 The goal of this video is to derive an expression for current in terms of drift velocity. What's drift velocity you ask? Well, just to quickly recap. We've seen that metals contain free electrons which move at extremely high speeds due to thermal energy. Now without a battery this motion is completely random and as a result because of a lot of collision that keeps happening on a large scale these electrons don't go anywhere. And as a result without a battery we don't get any current. But when you hook it up to a battery there's an electric field that gets set up which starts pushing the electrons towards the positive terminal. Now you will see that the electrons are slowly and steadily drifting towards the positive terminal with an average constant velocity. This average velocity is what we call the drift velocity. And we can assume that all electrons are moving with the same velocity. And we've talked a lot about this drift velocity in a previous video. So if you need a refresher feel free to go back and check that out. And so now the big question is how do we calculate the current from this drift velocity? So where do we start? How do we begin calculating current in terms of drift velocity? I think we can start by recalling what the definition of current was. We define current as the amount of charges flowing per second. So now let's think about what that current would depend on. Well clearly it would depend on the drift velocity. It will depend on the drift velocity VD. That's because if this number, if the speed, this average drift velocity of the electron is higher then we would expect more charges to flow by per second and so higher current. Lower means less charges to flow by, so lower current. What else do you think the current will depend on? Maybe it depends on the dimensions of the conductor. So let's put that in as well. So let's say that the area, the cross-sectional area over here, let's call that as A. And let's say this length is L. All right, what else? Well, another thing that might matter over here is how many electrons are there in the first place in this conductor? If there are a lot of electrons then in that one second I would see a lot of electrons going by and there will be higher current. So that also matters. And the way we represent the number of electrons available for conduction how many free electrons are there for conduction, we talk in terms of number density. So let's say the electron density is N. And since we are maybe looking at this for the first time, what this means is this number tells you how many electrons are there per meter cube. How many free electrons? Okay, when I say electrons over here, how many free electrons? Basically electrons available for conduction. There are electrons which are not available for conduction, which are bonded and they are stuck to the atom. But how many free electrons are available? So if N was, say, thousand, then that means there are thousand electrons available for conduction every meter cube. That's what that number is. So you would expect more the value of N, more would be the current. And just to give you some sense of what the reality is, if you take metals, good conductors, they have at room temperature about 10 to the 29. That's a freakishly large number. 10 to the 29 electrons available for conduction per meter cube. Incredibly high, right? Okay, so given these, how do we calculate current? I want you to take a crack at this first before I do it, all right? Because we can do this just from fundamentals, from first principles without using any formula, all right? And I'll give you, I hope you start. The way I like to do this is since current is amount of charges flowing per second through any cross-sectional area, let's start by looking at some cross-sectional area. So if we concentrate on this cross-sectional area, all I need to calculate is in one second, how many electrons are passing through this? If I know that number, I'll calculate, I got my current. For example, if I find out that 10,000 electrons are passing through this in one second, that means the current must be 10,000 E, right? Because each electron has charge E, so that would be the total charge per second. So can you pause the video and think about how to calculate or what would be the number of electrons passing through this cross-sectional area in one second? Give it a try. All right, if you're given this a thought, here's how I like to do it. So I like to concentrate on the electrons, which are very close to this particular area and a little bit left of it, okay? Let's consider this electron. Now, all the electrons are drifting towards the right. Now, in one second, where will this electron go? Well, it'll go through that particular area, it'll go through it and it'll end up somewhere over here. How much distance would it have traveled? Well, I know their velocity is VD. So in one second, it must have traveled VD amount of distance. That's the definition of velocity, displacement over time, displacement in one second. So this displacement or the distance traveled over here, it's the same thing, that should be VD. All right, why should I care about that? Well, that's because in that same time, there were some electrons which were a little bit left of this group, they too went through that area and maybe ended up somewhere over here. And similarly, there were some other electrons which are a little bit to the left of this group, they too went through that and they ended up somewhere over here. So what this means is all the electrons that you find in this tiny volume, this cylindrical volume, all those electrons must have passed through this area in one second. And therefore, if I