systems. Namely, one which have a formation of image of the first lens beyond the second lens.
Take the picture below as an example:
In this picture, the image from the first (convex) lens focuses at 80 cm, which is beyond the position of the second (concave) lens.
I don't think I have any problem calculating the image distance, etc., but I don't know how to draw the ray lines, similar to the picture below:
Thank you in advance!
The term "focusing" means something else than the OP suggests. It means that the different light rays coming that a single, specific point $P$ of the object emits to different directions re-converge back and reach the same place of the retina.
If the object, and/or its point $P$, is infinitely (very) far, then the light rays coming from $P$ are (nearly) parallel near the eye. But even if they're divergent, not parallel, the eye is able to refocus them so that they reach the same pixel of the retina.
So the single convex lens don't magnify anything. At most, they do exactly what the lens in the eyes do and what is needed for focusing – to redirect the nearly parallel light rays so that they intersect against less than an inch from the eye's surface, on the retina, again.
The diagram posted above is misleading because it suggests that the two parallel rays that are supposed to converge to the same point of the retina come from different ends of the objects we observe, $P_1$ and $P_2$. But that's not the case at all. If the diagram is fixed so that it makes sense, we see just one point $P$ and both (nearly parallel) light rays originate from the same $P$.
If we want to consider two points $P_1$, $P_2$ of the object, as the bottom part of the picture clearly wants, they must create two distinct dots $Q_1,Q_2$ on the retina, i.e. two different intersections of pairs of light rays!
So again, a single convex lens doesn't magnify anything. It just does what the eye has to do to focus, anyway: to make the light rays converge. To calculate whether an arrangement of lens (and yes, at least two lens or lens+mirrors are needed) are able to magnify, one has to consider the size of the image on the retina, i.e. different intersections of the light rays on the retina, separated from each other. To approximate the eye by a single dot isn't good enough to calculate the magnification!
If the concave lens was not there and the object was at a distance greater that its focal length the convex lens would form a real image.
Introducing a concave lens results in the incoming rays from the convex lens being refracted as shown in the diagram above.
The refracted rays are still convergent and so form an image $I$ at a distance $q_2$, your $d_v$, from the the concave lens.
After passing through the convex lens where the rays would have met is at a distance $p_2$, your $d_R$, from the concave lens and that can be thought of as a virtual object for the concave lens.
Best Answer
Ray lines can always be drawn using Snell's law.
But If you want to find the image only by basic properties(lines parallel to principle axis and passing through the optical center) of lenses then find the image for the first convex lens. Now if this image happens to be at the left of the second concave lens, you can take it again as an object for the second concave lens and can find the final image easily.
But in your case the image due to convex lens would be formed at 80 cm(>d=60cm) if the concave lens would not be there.But it would not still be problem. Just draw the first image as if the concave lens is not present there. Remember though you have found the image point by drawing only two lines, actual image will have all the possible lines between the object point and image point through the convex lens. Now out of these lines draw a parallel line with the principal axis from the image point and a line passing through the centre of the concave lens from the image point. Now these are the two lines you were searching for to make the final image.
Note:I am sure enough I don't need to say you where the dotted and solid lines should be.