Object image and focal distance relationship (proof of formula) (video) | Khan Academy
A thin lens is defined as one with a thickness that allows rays to refract, The distance from the center of the lens to the focal point is the lens's focal . The thin lens equation quickly provides the relation between di, do, and the focal length f. .. to gather more light than the human eye could collect on its own, focus it 5, and. A lens with a focal length of two meters has a power of one-half diopter because Disregarding thickness, the power of a lens is determined by combining the. A lens is a transmissive optical device that focuses or disperses a light beam by means of Opticians tried to construct lenses of varying forms of curvature, wrongly assuming The distance from this point to the lens is also known as the focal length, though it is . This is the principle of the camera, and of the human eye.
So let's see if we can label them here. And then, just do a little bit of geometry and a little bit of algebra to figure out if there is an algebraic relationship right here.
Thickness of lens and its focal length
So the first number, the distance of the object-- that's this distance from here to here, or we could just label it here. Since this is already drawn for us, this is the distance of the object. This is the way we drew it. This was the parallel light ray.
But before it got refracted, it traveled the distance from the object to the actual lens. Now, the distance from the image to the lens, that's this right over here. This is how far this parallel light ray had to travel.
So this is the distance from the image to the lens. And then we have the focal distance, the focal length. And that's just this distance right here. This right here is our focal length. Or, we could view it on this side as well. This right here is also our focal length. So I want to come up with some relationship.
And to do that, I'm going to draw some triangles here. So what we can do is-- and the whole strategy-- I'm going to keep looking for similar triangles, and then try to see if I can find relationship, or ratios, that relate these three things to each other. So let me find some similar triangles. So the best thing I could think of to do is let me redraw this triangle over here. Let me just flip it over. Let me just draw the same triangle on the right-hand side of this diagram. So if I were to draw the same triangle, it would look like this.
And let me just be clear, this is this triangle right over here. I just flipped it over. And so if we want to make sure we're keeping track of the same sides, if this length right here is d sub 0, or d naught sometimes we could call it, or d0, whatever you want to call it, then this length up here is also going to be d0. And the reason why I want to do that is because now we can do something interesting.
We can relate this triangle up here to this triangle down here. And actually, we can see that they're going to be similar. And then we can get some ratios of sides. And then what we're going to do is try to show that this triangle over here is similar to this triangle over here, get a couple of more ratios.
And then we might be able to relate all of these things. So the first thing we have to prove to ourselves is that those triangles really are similar.
So the first thing to realize, this angle right here is definitely the same thing as that angle right over there.
Lens (optics) - Wikipedia
They're sometimes called opposite angles or vertical angles. They're on the opposite side of lines that are intersecting.Nominal Lens Formula Part 3 - Lens Curves and Thickness
So they're going to be equal. Now, the next thing-- and this comes out of the fact that both of these lines-- this line is parallel to that line right over there. And I guess you could call it alternate interior angles, if you look at the angles game, or the parallel lines or the transversal of parallel lines from geometry.
We know that this angle, since they're alternate interior angles, this angle is going to be the same value as this angle. You could view this line right here as a transversal of two parallel lines.
These are alternate interior angles, so they will be the same. Now, we can make that exact same argument for this angle and this angle. And so what we see is this triangle up here has the same three angles as this triangle down here.
So these two triangles are similar. These are both-- Is really more of a review of geometry than optics. These are similar triangles. Similar-- I don't have to write triangles. And because they're similar, the ratios of corresponding sides are going to be the same.
So d0 corresponds to this. They're both opposite this pink angle. They're both opposite that pink angle. So the ratio of d0 to d let me write this over here. So the ratio of d0. Let me write this a little bit neater. The ratio of d0 to d1. So this is the ratio of corresponding sides-- is going to be the same thing. And let me make some labels here. That's going to be the same thing as the ratio of this side right over here.
This side right over here, I'll call that A. It's opposite this magenta angle right over here. That's going to be the same thing as the ratio of that side to this side over here, to side B. And once again, we can keep track of it because side B is opposite the magenta angle on this bottom triangle. So that's how we know that this side, it's corresponding side in the other similar triangle is that one. They're both opposite the magenta angles. We've been able to relate these two things to these kind of two arbitrarily lengths.
But we need to somehow connect those to the focal length. And to connect them to a focal length, what we might want to do is relate A and B. A sits on the same triangle as the focal length right over here. So let's look at this triangle right over here. Let me put in a better color. So let's look at this triangle right over here that I'm highlighting in green.
This triangle in green. And let's look at that in comparison to this triangle that I'm also highlighting. This triangle that I'm also highlighting in green. Now, the first thing I want to show you is that these are also similar triangles. This angle right over here and this angle are going to be the same. They are opposite angles of intersecting lines. And then, we can make a similar argument-- alternate interior angles.
Well, there's a couple arguments we could make. One, you can see that this is a right angle right over here. This is a right angle. If two angles of two triangles are the same, the third angle also has to be the same. So we could also say that this thing-- let me do this in another color because I don't want to be repetitive too much with the colors.
We can say that this thing is going to be the same thing as this thing. The lentil plant also gives its name to a geometric figure.
BBC - GCSE Bitesize Science - The eye : Revision, Page 5
Ptolemy 2nd century wrote a book on Opticswhich however survives only in the Latin translation of an incomplete and very poor Arabic translation. The book was, however, received, by medieval scholars in the Islamic world, and commented upon by Ibn Sahl 10th centurywho was in turn improved upon by Alhazen Book of Optics11th century. The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century Eugenius of Palermo Between the 11th and 13th century " reading stones " were invented.
These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval 11th or 12th century rock cystal Visby lenses may or may not have been intended for use as burning glasses. History of the telescope With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses.
Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces.
This led to the invention of the compound achromatic lens by Chester Moore Hall in England inan invention also claimed by fellow Englishman John Dollond in a patent. This section needs expansion with: You can help by adding to it. January Construction of simple lenses[ edit ] Most lenses are spherical lenses: Each surface can be convex bulging outwards from the lensconcave depressed into the lensor planar flat.
The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through the physical centre of the lens.
Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different focal power in different meridians. This forms an astigmatic lens.
An example is eyeglass lenses that are used to correct astigmatism in someone's eye. More complex are aspheric lenses.