Simple microscope

let's explore how a magnifying glass which is nothing but a convex lens helps us magnify things so let's explore this by taking an example suppose we wanted to examine the details of this letter a there are two ways to make it look bigger one we actually make the letter bigger somehow maybe order a jumbo size version of this page or something like that but if that's out of the option we could just go closer to it going closer to it makes the letter bigger inside our eyes as you can see but there is a problem once you go too close closer than the near point we can no longer bring the rays of light to focus onto a retina so right now we are closer than the near point and although the letter looks big it's blurry so what to do well we can just bring in a convex lens because of its converging power it helps focus the Rays back onto our retina and voila we can now see it clearly so the surprising thing is the magnifying glass doesn't really make things look bigger see even without the glass the size was pretty much the same all it's doing is making sure that it's in focus so in short without the lens we cannot go to close without blurring it and this is now the maximum size of the letter in our retina but with the lens we can go closer that makes the image bigger and the lens helps keep it focused and sharp so that's the principle of our magnifying glass so let's look at this in a little bit more detail imagine we have our eye and a principal axis drawn we're gonna we're going to consider that our eye is just a convex lens and this is the retina where the image is supposed to be formed now let's put an object let's put the same object that we had earlier the letter A let's keep it at some distance from the eye by the way this point D is the near point which means if the object comes closer than the near point will not get a sharp image we'll talk about that okay so anyways if the object is over here let's look at where the image will be formed we can draw a couple of rays to figure this out well one ray of light will start from the top of this object and who hit it right at the optics center because we know that goes undeviated anyway from the optic center goes on deviated and so this is one ray of white and another ray we cannot we can draw more parallel to the principal axis where will this go well this goes straight through the focus but where is the principal focus of this lens well our eyes are going to adjust its principal focus or its focal length in such a way that these two rays get converged right at the retina so even without knowing where the principal focus is we know for sure that this thing has to get focused at this point and that's the technique that we can use to figure out where the images inside the eye but this only works as long as the object is outside the near point if it goes inside the narrow point this one will look at that a little bit later but anyways notice this point is being focused over here similarly the point over here would get focused at at this point and as a result we can now reconstruct the image the image will look inverted and it looks somewhat like this so yeah the image inside our eyes are inverted our brains are going to correct it at everything but that's what it looks like and now here's the key thing how big this object this letter looks to us depends only on the height of the image formed in the retina if this height were to increase in the retina well it looked bigger to us if we were to decrease in the retina it looks smaller to us so since we want it to magnify we want we want this thing to be magnified you want it to look bigger to us we should try and figure out how to make this image image height in the retina bigger and so the big question is what does the image height depend on well if we ignore this blue ray for a while we can see what it depends on so if you just concentrate on this yellow ray can you see that the height of this image really depends on this angle over here let me just write that down depends on this angle over here if this angle were to increase then the height of the image will increase and by the way this angle is the same as this angle oh let's call this angle as theta so if this theta increases if this angle increases and this image size increases so to increase the image size inside our retina we have to somehow increase this angle theta so let's see what this angle theta depends on can we calculate that from this figure over here oh for sure we can if we call the object height as say H let's say this object height is H and let's assume this distance to be D I'll not write that down but let's assume this distance is D then we can if the angle for small angles we can use small angle approximation we can treat this like an arc you see that we can through this as the radius and the arc and so we can use the art formula and if we do that the arc formula tells us that this angle theta in radians is going to be the arc length which is H divided by the distance over here or the radius D and by the way this is not exactly equal to it's an approximation because in reality this is not an exact arc it's actually a right-angled triangle you can also think of it this way if you take tan theta you get H divided by D but if theta is very small tan theta is pretty much equal to theta and that's also how we can justify this approximation but anyways if you want to increase this angle theta we can either increase the height of the object we can make the object bigger so it'll look bigger to us or we can decrease the value of D this distance we can decrease it in other words you can bring bring that object closer to our eye that's what we did if we decrease it then also theta will increase and that's why they made sizeof increase so let's just go ahead and decrease that let's bring the object closer let's bring it all the way to the D point over here and now you can see that the object starts obtaining a bigger angle near our eye can you see this angle has increased this theta has increased and as a result let me write that that theta down over here and as a result this angle has increased this angle increases and therefore this image also increases in size look at this image that image looks bigger to us well that's exactly what we saw right as we went closer and closer that image started looking bigger and bigger to us but another thing that happens to us is that as this object comes closer our eyes get more and more stressed to understand why let's bring back that blue ray there a parallel to the principal axis if we just focus on these incoming rays as for now when the object was far behind they weren't very diverging the angle of divergence was small and as a result our eyes could easily focus that but as the object comes closer and closer and closer and comes right to the D point notice what happens to the angle of divergence these rays become more divergent the angle of divergence increases and as a result