 down L C oscillation. So, till now every chapter sorry every circuit we had resistance R somewhere sitting. First time in this circuit we do not have resistance only L and C are there and we are talking about a special kind of scenario here there is no battery fine. What has happened here is that we have charged the capacitor with a battery. So, capacitor has some charge now. Now we are discharging the capacitor against the inductor. So, basically this is a circuit drop the circuit first this is inductor this is capacitor. So, capacitor already has a charge of q naught at t equal to 0 and when it has charge q naught you have kept the switch open and after some time you have closed the switch. And now we are analyzing what is happening can you try to see what is happening I mean just write down the Kirchoff's loop equation for this loop first do that and let me know assume at any time t charge in the capacitor is q and write the circuit equation. If charge at any moment on the capacitor is q potential drop across capacitance will be q by c q by c. So, it will be q by c potential difference across capacitance and across inductor what will be the potential difference l d i by d t. So, q by c minus of l d i by d t equals to 0. How many variables this differential equation has 3 q i and t fine in order to solve it I need to reduce it to 2 variables how can do it can I say i is equal to d q by d t what is q. So, it is q by d t. What is q q what is q what is q d no i d t no I am asking what is this q where it is charge charge where in the capacitor this is not the charge that is flowing. Hold on i is not charge first of all i is what rate of flow of charge fine rate of flow of charge on this wire is i will that be equal to rate of charge that is going away from this capacitor it should be should be right. But, then if you say that q is the variable you are taking not what charge is flowing here then if q decreases current will increase. So, i is not d q by d t i is minus d q by d t because d q is a negative quantity charge in the capacitor when decreases d q is negative. So, i is minus of d q by d t fine. So, when is substituted here i will get d square i by d t square sorry d i by d t is equal d q by d t square d q by d t. We are d 2 q by d c it is d square q by d t square. So, basically q by c plus l of d square q by d t square is equal to 0. So, i will get what d square q by d t square is equal to minus of 1 by l c times q now does this remind you something like that what does it remind you. So, it is like that no you it said something yes q by capacitance no no not that from some other chapter does it remind you something. Ajay tell me how will you solve this this is second order differential equation have you seen second order differential equation in physics before what is x projection d v by d t and what is v d x by d t. So, a become d square x by d t square have you seen any relation between d square x by d t square and x. Sir, we did it when we were doing the series analytical solution I am saying beyond this chapter that is the only one. Have you seen this equation d square x by d t square have you seen this this is acceleration d square x by d t square equation is equal to minus omega square x have you seen this before what it is it is s h m s h m equation. So, the solution of this is in this form x equal to a sin of omega t plus phi where a and phi depends on initially what is the condition getting it fine now this equation looks very similar to that fine. So, mathematics does not care whether it is charge or x if this is a solution for this differential equation for this differential equation what should be the solution q is equal to maximum sorry q naught let us say q naught it is sin of what omega I am 1 by root l c times t plus phi or we can say you know q naught this is q max because this is its maximum value like amplitude is the maximum amplitude in s h m sorry maximum displacement from the main position similarly q m is a maximum possible value. Now initially what is given here at t equal to 0 the charge is q naught fine. So, when you put t is equal to 0 here you should get q naught. So, this is q naught is equal to maximum value of charge into sin of phi. So, it is q naught. Now charge cannot be created are you getting it charge cannot be created. So, on capacitor at any point in time charge cannot go beyond what was initially q naught fine. So, if q naught is should be the charge at t equal to 0 which should be its maximum value then what will be the value of phi 90 degree phi by 2 fine. So, value of phi is phi by 2 as at t equal to 0 q should be maximum. So, q m is what q m is q naught and phi is phi by 2. So, we got the solution of this differential equation as q is equal to q naught sin of omega t plus phi by 2 what is sin of omega t plus phi by 2 cos omega t. So, this is q naught cos of omega t q. Now can you plot a graph between charge and the time this is time and this is charge q charge on the capacitor this is the charge on capacitor. Can you plot the graph by the way this is omega is 1 by root L c. So, you will get a graph like this is it not a cosine graph you will get. So, you can see that charge on the capacitor positive the total charge in the capacitor is 0 actually you know that positive plate and negative plate. But if I look at just one plate what is the charge on the one plate it is positive initially then goes to 0 then negative then again become positive fine. So, the charge on the capacitor is fluctuating now q is this what about current how much is the current value of current how will you find i is what minus dq by dt i is minus dq by dt. So, if you do that you will get what q naught omega sin omega t correct how much is the charge stored in a capacitor what is the formula for that that is charge energy stored energy inside the capacitor is what q square by 2 c or half series square whatever you make want to write so at any moment energy on the capacitor will be q square by 2 c and at any moment what is the energy stored in the inductor how much it is half into l i square we have done this in moving charge magnetism. This is energy in the inductor this is energy in the capacitor fine now tell me is there any energy loss happening. Suppose initial energy in the capacitor was q naught square by 2 c is the energy getting lost somewhere or it is just coming back and forth and no it is oscillating what it is it is going back and forth right energy is not getting lost what is happening is that this electrical energy converts into magnetic energy and this magnetic energy again then converts into electric energy. So, there is a fluctuation going on between electric energy and magnetic energy fine now whatever is the fluctuation the sum of this energy plus that energy should be a constant should not depend on time prove it prove that this plus that is a constant everything is written it is independent of time total energy u c is independent is depending on time or not when you do q square by 2 c it depends on time cos square omega t will come but when you add it up with half it should be independent of time u c is q naught square cos square omega t divided by 2 c this all of you got u l is what half l i square. So, this is half l into this is i q naught square omega square sin square omega t if I add this up will omega t will go it does not seem like but if you pay attention omega is what 1 by root l c this will tell you omega square is 1 by l into c this if you substitute here you will get u l equal to q naught square by 2 c sin square omega t I take it now if I add this and that what will I get 0 0 sorry what will you get u q naught square by 2 c sin square omega t plus cos square omega t what is sin square plus cos square 1. So, this disappears whatever is a time sin square plus cos square will be always equal to 1. So, this will be equal to q naught square by 2 c. So, the total energy is independent of time this is what we have derived in idealistic scenario any doubt, but this is impractical this does not happen in reality why because an inductor will have some resistance right now energy may not get conserved in l c oscillation due to following reasons the first one is the inductor will have some negligible resistance which we are ignoring. So, you know energy can get lost from that resistance that is point number 1 in point number 2 even if suppose inductor has 0 resistance then also this inductor will have magnetic field lines and where while it is oscillating magnetic field getting converted to electric field over here the magnetic field lines could get lost this can emit radiation the inductor fine. So, that there will be radiation loss even if it is ideal inductor right. So, slowly and slowly energy will decrease and will just die down. So, this is l c oscillation first time we have studied a circuit in which there is no resistors. In fact, if you see there is no voltage supply also.