 Today we will be discussing the second lecture on charging and discharging of capacitors. In fact, in all the DC accelerators the basic principle of these accelerators is based on charging and discharging of capacitors and therefore it is necessary that we understand this aspect of accelerators very well. Now one of the things which we have to know is that how the capacitors are formed in DC accelerators and the basic equation which governs these DC accelerators is voltage generated on the capacitor V is given by the charge transferred to the capacitor divided by the capacitance value of that capacitor. So V is equal to Q upon C and depending upon the various geometries of the capacitors their values can be calculated. You can see here various geometries used for the capacitors here or the condensers and you can see that first one is the parallel plates, different geometries for example you can have two plates they will also form a capacitor and there will be a certain value of capacitance if that one or it could be a spherical there can be a sphere concentric sphere it can be surrounded by another sphere. So this will also form a capacitor, third one is hemisphere that it is not fully sphere but there is a hemisphere and then it is surrounded by another hemisphere also. In the case of Vandy graph tandem accelerators which you see in a little while from now is that this whole system is enclosed inside a pressure vessel and that acts like a second electrode which is so this becomes like as if it is two plates but they are hemisphere. Another geometry which is possible and which is used for example in the case of tandem accelerators and paletrons is cylindrical so that means there is a high voltage terminal like this and this whole thing is put inside a tank which is a continuous one. So this center portion which is the high voltage terminal acts like a capacitor so these are various geometries which are used and the capacitor values which is capacitance they are different for different geometries. In the case of Coq Coq Walton we are directly using capacitors and those capacitors are charged through diodes they can be charged through resistors also but charging through diodes is the advantages and that you will be in subsequent slides and these are some of the things which we have to understand before going to the accelerators. So now you can see here this first one is a Coq Coq Walton and it is a picture of one million volt Coq Coq Walton at Tata Institute of Fundamental Research which was set up in the late 50s then there is a line diagram of Vendigraph and you can see that the top portion is a high voltage terminal which here is a hemisphere surrounded by a pressure vessel and that pressure vessel is used to increase the increase the dielectric constant of the media so it is filled up with the gas insulating gas so that this high voltage terminal can be raised to higher voltages as I mentioned in the last lecture that in the case of Coq Coq Walton whichever open air type there the maximum voltage which could be raised to was about 1.25 to about 1.5 million volts that was because it was an open air type and there was a corona formation or the breakdown of the voltage because of high gradients and the gas surrounding was at one atmosphere air which has a certain dielectric constant and that is why the voltage was leaking sparking was taking place and we could not go to more than 1.5 million volt in the case of Coq Walton type of accelerators in the case of Vendigraph accelerator the whole thing is enclosed in a pressure vessel which is filled up with the insulating gas a mixture of nitrogen and carbon dioxide but later on a much improved gas like SF6 and that is used to fill up this whole system and as a consequence of that in the case of Vendigraph the voltage could be raised to about 10 million volts our later on there was an improvement and in the accelerator technology and going to the pelatrons where the charging was done using pelated chain and all that and voltage could be could be raised to the terminal to about up to 30 35 million volts the accelerators were designed for 35 million volts that was because the manufacturing accuracies of the terminal and of course the purities of SF6 and removal of sharp points so these are some of the things so you can see here that the first one is 1 million volt Coq Walton second is a line diagram of Vendigraph accelerator then the third one on the upper half is a 2 million volt tandem accelerator this was also modified in the late 90s and early 2000 2 is 6 million volt folded tandem ion accelerator and with a 6 million volt as the terminal so it could be improved and one of the improvement was because of the change of the insulating gas the nitrogen plus carbon dioxide decision was replaced by SF6 gas and of course the improvement in the manufacturing technologies later on then we also had a 14 million volt pelaton accelerator at TFR and this could go even to 15 million volts and these things will be discussed later on in some other lecture so basic principle which is important in the case of DC accelerators is that the voltage is equal to q by c q is the charge transferred to the capacitor having a capacitance value of c and so geometries may be different the capacitance value of the capacitor could be different and they can be calculated very nicely and the kinetic energy which is of gain by seeing the voltage difference of V or a charge state of n and total charge of q is given by q V and where it can be given in MVV the energy if the voltage in million volts and charge is q V is the charge state now this is just a diagram for the comparison purpose and to demonstrate that how important is this equation that is V is equal to q by c so you can see here the first one is a copper Walton and how the charging of the capacitor