Relative & absolute R.I. connection

- In this video, we will find the expression for relative refractive index, and we will do that with the help of a numerical. So let's say we have been given this refractive index of glass, refractive index of water, and we are asked to find, what's the refractive index of glass with respect to water? Now before we began, let's just quickly recall what refractive index was. Remember that refractive index is a number that tells you how slow light is traveling in one medium compared to some reference medium, so this number helps us find out what's speed of light in a particular medium. So, I like to think of it mathematically, if you write it, if refractive index of some medium is given as n than it tells us that speed of light in that medium is the speed of light in the reference divided by n. It's n times slower compared to the speed of light in that reference, and if that reference medium is not mentioned like over here, than we will always treat it to be vacuum. Okay, so over here it would be this, tells us for example the speed of light in glass is speed of light in vacuum divided by 1.5. Same is the case over here. and if the reference medium is mentioned, for example over here water is mentioned than over here with a here on the numerator there will be speed of light in water. okay and if this is not clear to you than we have discussed a lot about this in a previous video, so it will be a great idea to go back and watch that video first and than come back over here. All right so with this, let's start, let's start finding this so we have to find what this number is? so let me just write it down ng of w is what we want to find and since I don't want to keep writitng n-g-w all the time let me just call this as something . I will call this x. Okay and we will start with the definition if this number is x, what does that mean, this means speed of light in glass. Speed of light in glass is speed of light in water speed of light in water divided by x. Makes sense, I am using the same definition, divided by x, Since, I want to find what x is I am going to rearrange this. So we can write this as x equals V of w divided by V of g. And so to figure out what x is. We need to know what the speed of light in water is and we need to know what the speed of light in glass is. and can we figure that out is the question and the answer is yes we can. Because I know what's the refractive index of glass. So I can use the same definition and figure out this speed and similarly I can find using this number what this speed is. Okay, so great idea to pause the video now and see if you can solve this further yourself. All right, lets do this let me continue that over here so x will be equal to speed of light in water Well, speed of light in water is speed of light in vacuum because the reference here is vacuum and that we usually refer to as c divided by 1.33, this will be the speed of light in water divided by speed of light in glass Again using the same definition is the speed of light in vacuum speed of light in vacuum, divided by 1.5. And we are pretty much done with the physics over here . All we have to do now is solve this fraction. Since, there is a fraction over a fraction I like to do it this way. I will write the numerator as it is, c divided by 1.33 multiplied by the reciprocal of the denominator . Okay see as 1.5 reciprocal of the denominator . This always avoids, it helps me avoid confusion see what I am doing basically is if I have 3 divided by 5 we can always right that as 3 into the reciprocal of the denominator, Isn't it? It is the same thing I am doing over here. So now that we have this . Notice that the c cancels out. c divided by c is just 1 and therefore our answer is 1.5 divided by 1.33. So that's it we just have to divide this and since I don't want to divide. I am just gonna go down and get my calculator and there it is so 1.5 divided by 1.33 and viola, there it is 1.127 and approximated as 1.13. Okay, there it is so this is gonna be equal to 1.13 and there is our answer that's the refractive index of glass x is that refractive index of glass with respect to water so this number tells us that speed of light in glass is speed of light in water divided by this number. it's that much slower in glass. All right, let's see if we can generalize this. Can we write a general expression for n-g-w in terms of n-g and n-w? Well, let's see. we will , let's see, what we calculated x is that refractive index, let's get the color right, refractive index of glass with respect to water and notice that that is equal to c divided by 1.33 so what's 1.33, that's the refractive index of water with respect to vacuum. and what's this 1.5, that's the refractive index of glass with respect to vacuum. So you see, what we got eventually . Eventually we got as 1.5 which is n-g. So let me just write that down I'm writing this down in terms of these variables . So we got this ng divided by nw and notice that is our general expression. We cant treat this as a general expression now. okay so what we see is that the refractive index of glass with respect to water is equal to refractive index of glass divided by refractive index of water. So, if we just remember this expression than we can solve problems like this in one step and you know I actually like to remember this because I like to think of it this way, whenever we have glass with respect to water the refractive index of glass comes up on the top and the refractive index of water comes down at the bottom. So I were to write this in more general terms I will generalize even more so if it was given in general what is refractive index of medium one some medium one with respect to some other medium two . You can write this as refractive index of medium one with respect to vacuum divided by refractive index of medium two . I am dividing by medium two because it is with respect to medium two. so we can think of this as the general formula that connects the relative refractive index . this is the relative refractive index.Right whenever second medium is not vacuum. it's relative with absolute refractive index noticed when the second medium is vacuum we usually call it as absolute values . So that's the connection And, before we wind up I just want to go little bit deeper to show you even more generalized result all right, so just stay with me and you will get the this. Imagine, we were not given the refractive indices with respect to vacuum. Imagine, we were given with respect to some other medium these two values were with respect to some other medium. I will show you that this formula still works okay, this expression still works. So, let's say this was given with respect to oil both of these were given with respect to oil, and not, and not vacuum and yes off course these numbers will change let's not worry about the numbers than when we solve this Till Here, everything would still be the same and while substituting the speed of light in water instead of c in the numerator we would have speed of light in oil in the numearator is'nt it because our reference medium is oil and the same thing would have happened over here . If there would be speed of light in oil in the numerator and noticed when we would have calculated those two anways cancel out so do you see regardless of which reference medium we use over here . They will get canceled out, when you are calculating their relative refractive indices which means even if this was even if this was with respect to oil this would still work . So long story short long story shorter I'm trying to tell you is if you know the refractive indices of two media with respect to another common medium Says let's call it as x or yeah some common medium x than if you divide them you will calculate what the refractive index of one medium is with respect to another medium . This is a generalized result. And so you see if you remember this, then problems like these can be solved with ease, with just one or two steps and off course if you ever forget this no problem you just go back to your definition and from there we can always derive this.