# Microscopes, Telescopes and Visual Angle

by Jared Rovny

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00:01 Now let’s talk about a few instruments that we can use these optics inside of. Let’s start with microscopes. If we have a very small object which I’ve drawn in a small horizontal green arrow here and we want to magnify that object, we could use a convex lens again and put that object inside of the focal length of the convex lens. Well, what happens is exactly what we described using our normal ray tracing methods with the lenses. One of the rays, we could draw as going away from the near focus and leaving parallel as it goes vertically. The other ray, we could start going parallel vertically and would head straight towards the focal length as it leaves on the other side of the lens. Tracing from our eye these two rays back to their source, we would find that the source would be a much bigger image on the other side of the lens. So, the image that you’re looking at when you look through a microscope is actually a magnified and a virtual image of the object like some small cell that you’re looking at. Telescopes are a different sort of optical instrument that is slightly more complicated. We don’t have to go into all the different mechanisms of a telescope and how it works but there are a few ideas behind the workings of a telescope that we should be familiar with. A basic telescope, and there are by the way many, many kinds of telescopes so be aware of that, but a basic telescope like this one would use two lenses to increase the size of an object. So, if we have a one object that’s very small, in fact, at a distance in terms of its appearance, we can magnify it through two lenses using the same ray tracing formulas that we’ve already discussed. The idea behind these two lenses is that one of the lenses would have a very great focal length. The other one would have a very small focal length.

01:47 What we do is we arrange the two lenses so that those two focal lengths, the focal points, land right on top of each other. That’s in the red dot here. If these two focal points are right on top of each other then the light that’s coming from the first lens will bend, go through their focal point for the second lens, and then it will be one of our normal rays passing through the focal point and leaving parallel to the second lens. For a telescope, we can also talk about the visual angle.

02:15 It’s hard to describe the size of an object you’re looking at through a telescope. So for example, if you’re looking at a star, it’s not something like a cell where you can just measure the size and think of it as being very close to your eye and figure out everything about the size of the image of the cell. Instead, we’re looking at very big objects that are very, very, very far away.

02:34 In this case, since we have things that are essentially at an infinite distance for the purpose of rays, all these rays will be coming in parallel to each other. We, instead, talk about the visual angle and this is how we measure the size of these objects that are at a far distance. The way a visual angle works is we think about the object in front of us as taking up some angle of our vision as we look at that object. So, if I looked at an object like we see here, I could see this red object, the one closer to my eye, takes up a much greater visual angle in my field of view and so appears to be a much bigger object. So be careful when we’re talking about objects and images that are created by lenses. This green object, as the light comes in, passes through both of our lenses might create an inverted object very close to our eye that might appear on paper as we can see in front of this little eye here to be a smaller arrow. But this small arrow is taking up a very big angle in our vision and is in fact taking up a big space of our vision making this star in this case appear much bigger. We can measure the magnification in terms of these angles by saying that the original angle of the object was maybe θ1 and that the new angle of the object is θ2. So, we can have a magnification for this object which would be the ratio of the two angles. This works best for small angles.

The lecture Microscopes, Telescopes and Visual Angle by Jared Rovny is from the course Geometrical Optics.

### Included Quiz Questions

1. Inside the focal length of the lens
2. Outside the focal length of the lens
3. Between the focal point and the center of the circle which the curvature of the lens is part of
4. Outside the center of the circle which the curvature of the lens is part of
5. A converging lens always creates a magnified image
2. A star very far away
3. The sun in the sky
4. A large boat on the horizon
5. The words on the page of a book you are reading
1. The angular size of the image is twice as great
2. The angular size of the original object is twice as great
3. The angular size of the image the square of the angular size of the object
4. The angular size of the object is the square of the angular size of the image
5. The angular size of the image equals the angular size of the object

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