Magnifying power of telescope

in a previous video we saw how just by using two pieces of glass we can magnify things which are very far away in other words build a telescope in this video let's figure out the magnification or the magnifying power of this telescope we usually define the magnification as the angle subtended with the telescope at our eye to the angle subtended by the object without the telescope so with the telescope you can see that the angle subtended by this is this angle let me make that moon a little bit thin so that we can see that angle better all right there we have it the mall is fine but I'm just making it thin so that we can draw the angle scale nicely so this is the angle subtended at our eye right now let's call that angle as theta prime all right and what would have been the angle subtended by the moon if we hadn't used the telescope well if we hadn't used the telescope the angle subtended would be these would be directly falling on our eyes right if you hadn't used a telescope so the angle subtended would be just this let's called that as theta not let me call it let's call steerer naught and so the ratio of these two angles will be the magnification produced by the telescope and why are we taking the ratio of the two angles well we've seen before because the angle is what decides what the rate what the height of the image in our retinas or how big it looks like so the ratio of this angle is really the ratio of the image size in our retina again we've talked about this before alright now you can go ahead and pause the video and see if we can somehow look at this diagram and figure out what this ratio is just give it a try alright let's see theta prime is this number that can be figured by looking at this triangle we can use tan theta is equal to theta approximation so if you take an ratio it's the height of the moon let's call it as the height H I because it's the image height of the moon divided by the edges inside which is the focal length focal into the eyepiece divided by what is Theta naught equal to well instead of K instead of taking theta naught over here we can calculate theta naught over here can you see they're the same angles vertically opposite angles all right and so we could now take this triangle and figure out what theta naught is using the same approximation so again it'll be the opposite side H of I divided by the focal length the focal length of the objective and so if you simplify this the height of the image cancels out and as a result we now get the magnification we write that over here the magnifying power of our telescope turns out to be the focal length F naught the of the objective divided by the focal length of the eyepiece so this means to get the maximum out of our telescope we need to have a very high focal length objective and a very low focal length eyepiece all right one SuperDuper last thing is I was always confused with this rare diagram you see whenever someone would say that the rays of light are coming from far away I would always have this picture in my mind so I would always think that the rays of light coming from infinity would form point images but but over here in a telescope diagram it's not a point image why is that and for that matter it didn't make sense to me why the magnification depends on the focal length of the objective because I would think that if the focal increases well the rays of light will get focused far away but it would still be a point image right I mean why would it depend on the focal length of the objective so what's going on well what's really going on is that this is just an approximation you see in reality whenever rays of light are coming from far away from any source those rays of light in reality are not parallel to the principal axis they're parallel to each other as you as you seen you're here but they're not in general parallel to the principal axis they do form a small angle which we calling as theta naught with the principal axis which is approximating in most cases I think till now in most cases we approximate that angle to be just zero because we don't care about the image size but it's not zero and as a result the image size that you're forming also has a finite size it's not really zero but it is a point image in the sense that it is much smaller than the actual size of the moon the moon is thousands of kilometers across right this is like a few centimeters maybe so it is a point image but it's not point size like this so we have to get rid of this approximate and as a result if you look carefully what do you think will happen now if we have way to increase the focal length if you were to increase the focal length this this principal focus would be somewhere over here and the blu-ray would come from here it would hit the hit the lens somewhere over here and now can you see the two rays would now meet up somewhere over here let me let me just show you that here they just take a look at this if the focal length was larger can you see that this ray and this ray are now meeting somewhere farther away and not just that the image size as a result would be bigger so something that we miss out in this approximation is that the image that we're getting really depends on the focal length bigger the focal length C over here bigger the focal length bigger is the image size this is so important let me just show you that quickly again when we're looking at our moon if you use a high focal length objective the one we used in the demo actually look at the size of the image formed now compare this to what will happen if we use a low focal length objective this is a low focal length objective can you see that the image size is much smaller now in both cases the rays of light are coming from infinity they're being focused at their respective principal fossae yet the size of the image it depends on the focal length and it's for that reason bigger the focal length bigger will this image size be making this angle bigger and so more magnification and for the eyepiece well if the eyepiece focal length is smaller then you can go closer and that's why smaller the eyepiece focal length more the magnification so the key takeaway is that in any telescope the objectives purpose is to bring the object closer and the eyepiece the purpose of the eyepiece is to magnify that image just like a simple magnifying glass also in any telescope the objective is usually made very large so that can capture a lot of light and so the image will be nicely illuminated so that we can see it clearly