Hey guys. In this video we're going to be talking about the microscopic view of current. So, we want to look and see what's actually going on to these electrons as they pass through a conductor at the microscopic level. Alright, 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 to, 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 and 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 e, E squared times the electric field times tau over the mass times the cross-sectional area of the conductor, okay? Where n is the number of free electrons per cubic meter, okay? Called the free electron density. Now, in a conductor, right? A conductor is going to have some volume, it's going to have a bunch of atoms, okay? And each of those atoms are going to have electrons associated with them. So, there's going to be a certain number of electrons divided by the volume of this conductor. So, that's going to be the total electron density but in conductors a certain amount of those electrons are called free electrons because they are free to move about inside of this conductor. So, out of the total amount of electrons a small percentage of them are going to be free electrons, if we only 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 going to be this whole thing right here, divided by the area. So, we're going to lose the area. So, it's going to be n, e squared, Tao over m, and I've pulled the electric field to the right, okay? No big deal.
Alright, let's do an example. A conductor has 1 times 10 to the 20 electrons per cubic meter, 1% of which are free electrons, if the electric field inside the conductor is 5,000 news for Coulomb and the mean free time is 5 microseconds, what is the current density in the conductor, okay? We just saw that the current density was n, e squared, Tao over m times E, okay? Where n is the free electron density, okay? We're told, that in total is 1 times 10 to the 20, and in free is 1% of n total, okay? So, out of that one times 10 to the 20, 1 for every hundred electrons is a free electron, okay? So, this is just 1/100 of 1 times 10 to the 20, which is 1 times 10 to the 18, okay? You just divide that by 100, okay? So, you lose two exponents of 10, okay? Now, what we want to find is the current density. So, all we need to do is plug in these values, we know what n is, right? 1 times 10 to the 18, we know e 1.6 times 10 to the negative 19 squared, the mean free time is 5 microseconds or 5 times 10 to the negative 6 seconds, the mass of an electron is 9.11 times 10 to the negative 31 and the electric field is 5,000 Newtons per Coulomb, plugging all of this in, we get a current density of about seven times 10 to the 8 amps per square meter, okay? Now, we can find that the resistivity of a conductor by looking at this microscopic picture, the resistivity of this conductor is going to be given by this equation, okay? 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 going to define a new quantity related to the resistivity called the conductivity, if the resistivity is the inherent resistance to flow of electrons, right? To the flow of current then conductivity is the inherent benefit, right? The inherent strengths at which this conductor conducts current, okay? It's the opposite of resistivity and it's just one over the resistivity, okay? So, this is just going to be n, e squared, tau divided by the mass of the electron, okay? Let's do another example to wrap this up.
Copper has a conductivity of 5.8 times 10 to the 7 one over ohm, meters, if the density of free electrons in a copper conductor is 5 times 10 to the 17, what is the mean free time for the electrons, okay? We know the conductivity is n, sorry, not n squared, is n, e squared, tau over m, if we want to find the mean free time, we have to solve for tau. So, tau 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 5.8 times 10 to the 7, the free electron density, which is given to us, we don't need to calculate it, it's already given to us, is 5 times 10 to the 17, and the electric charge 1.6 times ten to the negative 19 squared, plugging all this in, we get 4.13 times 10 to the negative 3 seconds. So, about 4 milliseconds is the average time between collisions for these electrons, okay? This wraps up our discussion on the microscopic view of current in conductors, thanks for watching guys.