our eyes have to work harder to now again focus them right on the retina and so although coming closer to the object increases the image size it also ends up stressing your eye and therefore if you were to bring this object even closer trying to make this angle even bigger let me just draw that over here not out this angle would increase even further giving you a bigger image but now the Rays are sore divergent your eyes will not be able to focus it right at this point the two rays will not meet it your eyes won't be able to do that it ran out of power per se and as a result the Rays will not meet over here they might meet behind the retina or whatever but anyways the point will not be focused and as a result even though you get a big image you end up getting a blurry image again that's what we saw the moment you go closer than the deep point in the near point the image becomes bigger in size but becomes blurry and therefore with our naked eye this is the limit this is the maximum angle so we can write that down so let's write that down over here the maximum angle we'll call it is theta naught that's the maximum angle that the object can subtend to our naked eye without blurring it without blurring it oh that would be in this pose in this case and that would be the height of the object divided by this distance that's there that's the near point distance let's call it as capital D itself and so this is that maximum angle this angle so let's call this theta naught this is that maximum angle or this angle which corresponds to the maximum size you cannot get any bigger angle for a naked eye and still get a sharp image so if you want to make that angle even bigger then you need some help over here to converge that beam of light and we get that help from the magnifying glass so a magnifying glass as shown before is just a convex lens and let's say that we bring our object exactly at the principal focus of this convex lens what's going to happen well we're dealing with very thin lenses over here they're not as thick as what I've shown in the picture over here that's exaggerated but because of these thin lenses we can pretty much assume that the optics center of this lens is at the same location as the optics center of this lens and if you do that assumption then this array that was passing through the optic center oh pretty much remains the same and as a result well the angle subtended by the object still remains the same at the eye so the size of the object size of the image in the retina won't change at all it's pretty much the same as we had even without the lens but now the lens will help converts the beam of light so when this this array of light that will change let's draw that so this ray will now change a little bit because of the lens as it goes through the lens the lens will converge this and guess what since the object is right at the principle focus after refraction the rays of light are going to become parallel to each other and so as a result after refraction this ray is gonna become parallel oops it's gonna become paddle like this oops okay fine it's gonna become paddle like this and so notice the incoming rays for our eyes are parallel rays and guess what our eyes can easily convert this parallel rays of light because remember parallel rays come from object which are very far away and when you look at things very far away you can easily converge them that's the easiest one to converge and so our eyes now can converge this beam of light with ease at this point and because it now is able to converge the beam of light this thing will now become very sharp so look at this this will now become very sharp and again that's exactly what we saw earlier when we introduced the lens that's image did not change in size but it became sharp so now what is this new angle going to be well that's bigger than theta naught let's call it as theta dash and that theta dash is going to be approximately well we can use the same formula this also should be approximate okay so this is approximately all right okay well this time is going to be height the same thing height divided by what's the distance well that's the focal length right because this let's is a pretty thin so we can imagine we can pretty much neglecting Ness so the distance is pretty much the focal length over here and so that is f and so now we can go ahead and define this thing called as magnification that this lens is producing so if this height due to this lens introduction of the lens say is two times more than this then we can say the magnification is two so magnification over here is going to be the height of the image in the retina with the lens to the height maximum height without the lens but guess what the ratio of these Heights is the same as the ratio of these angles right because if this height or a double that means this angle also pretty much pretty much doubled and therefore when it comes to a simple microscope or a magnifying glass we define magnification magnification as the height of the image in over here divided by the height of this image but that's the same pretty much the same as the angle subtended over here divided by the angle over here and that turns out to be H by F divided by H by D so that's Hetch divided by F divided by H by D it's divided by capital D and that is approximately equal to this cancels you get D divided by F and it's a little bit crowded but this is now the magnification of our simple microscope well let's end this with a couple of details of this formula that we derived only works when the object is kept right at the principal focus because only then we can do this and you may have some questions like what happens if the object is kept somewhere here or what if we bring the object inside will they still magnify what will happen we'll talk more about them in future videos and the second thing is because the rays of light are parallel after refraction they're parallel when they're come and hit your eyes your eyes are in their most relaxed state so this formula when you're using this magnification this this setup your eyes are relaxed eyes are relaxed so if you're reading some literature using a magnifying glass this is the setup that you should do and one last thing that surprised me was whenever I was using a magnifying glass I used to use it like this I used to keep the glass far away from my eyes and closer to the object well guess what you shouldn't do that because if you do that then your eyes are far away from the object and the angle subtended will still be pretty small but in order to use the magnifying glass properly you should keep the glass very close to your eyes that's what I mean that's when you can really get close to the object increasing that angle increasing the angle subtended and that's why you might you may have seen in movies detectives when they're using a magnifying glass that's really how they use it you have to keep it close to your eyes and then closer to the object