Microscopic View of Current: Study with Video Lessons, Practice Problems & Examples

The drift velocity of electrons in a conductor is influenced by collisions with atoms, defined by the equation vd = emEτ, where e is the electron charge, E is the electric field, τ is the mean free time, and m is the electron mass. Current density J = nee2τEm relates to free electron density. Conductivity is the inverse of resistivity, highlighting a conductor's efficiency in current flow.

Microscopic View of Current

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 with 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 τ to. Okay. Where τ 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 ne2Eτ/mA. 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, is 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 that's going to be ne2τ/m. And I've pulled the electric field to the right. Okay. No big deal. Alright, let's do an example. A conductor has 1×1020 electrons per cubic meter, 1% of which are free electrons. If the electric field inside the conductor is 5,000 Newtons per 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 ne2τ/mE. Okay? Where n is the free electron density. Okay? We're told that in total is 1×1020, and in free is 1% of ntotal. Okay? So out of that 1×1020, 1 per every 100 electrons is a free electron. Okay? So this is just 1 100th of 1×1020, which is 1×1018. Okay? You just divide that by 100. Okay, so you lose 2 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? 1×1018. We know e, 1.6×10negative 19 squared. The mean free time is 5 microseconds or 5×10negative 6 seconds. The mass of an electron is 9.11×10negative 31. And the electric field is 5,000 Newtons per Coulomb. Plugging all of this in, we get a current density of about 7×108 amps per square meter. Okay? Now we can define 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 strength 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 ne2τ/m. Let's do another example to wrap this up. Copper has a conductivity of 5.8×107, one over ohmmeters. If the density of free electrons in a copper conductor is 5×1017, what is the mean free time for the electrons? Okay? We know the conductivity is in, sorry, not ne2, is in e2τ/m. If we want to find the mean free time, we have to solve for τ. So τ is just m×σ/ne2. The mass of the electron is 9.11×10negative 31. The conductivity is 5.8×107. The free electron density, which is given to us, we don't need to calculate it, it's already given to us, is 5×1017. And the electric charge, 1.6×10negative 19 squared. Plugging all this in, we get 4.13×10negative 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.

Here’s what students ask on this topic:

What is the drift velocity of electrons in a conductor?

The drift velocity of electrons in a conductor is the average velocity that electrons attain due to an electric field. It is given by the equation:

$v{d}_{}=\frac{eE\tau }{m}$

where $e$ is the electron charge, $E$ is the electric field, $\tau$ is the mean free time between collisions, and $m$ is the electron mass. The drift velocity is typically much slower than the speed of electrons in a vacuum due to frequent collisions with atoms in the conductor.

How is current density related to free electron density?

Current density $J$ is related to the free electron density $n$ by the equation:

$J=n{e}^2\tau \frac{E}{m}$

where $e$ is the electron charge, $\tau$ is the mean free time, $E$ is the electric field, and $m$ is the electron mass. This equation shows that current density is directly proportional to the free electron density, the square of the electron charge, the mean free time, and the electric field, and inversely proportional to the electron mass.

What is the difference between resistivity and conductivity?

Resistivity and conductivity are inverse properties of a material. Resistivity ($\rho$) measures a material's inherent resistance to the flow of electric current, while conductivity ($\sigma$) measures a material's ability to conduct electric current. The relationship between them is given by:

$\sigma =\frac{1}{\rho }$

High resistivity means poor conductivity and vice versa. Conductivity is often used to describe how well a material allows the flow of electric current, while resistivity is used to describe how much a material resists the flow of electric current.

How do you calculate the mean free time of electrons in a conductor?

The mean free time $\tau$ of electrons in a conductor can be calculated using the conductivity $\sigma$ and the free electron density $n$ with the equation:

$\tau =\frac{m\sigma }{n{e}^2}$

where $m$ is the electron mass, $e$ is the electron charge, and $\sigma$ is the conductivity. By rearranging this equation, you can solve for $\tau$ if the other parameters are known.

What is the significance of the free electron density in a conductor?

The free electron density $n$ in a conductor is the number of free electrons per unit volume that are available to conduct electric current. It is a crucial parameter in determining the electrical properties of the conductor, such as current density and conductivity. Higher free electron density typically means better conductivity, as more electrons are available to carry the current. The free electron density is calculated by considering the total number of electrons and the fraction that are free to move within the conductor.