find how much charge is present over here, that should be the current because that's the charge that went through this area in one second. So now again, if you haven't tried it earlier, now would be a great time to figure out how would you calculate this, how much charge is present over here. So give it a try again if you hadn't tried earlier. All right, so how do I calculate how much charge is there? I will first calculate how many electrons are there? How do I do that? I already know that a meter cube has n number of electrons. So these many meter cubes will have how many electrons? Well, I first need to know how many meter cubes this is, or I need to know how much volume this is. Can I calculate the volume of this cylinder, this tiny cylinder? Yes, I can. I know the area of the cylinder is A and I know the length of this cylinder is VD. And so all of this is happening in one second. Let me just concentrate only on that. So let me just get rid of everything else. If I concentrate only on that in one second, in one second, everything has happened in one second. The volume covered by the electron, this volume, this volume, that is A, that is A times VD. And why did I calculate the volume? Because now that I know how many meter cubes this is, this many meter cubes, I can calculate how many electrons are there. One meter cube has n number of electrons, these many meter cubes has how many electrons? Oh, it has n times that number. So the number of electrons, which I'm just going to call hash over here, that number should be n times this value. Does that make sense? Hopefully, this is making conceptual sense. So these many electrons passed in one second through this area. So what is the charge over here? Well, each electron has charge E. So this total must charge, have a charge E times this number. And so the total charge, which I'll write over here, should be E times this number, that is n A times VD. And this is the total charge that passed through it in one second. That by definition is our current. So we have calculated what current is through this particular conductor. And again, I hope this you don't have to mug this up. So you can kind of see this is the total electrons. This is the volume covered by electrons in one second. And so this is the total electrons passing through the area in one second. And so this is the total charge passing through the area in one second, giving us the current. Do you know why I get excited about this equation? I'll tell you why. This is the current. This is the wire that carries current to my tube light. I know that the wire carries roughly around one 1.5 amperes of current. I know the cross sectional area of that wire. I can Google that, but you can even calculate it. It's roughly around 10 to the power minus six minus seven meter square. And as mentioned earlier, this is wire cop. This is fire made of copper. So I also know the value of electron density. That's about 10 to the 29 per meter cube. And I also know the value of E, which means if I plug this in, I can figure out the drift velocity of electrons over here. And if you do that, and I'm pretty sure you can do that yourself. I'll get an answer about a millimeters per second. So with just a pen and paper, we now have the capability of calculating speeds of electrons, electrons in wires. If that doesn't blow your mind away, I don't know what will. And this now raises a very interesting question. Look at the speed of the electrons. They are drifting so slowly at millimeters per second, which means if we go to my room, then it should take hours for the electrons to go from the switchboard all the way to my tube light, hours, literally. But then why is the effect of electricity almost instantaneous? Why does the tube light turn on immediately, almost immediately as I turn on the switch? What do you think is happening? Can you pause the video and think a little bit about it? Well, the answer is you don't need electrons to go from the switchboard all the way to the tube light to turn it on. Electricity doesn't work that way. There are electrons everywhere in the wire. And so when I turn on the switch, an electric field gets set up instantly in the wire. And this electric field starts pushing the electrons everywhere. And so electrons start moving everywhere almost at the same time. And as a result, the electricity effect is instantaneous. It's not because electrons are moving very fast. They don't electrons are moving super slowly. They crawl at very low speeds. But it's because electric field gets set up almost instantaneously. All right, I want to end this video with a puzzle question. Imagine we had a conductor whose area changed like this. And let's say now we hook it up to a battery. My question to you is where do you think would the current be higher in this part of the conductor or in this part of the conductor? Well, when I was first asked this question immediately after learning this formula, I thought, OK, it should be higher over here because more area means more current. Less area means less current, right? But that's wrong because the current has to be the same. Remember, charge conservation, whatever amount of charges flow in per second, the same amount of charges must flow out per second. If there were more current over here, that means some charges are being lost and charge conservation says that cannot happen. So from charge conservation, which is a fundamental law of electrostatics or, you know, of electricity, the current has to be the same. So the puzzle question for you is how can you justify this using this equation? All right, how from this equation you can convince yourself that the current here and here must be the same?