 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. And 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 abruptly by defining something new? Well, the reason I'm doing that is because this term will be important later on, and I don't want to introduce this in the middle of a derivation and distract ourselves. So that's why I'm doing that. So what exactly is this relaxation time? Funny word. So electrons are constantly bumping into these atoms. 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. So relaxation time then becomes the average time between two successive collisions. So if this number was say one minute, as an example, 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 maybe few seconds. But if you average that out, then that number will be one minute. That's the meaning of 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 our drift velocity. That's what I meant. How do we calculate this? Well, what is our definition of drift velocity? It's the average velocity with which the electron is drifting forward. 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 a 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 divided 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. v1, v2, v3, v4, v5, v6, all the points. I'm just going to represent it by a single number, v here, divided by n. 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. Now, not a trick. Here's what I'm going to use. I know between collisions, electrons are going with a constant acceleration. Because between collisions, the only force on them is due to the electric field. That's a constant. And so the force is a 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 at. 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 at, I will get the velocity at this point. If I want velocity at this point, it will be u plus a into t. Time will be now 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 to u plus at, I'll get a velocity at this point. So in general, all I have to do is do a summation of u plus at, u plus at, divide by n. And again, we need to be very careful. What is u here? u is the velocity just after the previous collision. 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, divide by n. Let's see if you can simplify it even further. Let's start with this one. What happens when you add up all the initial velocities? Well, 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, million randomly oriented vectors? Well, you'll 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, an 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 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 velocities will just cancel out. Now you may ask, well, yeah, but how do you know it's going to be zero? 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, 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 will 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 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, 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 times and ask it, hey, electron, how long has it been since your last collision? That average 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. Some cases 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, which I definitely don't want to do in this video. But 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, 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 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. And 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.