takes place here second one is a upper half of the radiograph accelerator where only the high voltage terminal is shown and the third one is a two-stage two stages either tandem or the pelaton and the whole column section including the high voltage terminal is shown to be enclosed in a pressure now in all these three cases which are the main accelerators you can see that there is a there is a capacitor formed and geometries you can see in all the three cases are different in the first case they are like parallel plates shown to be parallel plates second case it is a combination of a cylinder up to for example up to this place and then it is a hemisphere third place it is a concentric cylindrical so the capacitance values of these three cases will be different and they can be very nicely and very accurately calculated now so it is important in the dc accelerators to know that how this charging of the capacitors take place and that is the that one has to understand very well as i mentioned earlier the energy gain is q times v and v is the voltage different q is the charge which is equal to n times e here the n is charge state and the kinetic energy will be n times v v is the voltage so this kinetic energy will be in m a v if v is in million volts and is a charge state and e is the unit charge now so i have here involved another parameter which is charge state so let me explain what is the meaning of charge state because that is so suppose we have a oxygen oxygen atom and oxygen atom has eight electrons now if you remove all the eight electrons from the oxygen then it becomes a bare atom and we call it o eight plus that means all all the electrons have been removed and it becomes eight plus positively charged ion now and if you subject this oxygen eight plus to a voltage of two million volts then the kinetic energy will be eight into two is equal to sixteen m v so you can see that in the case of heavy ions the same voltage can lead to much higher energies had it been proton which can have only h one plus the energy could have been only two m e v while for heavy ions where all the electrons can be removed the energy could be much higher and this property of the ions heavy ions has been exploited in the case of tandem and peloton accelerators and that is why there the much higher kinetic energies are possible now i have said that oxygen eight plus means that all the eight electrons have been removed here that is the meaning of epsilon eight blood so it's a positively charged and it is eight plus charge now if you consider me a very simple case of h plus that means hydrogen has only one electron and that only one electron can be removed because it has only one electron that means h plus means the only electron is available with hydrogen atom that has been removed and if we add then you will ask a question that can we add one electron to the hydrogen atom i think yes is it possible and that is what is required in the case of peloton or even tandem accelerator where the initially in the first half of the accelerator the negative ions are injected and accelerated so if you add one electron instead of removing it to the neutral atom then it becomes h minus h minus means that it is the one extra electron has been added to h pattern similarly in the case of oxygen suppose you have oxygen atom and you add one electron you attach one electron extra electron that will be called as a oxygen minus which means one electron extra electron has been added now if you take how the configuration will look like and now in the case of let's say hydrogen which has only one electron so it will be in the first cell which is s cell so it will be configuration or orbital configuration will be one s one because there is only one electron however if you remove that electron then it becomes h plus and the configuration will be one s zero because there is no electron now but if you add one extra electron then it becomes one s two s avatar can have only two electrons so it will so suppose there is heavy ions then you add one more electron then it will go to the next orbital so this is how the orbital configuration in the case of hydrogen will look like now coming back to charging and discharging which is responsible for voltage generation in the case of dc accelerators so all the dc accelerators and i am going to talk about mainly for coctrot valton wendigraf tandem and peloton they are all based on the voltage developed on the i voltage terminal and i voltage terminal is nothing but a like a capacitor and we have to charge so we have to transfer the charge to that capacitor and according to that equation which is given in the bottom v is equal to q by c the voltage will be generated so therefore it is very important that we should know well to calculate the value of capacitance as well as how charging takes place how efficiently charging is done charges transfer to this capacitor these things will define or determine the efficiency of the accelerator so if we take for example one capacitor and keep adding charge to this capacitor is potential will keep rising according to that equation so we can say effectively suppose there is a capacitor and its voltage is about v therefore that means you have transferred q amount of charge to that so you can write that charge q is proportional to the voltage generated on that capacitor this effectively means that the q is equal to c into v then c is nothing but a constant of proportionality and is called capacitance of the capacitor so you will see that the basic equation in all these accelerators is v is equal to q by c that is the basic relation which we have to remember and how accurately we can determine q value q is charged and the calculate the c value the voltage determination will be that much accurate so let me now see that how the charging takes place if you