 Now coming back to the charging and discharging and so this is the circuit we have used and the capacitance value of C of the capacitor shown here it is charging so let us understand the physics what is the physics of it and if you understand this then I think the understanding of different types of DC accelerators will become very simple so let us say that here epsilon is the emf of the voltage source here or we are using battery here when the circuit is closed or completed then the capacitor starts charging or it starts storing the energy suppose I is the current passing through this circuit which is shown here above and at any time how much charge will be transferred how much will be the current flowing that and as I explained that suppose Q is the charge transferred or stored in the capacitor and I is the current passing through this circuit then you can use this equation that I into R that is the R is the resistor value of the resistance and plus Q upon C that is the voltage developed across the capacitor is equal to the emf value or the epsilon so this equation will be for a closed circuit this explanation can be written let us say this is equation 1 now when the slowly slowly this capacitor will be charged to the full charge value of Q naught then once it is charged then the current flowing through this circuit will be 0 and in that case the total value total value of voltage developed across the capacitor will be equal to the epsilon of that source so you can write that when the capacitor is fully charged the current stops flowing and under that condition Q is equal to Q naught that is the maximum value and in that case once if this condition is satisfied or the capacitor is fully charged then the current flowing is 0 that means I equal to 0 and Q naught of course here is a maximum charge with this C can accommodate under that condition Q naught that in maximum charge divided by C is equal to epsilon so let's say this is equation number 2 now this from this equation 1 and equation 2 you can yeah equation and 1 equation and 2 you can write that I into R that is the voltage across the resistor plus Q upon C at any time T we are writing this is equal to Q naught divided by C that is epsilon that is the emf of the source this is now if you rearrange it you can write that 1 upon C into Q minus Q naught is equal to minus I R further rearrangement of this can be written as Q naught minus Q is equal to RC and here I have replaced the current passing through the resistor as DQ by DT current is nothing but the rate of charge transferred to the capacitors so this is at this condition is satisfied at any time T and for a current of I which is flowing through the circuit if you arrange it rearrange again with this condition put in the equation then you can write that DQ divided by Q naught minus Q is equal to DT upon RC now this can be integrated this equation with the appropriate conditions of the limits and what are the limits limit is that at T equal to 0 there were no charge across the capacitor so Q is equal to 0 but let's say at time T when the condenser is fully charged Q is equal to Q naught now if you integrate it with this left hand side and right hand side and that is written in this one and you can write with the limit of 0 to Q this equation above equation and then you will see that you get is almost like a equation this is almost like a equation that 0 to x dx upon x is upon x is equal to nothing but this integration is log x and of course limits are same so this is this is somewhat similar to that and if you do this integration then you will see that we get minus log as this the base is E not the not the base is not 10 but it still LN it is not log base E and this is Q naught minus Q plus that is equal to 1 by CR into T so you have integrated this equation so this can be rearranged and it can be written as Q naught minus Q divided by Q naught is equal to E power minus T divided by CR now this is a basic equation for charging and we have to remember this so this is a basic equation which will come again now this relation gives the value of charge on the capacitor at any time T so this is a basic equation for charging the capacitor now here what are the conditions what are the lower and upper conditions what are the different conditions which can apply here and what is this CR this CR is basically called the time constant of the circuit and that is a very important parameter normally it is denoted by tau tau is equal to CR and that is time constant of the circuit it's a important parameter and when you are we are designing an accelerator we have to keep in mind this tau value and it has to be appropriately chosen now first condition is that when CR is very small as compared to 1 then the above equation can be written in such a way and the consequence of that will be that Q will be Q naught very rapidly because that T is much CR or tau is much smaller than 1 or much smaller than T you can write so if you write then the charge transferred to that the capacitor very quickly it will be done so the voltage will be generated very quickly and that is what we want but then there is a some limitations on that other condition is that if CR that means the tau is much much greater than 1 then the capacitor will take very huge time to charge or to reach the full value and of course the third condition will be that T is equal to tau that means what will happen in a time which is equal to the time constant of the circuit how you define that so if you take that above equation Q is equal to Q naught and within bracket 1 minus e power minus 1 so e is 2.713 or so if you put that now then Q becomes 0.632 Q naught that means if you write Q divided by Q naught is 0.632 or you can call it that it is 63 percent Q is nothing Q is roughly 63.2 percent of the maximum charge can be raised to so the time constant is CR which is CR or tau is the time during which charging becomes about 63.2 percent and that is shown here that within 2 within tau here in this figure we have plotted Q as a function of time so within the time tau the Q has been charged to or the C has been charged to 63.2 percent of the Q naught which is the maximum charge and we would like to have this time constant very small and of course if you have this as very small then there are other limitations now so this is charging capacitor has to discharge in order to generate the voltage of the terminal so it is charging on the capacitors which are on the left hand side and the discharging to charge the capacitors on the right hand side which are responsible for generation of the voltage across the accelerating tube so here we have to understand that how the relationship goes for discharging of the capacitor so you again take it and again take the same circuit which we have used earlier and see that how the discharging is taking place now here what we have done we have now opened the key which I showed you earlier can we just so the circuit earlier was this is a resistance and there was a key here so now this key has been opened here this is a capacitor so now this is open so what will happen with time this capacitor will lose the charge and the discharging of the capacitor will start and we should also understand that how the discharging of the circuit the capacitor takes place during the discharging the epsilon will be equal to zero because it is disconnected and equation we can again write the same equation which we followed earlier we can write