Microscopic View of Current

by Patrick Ford
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Hey, guys, in this video, we're gonna be talking about the microscopic view of current. So we wanna look and see what's actually going on to these electrons as they pass through a conductor at the microscopic level. All right, let's get to it. The speed of electrons through conductors is what we would call the drift velocity. Okay, we know that this is going to be slower than if the electrons were free to move through a vacuum. Okay, because the electrons bounce around off of the different atoms. So if there's an electric field e inside of the conductor, okay, And that electric field is due to the potential difference across the conductor, then the drift velocity of these atoms, right? Sorry of these electrons as they bounce around inside of the conductor is going to be the charge of the electron e times that electric field over the mass of the electron times. This new thing, which we give the Greek letter tau too. Okay. Where tau is the average time between collisions. Okay, so maybe this time is longer than this time, which is longer than this time, but shorter than that last time. Until the next collision. But on average, we call the time between collisions the mean free time, right? Mean being average free being when it's not in a collision when the electron is in between atoms, So the mean free time. Okay, now current can be calculated in this microscopic view. With this equation, all I have to do is plug in the drift velocity. This is in e squared times the electric field times Tao over the mass times the cross sectional area off the conductor. Okay, where in is the number of free electrons per cubic meter called the free electron Density? In a conductor, a conductor is gonna have some volume. It's gonna have a bunch of atoms, okay. And each of those atoms are gonna have electrons associated with them. Okay, So there's gonna be a certain number of electrons divided by the volume of this conductor, So that's gonna be the total electron density. But in conductors, a certain amount of those electrons are called free electrons because they're free to move about inside of this conductor. So, out of the total amount of electrons Ah, small percentage of them are gonna be free electrons if we Onley, count up the number of free electrons and divide that by the volume. That is the free electron density. Okay. And the current density, which is a value that we've seen before Just the current divided by the area of the conductor is gonna be this whole thing right here, divided by the area. So we're gonna lose the area, so it's gonna be in e squared towel over em, and I've pulled the electric field to the right. Okay. No big deal. All right, let's do an example. A conductor has one times 10 to the 20 electrons per cubic meter, 1% of which are free electrons. If the electric field inside the conductor is 5000 in super cool. Um, and the mean free time is five microseconds. What is the current density of the conductor? We just saw that the current density was in e squared towel over m times e. Okay, where in is the free electron density? Okay, we're told that in total is one time sent, the 20 and in free is 1% of in total. Okay, So out of that one times 10 to the 21 per every electrons is a free electron. Okay, so this is just 1 1/100 of one times 10 to the 20 which is one times 10 to the 18. Okay, you just divide that by 100. Okay, so you lose two exponents of 10 Okay? Now, what we wanna find is the current density. So all we need to do is plug in these values. We know it in is right. One times 10 to the 18. We know E 1.6 times times the negative 19 squared. The mean free time is five microseconds or five times 10 to the negative. Six seconds. The mass of an electron is 911 times 10 to the negative 31. And the electric field is 5000 Newtons per Coolum. Plugging all of this in, we get a current density of about seven times 10 to 8 amps per square meter. Okay, now we can defy. We confined the resistive ity off a conductor by looking at this microscopic picture, the resistive ity of this conductor is going to be given by this equation. OK, all the same things here. This is the mass of the electron. The number sorry. The free electron density. So the number of free electrons per cubic meter, the electric charge squared and the mean free time. Now we're gonna define a new quantity related to the resistive ity called the conductivity. If the resistive ITI is the inherent resistance to flow of electrons right to the flow of current, then conductivity is the inherent benefit, right? The inherent strength at which this conductor conducts current. Okay, it's the opposite of resistive ity. And it's just one over the resistive ity. Okay, so this is just gonna be in e squared Tao divided by the mass of the electron. Okay, let's do another example to wrap this up. Copper has a conductivity of 58 times 10 to the seven one overall meters. If the density of free electrons in copper in a copper conductor is five times 10 to the 17 what is the mean free time for the electrons? Okay, we know the conductivity is in. Sorry. Not in squared is in e squared Tao over in. If we want to find them mean free time, we have to solve for towel. So Tao is just m times. Sorry, Sigma divided by N E squared. The mass of the electron is 9.11 times 10 to the negative 31. The conductivity is 58 times 10 to the seven. The free electron density which is given to us we don't need to calculate. Its already given to us is five times 10 to the and the electric charge 1.6 times 7 19 squared. Plugging all this in we get 4.13 times 10 to the negative three seconds. So about four milliseconds is the average time between collisions for these electrons. Okay, this wraps up our discussion on the microscopic view of current and conductors. Thanks for watching guys.