Drift velocity - formula & derivation

electrons inside any conductor like a wire are continuously moving at extremely high velocity randomly moving but because they continuously keep bumping into atoms they don't go anywhere as a result without a battery we don't get any current but when you do hook it up to a battery an electric field gets set up and now the electrons start getting accelerated in the opposite direction due to which they start drifting in the opposite direction of the electric field with a constant velocity which we call the drift velocity so the goal of this video is to actually show that this velocity is a constant and to figure out what is the expression for this drift velocity so before we begin i want to start by defining a new term called relaxation time you may be wondering why am i starting a droplet by defining something new well the reason i'm doing that is because this this term will be important later on and i don't want to introduce this in the middle of a derivation and you know and distract ourselves okay so that's why i'm doing that so what exactly is this you know relaxation time funny word so electrons are constantly bumping into these atoms right now the time between the collisions is where we like to think that the electrons are relaxing they're relaxing and nicely drifting in the electric field okay so relaxation time then becomes the average time between two successive collisions so if this number was say one minute as an example okay if this number was say one minute then what it means is that in some collisions last for say three minutes or four minutes some collisions will last for say half a minute or you know maybe few seconds but if you average that out then that number will be one minute that's the meaning of you know relaxation time and the symbol we use for this is tau so with that out of the way let's now go ahead and define our drift velocity so derive or drift velocity that's what i meant okay how do we calculate this well how what is our definition of drift well how definition of drift velocity it's the average velocity with which the electron is drifting forward okay just to be clear we are averaging velocity of a single electron over time i am not averaging velocities of all the electrons that's how i'm doing a single electron over time so let's say i consider this as my start time and let's say this collision after this collision and let's say from here to here imagine there are million collisions happening in between so drift then would be you calculate velocity at every single point every single point add them all up and divide by n does that make sense that would be the drift velocity so let me write that down mathematically i can say drift velocity will be summation of velocities at every single point all right v one v two v three four five is equal to all the points i'm just going to represent it by a single number v here divided by m so this means v is velocity at any point you take any point that is what your v is going to be so velocity v is velocity at any point all right now how do i go ahead how do i calculate that well here's my trick not a trick or here's what i'm going to use i know between collisions electrons are uh going with a constant acceleration right because between collisions the only force on them is the elect due to the electric field that's a constant and so the force is the constant the acceleration will be a constant and that means it's a uniformly accelerated motion and i know that in a uniformly accelerated motion velocity at any point will be just the initial velocity u plus a t okay so if i wanted to know what the velocity at this point was i'm just going to draw a couple of points at this point was i would just have to know what the velocity was just after the collision let's call that velocity as u and then i need to know how much time has elapsed since then let's call that as t and if i do v equal to u plus a t i will get the velocity at this point if i want velocity at this point it'll be u plus af into t time will be enough from here to there if i want velocity at this point now i will not consider this i will now consider this as my initial velocity just after collision here what my initial velocity is this u and this u will be obviously different and again how much time has elapsed since then different t and now if i do v equal u plus a t i'll get a velocity at this point so in general all i have to do is do a summation of u plus 80 u plus a t divided by n and again we need to be very careful what is u here u is the velocity just after the previous collision okay this is just after previous collision and what is t t is the time since the previous collision right hopefully it's making sense time since previous collision all right so what will that equal so if i now calculate the drift velocity that's going to be i can distribute this summation so i'll get a sigma u divided by n plus a times sigma t i'm just going to use pick here divided by n let's see if we can simplify it even further let's start with this one what happens when you add up all the initial velocities but remember in this model just after the collision the velocity is random right that's the whole idea behind this random motion of the electrons isn't it so u has a completely random value random direction and so what happens if you add up all randomly oriented uh million randomly oriented vectors well you will get zero isn't it or something which is very close to zero and how does that work out you may ask well think of it this way if there are million collisions sometimes after a collision an electron might go to the right with some speed but similarly after some other collision electron might go to the left with some speed because it's happening randomly they're both equally probable and they will cancel out similarly after some collisions you know the electron will go up with some speed and similarly after some collisions the same electron will go down with that sums with that same speed and again when i'm doing adding them up they will cancel out and that's why there's a very good chance that most of these you know velocities will just cancel out now you may ask well yeah but how do you know it's going to be 0 that's a good question how do we know after a million they will just cancel out perfectly well they don't have to cancel out perfectly you see if this number becomes very small it's way smaller compared to this number that's enough for us then we can just assume it to be zero and i'll tell you another way in which you know we can convince ourselves that this number has to be close to zero if there was no electric field let's say if we did not put any electric field then there would be no acceleration this term would be zero right and without acceleration without electric field we know electrons don't move anywhere we know that there is no current right on an average and so that means if a is zero drift velocity has to be zero and so that means this number has to be zero does that make sense hopefully that hopefully that made sense okay so we know that that this number this number has to be zero all right let's look at this one what about the acceleration can you figure that out well yes we can i know the electric field is e from that i can figure out what the force is and then i can use newton's second law and calculate what the acceleration is it'll be a great idea to pause the video and see if you can do that part yourself first all right let's do it so acceleration is force per mass right and what is the force equal to well don't say mass times acceleration well the force is due to the electric force right and electric force is q times e and we know what q is for electron that's small e and so we'll write this as q times e this is the force divided by m so we got that final question is what is this well remember what this is just the average value of t and what was t t is the time since the previous collision so this number this number is the average time since the previous collision and just to clarify if this number say was i don't know two minutes as an example let's say if this number was two minutes then what that means is if you were to take an electron at different different different different times and ask it hey electron how long has it been since your last collision that average you know answer that it gives that number will be two minutes in some cases it would have gone for four minutes without a collision in some cases it would have gone three minutes without a collision sometimes you might have gone 10 minutes without a collision rare cases but if you add up all of that then that will be your average time since the previous collision and now here comes the trickiest part of the derivation it turns out that this value average time since previous collision is exactly the same as the relaxation time all right it turns out to be exactly the same now i thought a lot about you know how can i how can i convince you you know logically but unfortunately it's not as simple you know there's no simple way to actually see how this is true the best way would be to do you know doing rigorous mathematics um which i definitely don't want to do in this video but we we can spend some more time you know trying to understand why this is true in a future video all right exactly why these two times are exactly equal to each other and what does it mean and everything you know in a separate video altogether but as of now you know just accept that this value will also turn out to be the same as tau and if that is true we now have found our drift velocity expression the drift velocity becomes what is it what do we get we know a is e into e divided by m times this number we are accepting that it is tau all right so here's the expression for drift velocity and let's note a couple of important things the first thing you see is that this number is independent of time all right the relaxation time is a constant for a given material and a temperature and so the drift velocity truly is a constant doesn't change with time and it's for that reason we always assume that electrons are flowing with a steady speed when we have an electric field so that makes sense the second thing to note is that the drift velocity this this average constant velocity depends on the strength of the field the stronger the electric field the electrons will drift faster which kind of makes sense right more the electric field would expect more current so that makes sense the third thing to note is that the drift velocity also depends on the relaxation time how does that make any sense well if the relaxation time were to increase that basically means that there is more time between successive collisions on an average and that basically means electrons can now spend more time getting accelerated and gaining more speed so the average now starts increasing so it makes perfect sense that you know drift velocity should increase with relaxation time