that ri that is i is the current plus q by c is equal to zero now since since i equal to dq by dt that we can again write we go through the same mathematics and we can again write this equation and so the ultimately by rearranging we get dq by q is equal to minus dt by cr cr is again called the time constant of the circuit and we if we integrate it again now the limits will be different earlier the limits were from zero to q here now because we are discharging and at t is equal to zero the q is q naught and from that at any time t it will come to q so the limits are q naught to q dq by d q divided by q is equal to that is shown here so now you will get a different equation which is q is equal to q naught e power minus t by cr earlier equation in the charging this this term was one minus this here it is directly q is equal to q naught e power minus t by cr or we can also write if we write cr as tau which is a time constant then q that we charge at any time across over the capacitor is q naught that was the maximum value with which we started into e power minus t by tau where tau is cr again the time constant of the circuit so we can again apply those three conditions here and one of the condition was that if what will happen at time when it is because one of the most important parameter of in designing this circuit is cr value and therefore we should understand this that what happens when when t is equal to cr or t is equal to tau when t is equal to tau or cr then q will become q naught e power minus 1 or it is equal to q naught by e and as I said that e value is 2.713 therefore it will become 0.368 q naught or you can write q divided by q naught is 0.368 or it is nothing but 36.8 percent so here in the case of discharging the time constant tau which is equal to cr is defined as the time during which charge on the capacitor falls to about 36.8 percent of the maximum value to which it was charged so you will see that sometimes the conditions are contradictory so when you want to understand these things these two equations for charging and discharging have to be considered and they have to be time proper values of rc and tau or the tau has to be taken because you cannot have a very large value of resistance because then otherwise the condition will be different and I or you cannot have very large capacitance or very small even so there has to be optimum value and these two equations play very important role in developing the voltage across the capacitors and of course the across the terminals which are used to obtain the i voltages in the dc accelerators so these are two equations which have to be taken into account now you might wonder that I so far in the circuit analysis I have used resistors and here I am showing that in the case of cocteau-tvolton accelerator I was using the diodes and that is because of the iv characteristics of the diode and which are very helpful and you can see that in the forward direction the r value is very small in the p-n junction of the solid state diodes that means one side is positive other side is negative and under that forward bias r is very small because the current is easily passing through and that is shown here on the right hand side that with a small voltage itself which is a positive voltage the current passes very high current passes and that is why it is for forward direction and in the forward direction r becomes very small on the other side on the other side on the negative side when the polarity is reversed it is called reverse wise and under that condition the r is very large and this helps in retaining the charge across the capacitor or retaining the voltage which has developed and these properties of diodes are very useful in developing the voltage so we have to the performance in developing the voltage across the voltage generators is very important and the diodes play a very important role because of their characteristics. Now in the beginning I told told you that in the case of subsequent accelerators which are improvements we use in the beginning for example when Vendigraph was first based on nitrogen 80 percent and carbon dioxide 20 percent mixture was used for insulation of the in order to reduce the corona formation or to reduce the discharge this was used and later on SF6 gas which is a sulphur hexafluoride gas has been used and it is found to be much more useful and that is because you can see the properties here that dielectric constant of SF6 is about 2.5 times of that of air that means nitrogen plus oxygen carbon dioxide is nothing but air slightly better than air so that the dielectric constant is about 2.5 times and that precisely reason why the voltage in the case of tandem accelerators or pelatons where SF6 is used you can go to much higher voltages because of this dielectric constant but there is a disadvantage that this gas is 5 times heavier than the air although voltage wise it has advantage you can go to much higher voltages but since it is 5 times heavier and therefore if there is a little bit leak of SF6 gas it will displace the oxygen in the atmosphere or in that lab and that can create some health hazards and of course some of the disadvantages which are responsible for they are not using nowadays is you can see that it is very heavy displaces oxygen and can create oxygen deficiency so it can be a health hazards reliable one of the thing is that of course there will always be some leakages but that has to be monitored and it so happens that reliable SF6 monitors are not easily available and therefore but you have to use those monitors and therefore most of the time when SF6 monitors SF6 monitors are not available you try to use oxygen deficiency monitors because it displaces oxygen so this is equivalent but not exactly same so what you do is that any leakage takes place it displaces oxygen and therefore there will be oxygen deficiency this is also not correct which is also not right and it will it can again cause the health hazards but you can use the oxygen deficiency monitors which are easily available to indirectly find out if there is any leakage of SF6 so another disadvantage is that whenever arcing or corona takes place inside the high voltage terminal it SF6 breaks and one of the component which is S2F10 and this is a very highly toxic component and that is that can create not only health hazards but also it eats away some of the components of the accelerator and therefore this is one of the advantage this advantage not advantage disadvantage why it has been banned now this uses of SF6 gas is banned and it is discouraged people are discouraged to use SF6 now in accelerators and that is because it can be used as chemical warfare agent and it doesn't give a skin irritation it is inert suppose somebody uses in chemical warfare people will not know that it has been done and little warning of exposure so because it doesn't give any skin irritation then it it does not give any warning because you will not even know that SF6 has been used so it since it can be used internationally there is a move that it should be completely banned from usage in the accelerator so this this is what I feel now if you understand this this charging and discharging and importance of time constants importance of usage of diodes in the DC accelerators then I am sure that you will be able to design and even operate the DC accelerators very nicely details of these in the Vendigraf, Tendon and Palletons will be given in subsequent lectures thank you