Ohm's law - derivation (using drift velocity)

the goal of this video is to derive ohm's law from the microscopic point of view by looking at electrons and this is not a standalone video we have done a lot of work before this in previous videos so we're just going to pick up from where we left off so before we start let's quickly you know recap whatever stuff that we've already seen we saw that paul drouda a german physicist was the first person around 1900s to start giving an explanation a model to explain this and his idea was he treated electrons as tiny tiny particles tiny balls of matter and he said that when you you know when you put when you put a battery there's an electric field that accelerates the electrons but the electrons don't travel in straight lines because there are a lot of crystal ions metal ions over here these electrons are continuously bombarding continuously bouncing off sorry continuously bouncing off these different different ions and so it has a very zigzaggy kind of motion and so here's how we visualized it we electrons are moving with extremely high random velocities thousands of kilometers per second thermal motion we say but that's not going to get them anywhere because they keep bouncing back and forth however in the presence of an electric field we do see that slowly and steadily it starts moving across the conductor and it's this motion that contributes to the electric current and we call this the drifting motion and we figured out what that drifting velocity was and so coming back so what we did is we said look we are no longer imagining that the electron we can forget about this we can forget about this and instead we can imagine that all the electrons are moving with a constant velocity and that's what we call the drift velocity they're not doing that but that's what contributes to electric current so this is how we assumed and we figured out what the drift velocity was we found that their velocity is this number and the way we calculate is this is the acceleration which is force acting on the electrons divided by mass so this is the acceleration of the electrons multiplied by the time and this time is the average time between two successive collisions we call that relaxation time and so the surprising thing over here was even though the electrons are continuously accelerated by the field the net motion is constant velocity it's it doesn't change we can assume that it's moving with a constant velocity this is the most important part of this entire derivation then we used this model to figure out what the electric current was in terms of drift velocity and we forgot that the electric current equals this number and again to give you a little sense of what this number is saying this n represents the number of electrons per unit volume it's the number density of the electrons and so this over here tells you how many electrons are passing per second so this if you think about it is the volume of the electrons per second volume travel per second and this is the total electrons per second and you multiplied by the charge to give you total charge and again this is derived previously so you can go back and revise if it's necessary now with these two we have to figure out or we have to derive or prove ohm's law all right so where do we start well since ohm's law has current and voltage in it and i see current let's start from here i know the expression for current and i can substitute the expression for vd over here so if i do that i'll get i equals e times n times a that is the cross sectional area of this conductor times v d and that v d i'm going to substitute from here so that's going to be e times capital e divided by m into tau so tau is the relaxation time all right so we can go ahead and simplify that now that'll give me let me write this and here then i get an e squared and i have my a let me write yeah okay then i have a i might have my tau divided by m sorry same color times the electric field all right so we have brought the current into the picture but i need the voltage now i want to bring the voltage into the picture voltage is basically the potential difference across this conductor but what i have is the electric field inside the conductor so i know that the battery creates an electric field from the positive terminal to the negative terminal so here is that electric field in this direction and that electric field i know but can i calculate what the potential difference is from this electric field well yeah there is a connection between the field and potential difference why don't you pause the video for a while and see if you can recollect that yourself all right the electric field is the negative potential gradient let me just write that down over here electric field is the negative potential gradient and you know a lot of jargons over here what this basically means let me take an example it'll make a lot of sense so if i said the electric field was something like you know 10 units in this case units would be volt per meter you can see newtons per coulomb is the usual uh unit but you can also say volts per cool over volts per meter so if the electric field was 10 volts per meter what this is saying is that if you travel along the electric field every meter you're going down potential of 10 volt okay electric potential 10 volt so one meter you go forward 10 volt you drop another meter you go forward another 10 volt it drops and the negative sign is basically saying that along the field potential drops just like in gravity as you go down along the field the gravitational potential drops it decreases same concept over here okay so that is the connection between electric field and the and the potential difference so in our case we know the potential difference across the conductor let's that itself is what v is so that potential difference we know is v and let's say the length of the conductor is l the length of this conductor is l and we're going to assume that this electric field is pretty uniform and if we do that then we don't need the differential sign over here we can just say electric field has to be the potential difference divided by length in magnitude all i'm considering is in magnitude okay these are all magnitudes i'm not considering signs over here and so if i plug that in i will now get what is this i'll get n e squared times a tau divided by m into e is v divided by l so potential difference has now come into the picture and notice we're almost done we've gotten v is equal to ir form right we have i here and we have v over here and indeed current is proportional to voltage so this itself is the proof of ohm's law and why is this happening why is current turning out to be proportional to voltage it's coming out because of this expression it's all coming from here because electrons are traveling with a constant speed drifting with a constant speed which is proportional to the strength of the electric field everything boils from there that's why i said this is the most important part of the derivation now all we have to do is put this in a nice form over here so let me shift all of this on this side so it's just going to become the reciprocal and if we do that this is what we end up with you can just check i just rearranged it so we have now the familiar form and so whatever is in the bracket notice that's all a constant it does not depend on voltage or current this is the mass of the electrons the charge on the electron relaxation time which is dependent on the temperature and the material something we'll speak about in the future videos n is the number density again independent of these two and a length in the area are the dimensions of the conductor so notice this whole thing is a constant and that's basically what we are going to call our resistance resistance and so we have derived ohm's law and so of course we'll talk about this expression in another video separately i don't want to rush it in this video but one thing you can see you can get some insights now where is this resistance coming from the resistance comes from the collisions of the electrons with the different different atoms and you can see it from this formula you see there is a tau in the denominator tau is the relaxation time right it's the time average time between successive collisions now imagine if there were no collisions at all then tau would be infinity right if there are no collisions at all then the time for the next collision would be infinity so if there were no collisions tau would be infinity and if denominator was infinity this resistance would have been zero so you can now get this amazing thing right at the microscopic level we are able to understand what causes resistance it's the collisions between the electrons and the atoms and so i guess to close the video i just like to say one mind-boggling thing for me is we are used you're using newtonian mechanics treating electrons as tiny particles which you might know by now is not accurate for microscopic particles you are supposed to use quantum mechanics but even then with these models and assumptions we are still able to get such a nice insight into what's really happening inside the conductor and we're able to derive ohm's law and that's that's pretty amazing