Magnification formula for mirrors

in a previous video we took a concave mirror of focal length 2 centimeters and we kept an object 6 centimeters in front of it and we asked ourselves where its image would be and to do that we use this thing called the mirror formula which basically connects the image distance the object distance with the focal length and by substituting we figured out that the image would be 3 centimeters in front of the mirror in this video we would like to figure out what is the size of this image how big this image is and we also want to figure out whether it's a real image or a virtual image now since we want to figure out the height of this image without drawing ray diagrams we might guess that we need to use a formula and if you look at the formula that we know now that's the mirror formula notice that this has nothing to do with Heights there is no heights in this formula at all which means this is only useful to figure out how far the images from the mirror it has nothing to do with how big the images so we might guess that there is another formula which might we might have to use which is concerned with the heights of the images and the objects and there is an another formula is called the magnification formula we'll talk about why it's called the magnification in a while but notice that in this formula the left hand side you have H head stands for height H eye is the height of the image h o is the height of the object and you can see that this ratio the ratio of the heights the magnification formula is telling us that it's equal to the negative ratio of their distances now before we go ahead and substitute anything let's talk about why it's called the magnification and let's only concentrate on the left hand side so let's forget about the right hand part of it only look at the left hand side this left hand side this particular ratio this ratio itself is magnification what does that mean well let me give an example and then it'll make sense so let's say that number this is the ratio this ratio was let's say 2 what does that mean well if you substitute this as 2 this means that the height of the image equals 2 times the height of the object so equals 2 times the height of the object so this automatically means that the image is twice magnified compared to the object isn't it and that's why I notice if this ratio is 2 if this number is 2 it means that the image is twice compared to the object so don't you see that the number itself is telling you how magnified the image is compared to the object and that's why the ratio is called magnification now this magnification is sign-sensitive remember our science when it comes to the height we would usually take the above principal axis as positive and below the principal axis as a negative so if you think of this carefully in this particular example when M equals 2 we see that height of the image is twice compared to the height of the object this also means that the image height and the object height have the same sign did you see that so this means that if the object height is positive like in this example the image height is also positive since both are positive it means they have the same orientation or in other words image is erect and we've seen before when image is erect it's always virtual image so this also tells us this number not only tells us how magnified the images but by looking at the sign of this number we can understand whether it's real or virtual if this sign is positive like in this example it means that the image height and the object height have the same orientation and this automatically means this is a virtual image let me take another example just to be clear let's say in the second example we got this M as - point three now could you tell what this means I want you to pause the video and think about what is the meaning of this all right let's do this so this tells us that the height of the image is - point three times the height of the object isn't it - point three times the height of the object if you substitute so notice the negative sign is now telling us that the height of the image has the opposite sign compared to the height of the object see if the if this is positive this is negative if this is negative this would be positive so the negative sign is immediately telling us that the image and the object are inverse in relation as in the image is inverted compared to the object and that can only happen when it comes to real images so just by looking at this sign we can now say this is negative it is real and it's also telling us that the image is point three times compared to the object point three times magnified means it's actually smaller than the object isn't it so notice just by looking at the sign of this magnification we can tell whether it's real or virtual and by looking at the number we can tell how big the image is compared to the object and that's exactly what we want to find out in this particular scenario we want to find how big this image is and we want to find whether it's real or virtual so all we need to do over here is to figure out what the magnification is in this particular case so how do we calculate the magnification in this example well notice that we don't know the height of the object or the height of the image all we know over here is the object distance and the image distance and the focal length so somehow magnet there must be a connection between the magnification and these distances and that connection is what we call as the magnification formula and so now let us look at the right hand side and we can see the connection so notice this is telling us that the magnification is equal to the negative ratio of the image distance and the object distance now just to clarify how I like to think about I don't like to think of magnification equals H Rho as a as a formula don't think of that as a formula think of that as the name this ratio itself is called as magnification and whenever I say magnification formula I'm talking about the connection between the magnification and the distances of the object and the image not it so now that we know the image distance and the object distance we can figure out what the magnification is and so from the magnification now we can figure out the properties of the image so let me do that over here so let's get rid of that mirror formula all right so we need to calculate the magnification M and all we need to do to calculate this value notice is figure out our substitute emails distance and the object distance but we have to substitute with signs quickly recall what the science was we take the incident direction as positive so you start from the pole and you go to the incident direction here the incident direction is towards the right so we'll take the right side all positions on the right are positive positions all the positions on the left are negative positions so great idea to pause the video and see if you can try to do this yourself first all right let's do it so magnification M is going to be equal to minus V over u I'm just gonna use this formula minus V over u what is V V is 3 centimeters that's the image distance but what is the image is on the negative side so minus 3 centimeters divided by u what's u u is the object distance an object is also on the negative side so it'll be minus 6 centimeters now let's see what we can simplify centimeter cancels out the negative sign cancels out 3 goes 1 times 3 goes 2 times so the magnification turns out to be minus 1/2 which we can write as minus point 5 so we found the magnification and so the next thing I want to do is write the magnification in terms of the height so we that this means M is minus 0.5 means the image height is minus point five times the object height minus 0.5 times the object ID and this tells us everything about the image first of all there is a negative sign that means it's a real image that means it's inverted just like what we discussed before and another thing is image height is 0.5 times the object height which means it is half the object height isn't it so it's diminished it's smaller in size and it's half the height of the object so if the object height is 10 centimeters this is 5 centimeters if the object height is 20 centimeters this is ten centimeters so we have everything that we wanted and we have solved our problem so to quickly summarize what we learned in this video we learned this thing called as the magnification formula we define magnification as the ratio of the image height and the object height and the sign of the magnificient tells us whether it's a real or a virtual image and the number tells us how magnified the image height is compared to the object height and if we ever figure want to figure out magnification in terms of when the object and the image distance are given to us we use this magnification formula and by the way if you're interested in figuring out where this formula comes from then there is a bonus video where we derive this you can just go ahead and check that out