know the all the parameters so let us take a very simple simple circuit which consists of a capacitor c capacitance value of c and there is a resistance of value r and we put a battery with a electromagnetic force epsilon then if you close this close this circuit suppose I put a battery I put a key in between and close that key then the circuit will be closed and the current will start flowing and charge will be collected on the capacitor that means the energy will be stored so it effectively it will look like as if it will look like as so this let's say is resistance and this is a battery and I put a capacitor and I put a key here so as soon as I close this key that means it becomes like this it becomes like this then there is a current flowing now if you if you do that here epsilon is a emf that is electromagnetic force of the voltage source now you can see that in this case of cork trot valton kind of multiplier which is shown above and so first one and the second one are similar only details are given in the second one the voltage is basically a voltage multiplier and it's a circuit where the charging of capacitors takes place in parallel and discharging them in series and that is how the voltage across the across the capacitors are are generated and if you if you connect a accelerating tube accelerating tube is basically a set of electrodes where the gradient is formed and that can be used for acceleration of the charge particles here you can see that I have used instead of resistance we have used here the diodes and there is a reason for that because diode has a property that in one direction in one direction the diode you can see I'll explain diode normally in the beginning it was it was always shown like this that there is a electrode here and there is a cathode so there is a current passing through this so this was earlier was a diode circuit diode element but now this was when the vacuum tube was used but today the diode is shown like this which is a solid state device pn or now this has a property that if this is positive and this side is negative that means you have a you connect the battery like this then there will be awkward there will be resistance here then it will start current will start flowing in this one if it is positive now in this configuration if this is positive and this is negative so that means when this is positive and this is negative it is called forward that forward bias and in forward wise there will be current passing through that now the if suppose I reverse it that means either I make it negative and positive here that means in that case this will become negative and this will become positive then there will not be any current flowing in this circuit and that time it will be called reverse wise and this property is reverse bias and this property is very effectively used in generating the voltage in the case of cockroach valton type of accelerator now you can see here that in this if you see the complete circuit this is one unit and in the cockroach valton kind of thing you can have several units for example here two units are shown that a one unit means two capacitors and two diodes they form the one so here you can see that we are having four capacitors and four diodes so there are two units in this case also in this case you can see there are two diodes and two capacitors so it is a one unit it is a multiplier and when there is a here the transformer here which gives the voltage at the output which is V sin omega t let us say sinusoidal wave is given and like this okay so you can see that when this is positive then this will not conduct it will be in reverse bias but let's say this is negative and this is positive although it is grounded but so when this is negative and this is positive that means it is ground so this is negative then this diode will be in forward direction and there will be current flowing in this circuit so as a consequence of that it will be forward biased and there will be charging taking place of this capacitor and the voltage will be generated across this capacitor to a maximum of this sinusoidal wave that means if it is V sin omega t then V will be generated now in the in the reverse direction that means when the sinusoidal wave goes to the negative cycle then this will not be conducting because then this will be negative and this will be positive so this will not be conducting but in the first half negative half is already the voltage across this first capacitor is generated to a maximum value of V now when we go to the other half that means this is positive and this is negative this will not conduct so there will not be any current in this one but this will have this diode will start conducting so that means this is negative and this is positive so this will conduct and then this will be charged to 2v because this is one this is already charged to V and there will be addition of V sin omega sin omega t and the peak value is V so this second one is charged to 2v so this is how it is generated same thing you can see here that when this is ground and this is negative here then the charging of this capacitor will take place through this circuit that is nicely explained in another transparency also and when reverse takes place then this will charge through this capacitor so this will be charged twice so you can see that if you if you have n number of stages and each stage minded is having two capacitors and two diodes so if suppose you are having n number of units n number of stages and then the total voltage at the output here on the top will be two times n into V n is the number of stages but if when it is connected to a accelerating cube there will always be a current flowing this so the total voltage will be slightly reduced due to loading in the ion current which is passing through the accelerating cube.