 Write down the next topic resonance in series LCR circuit what is the current you get if you supply voltage V0 you get I0 equals to V0 divided by yes omega L minus 1 by omega c square plus R square what is this this is the peak current fine now this peak current is fixed for a given frequency but if you change frequency omega will peak current change will impedance change if you change omega it will change right and there will be one frequency for which xc will become equal to xl and that moment you will have maximum of the maximum current this is the maximum current at a particular frequency this maximum current can also be varied if you change omega so the maximum currents maximum will happen when xc become equal to xl I0 will be maximum so how much the I0 will become V0 by R fine and what frequency is it when xc become equal to xl how much omega L equals to 1 by omega c so omega becomes so omega becomes 1 by root l into c so if you have this frequency then current will be the maximum possible fine this is the resonance condition this is where the resonance happens now we will be using this resonance condition in one of the instruments okay for example see for example if you have a radio have you ever seen that analog radio where you rotate the knob and this thing goes from 90 megahertz and it goes to 100 you have not seen it that radio when you rotate the knob what happens is that you change the value of capacitance you do not change your omega actually omega is fixed okay so when you change the knob capacitance changes and there will be some value of capacitance for which one by omega c becomes equal to omega l in that circuit so in that circuit resonance will happen then and once the resonance happens it will happen at a particular frequency and if you are getting signal of that frequency suppose you want to listen a song like 92 FM okay that has a fixed frequency so if at that frequency resonance happens then the peak current will be corresponding to that particular frequency get it and other frequency current will be minimal so the other frequency is your noise and the peak current where the maximum the maximum current comes where x c becomes equal to x l this is where the point where you will get the maximum signal and if that resonance happens at 92 FM you will get a station 92 FM in your radio fine I mean it will not be understood just like that I mean I am just giving you a rough picture of how it functions and why it is practically used right now in order to get into the details of exactly how radio is built and how it operates and all it is a month long thing fine so if you plot a graph between frequency omega and the peak current let us say I0 okay you will get a graph something like this what you think this frequency resonance frequency at this frequency the maximum current is the maximum fine so this point the current correspond to V0 by R okay now tell me one thing you want this kind of resonance or this kind of resonance which one will be best for you 1 or 2 for 2 2 2 right in order to catch the signal 2 is the best why in 1 you are still catching frequency from here to here with a substantial amount of current so you will have more noise but here it is a sharp resonance the resonance is sharp over here so if you catch this signal you will be able to sharply catch it the other frequency or the noise will be minimal over here getting it see basically you are catching everything are you getting a point you are catching all the frequency but this particular frequency has a maximum current and other noise in this curve 1 will have substantial current you will hear a lot of background noise but in curve number 2 other frequencies have very less current if you decrease little bit your current will just go down from here to here fine but if you decrease a bit current from here will go there are you getting it if you change frequency so that is why 2 is preferred over 1 why because it is more sharp okay now in order to understand the quality of this resonance 1 is qualitatively you just look at the graph okay this is better and that is better okay but then it is as good as saying that some object is heavy and some object is light does it make any sense this does not convey lot of meaning okay something which is heavy to you may be lighter for someone else fine this is just qualitatively we are saying something is better something is not it is your own perspective fine there might be a better resonance than this okay so we need to assign a number to what is the quality of resonance fine until as we give a value like for example we give okay this is 5 kg and that is 10 kg that makes much more sense that is just saying that something is heavy and other thing is light okay so that is why we need to come up with a factor called quality factor which determines what is the quality of the resonance in these subjects so next I mean in continuation of this only next subtopic is quality factor this is also just referred as Q factor or it is also referred as sharpness of the resonance does it make sense sharper the resonance better the quality of resonance fine now how you define sharpness of anything how it get determined suppose you have two object this is one object okay and you have another object which is more sharp two guys two guys one guy is just four feet okay the width is suppose one and a half feet other guy is seven feet width is two feet whom you will think that it is fatter and who will you think it is thinner you will say four feet also it is a comparison between length and width that matter you will just see the width so are you getting it so this is this is sharper in a way even though this width is more than this so you compare what is the width relative to what is the length okay now in order to find out the sharpness of resonance we need to first define the width getting it so suppose I want to find out sharpness of this curve suppose this is the curve for which I want to determine what is the sharpness first I need to define what is the width of this now how do you define the width it is continuously changing what do you think how should we do that okay but I want to keep it simple see you can assume any width this one this one this one this one if it is sharper at every point their width will be less than the width of this yes or no right so you can take any point and say that is the width and what makes more sense which width we should take like some somewhere where exactly do you think it is we not by root 2 r isn't that right isn't that right i know it is we not by r i mean we not by root 2 r we will be taking up I mean it is somewhat related to what we have been doing till now okay it is like rms value of the peak curves so you will draw a horizontal line at a current of v not by root 2 times r fine and you will say that this is the width okay and since it is symmetrical about this line I will say that this one is delta omega and that is also delta omega total width is 2 delta omega any doubt now tell me this is how you define the width how you define the length one thing you will come in your mind okay this looks like a length this is like a mountain so I will say that this is the length okay but that is now it should be that's not how it should be when you found out the sharpness what you did you divided length from the width so you divided by omega you have to keep it dimensionless you have defined you have you have divided the length from other length to find the sharpness okay here also you divide 2 times delta omega from omega see if if the resonance is happening at 5 hertz and delta omega is 3 hertz it's a very blunt resonance it's not a good resonance is it but if frequency is 1000 and delta omega is 5 then it is good resonance you have to compare with respect to omega how much is delta omega so the more the difference the better the quality is no less is the difference less is this width better is the quality fine so we define see the thing is it's not a something which is given by God it is our own creation we have assumed that okay let's call Q factor as omega divided by 2 delta omega are you getting it you can say no I'll call it omega divided by omega square divided by delta omega square it's your own wish but everybody has agreed to you know to call quality factor as omega divided by 2 delta omega everybody has agreed to say that okay Newton's second law of motion is force equal to mass and adhesion if Newton would have said that no force equal to mass into adhesion square you I mean all the things are still valid I mean you can start from there and build the entire mechanics backwards because force equal to mass and adhesion that is a momentum is able to be and that is why you are able to solve these questions like that had it been m into a square the entire mechanics you would have been studying in a different way fine so it is our own creation what do you call similarly work done work done is not I mean the formula is not the God's creation formula is our own creation similarly here we will just assume a formula which makes physical sense as the formula of any physical variable but we derived it right so that that will be the the law right so we can't change that huh because we have all agreed to assume saying it's a hypothetical situation we have all assumed to call q factor as omega divided by 2 delta omega if I say no if it is it should be omega divided by 2 delta omega whole square can you object it no still all the observation all the arguments still makes sense lesser the delta omega better will be the quality factor still I'm scoring it okay so I'm taking the simplest of all expression for quality factor which is this right now omega not divided by 2 delta omega okay omega not is a resonant frequency is this getting it okay now can you get the value of frequency for these two one this is point number one and this is point number two current is what I not is right now first we not divide by 1 by omega c minus omega l whole square r square this is the this is the plot for this is the curve equation are you getting it this is the relation between current and voltage this is how the current will change if you change the frequency this is the equation of the curve okay you want to find out information about these two points you want to find omega for that what you should put I not as we not by root 2 r because we not by root 2 r if you substitute this yeah if you substitute this current in that curve equation you will get these two point or not right so I'm going to substitute I not as V not by root 2 r now can you simplify a bit don't just theoretically talk do it okay so if you square it if you square both sides you will get right I mean before squaring I can cancel out V not okay and then take it in a numerator and then if I square I will get 1 by omega c minus omega l whole square plus r square is equal to 2 r square all of you got this okay so you will get 1 divided by omega c minus omega l is equal to I mean plus minus r but let's take it the positive okay if you take minus r you'll get one point if you get plus r you'll get another point so even if you get one point it is symmetrical you automatically get the other one okay now these these two values when you then you solve you get omega for one or two I'll get again this equation is satisfied by these two frequencies fine now there is a small trick over here which is not very common but then you remember once you use it once so omega you write in terms of omega not how will I write omega is equal to omega not plus minus delta omega write this down from A y A of change what the same equation is used in date of change what A y that x plus no no no this is not that don't get into all that delta omega delta omega will assume is very less compared to omega not we are assuming that we are assuming that the sharpness is good enough for us to ignore delta omega as compared to omega not getting it so if I substitute this in that and if I use this approximation can you get the value of delta omega try to do it you have to first write down something do it yourself and then speak about it okay otherwise you'll not learn anything you'll if you don't do it yourself once you get stuck then you ask don't just keep on asking few things which are which you haven't arrived there it's there in your textbook also you can refer that don't just look at the board and okay now now do you remember binomial approximation this minus one will come inside do you know this so remember this approximation this is widely used 1 plus x raised to power y is approximately equal to 1 plus x y if mod of x is very less than 1 I mean if it x is like 0.001 then you can use this approximation okay have you learned binomial theorem okay it's a part of that fine I'll rub this part so if I apply binomial approximation of there I'll get 1 by omega not c 1 minus plus this is equal to r okay so if I further simplify it I'll get 1 by omega not c minus omega not l this is omega not minus plus minus plus this you understand how much is this what is the value of this but he is 0 and at omega not xc will become equal to xl or not this term is 0 and how much is this this particular thing and can you convert this in terms of l omega not square is what 1 by lc so it becomes delta omega l so you will have 2 times delta omega l is equal to r okay and if I just consider the magnitude then 2 times delta omega is nothing but fine so we have got a simple expression for the width so q factor will be what q factor should be omega not divided by 2 delta omega right so this will be omega not l by r and this is dimensionless no dimension because this is frequency divided by frequency let me know if you have any doubts on this yeah I mean see we are anyway taking mod at the end you could have just taken plus everywhere plus you would have arrived the same way we want the magnitude of this any other doubt fine so this is another difficult derivation probably this is one of those derivation people don't do it properly this one and analytical solution of lcr