Microscopic View of Current: Videos & Practice Problems

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

The microscopic view of electric current focuses on the behavior of electrons as they move through a conductor. This movement is characterized by a concept known as drift velocity, which is the average velocity of electrons due to an electric field. Drift velocity is typically slower in conductors than in a vacuum because electrons frequently collide with atoms in the material. The relationship between drift velocity (\(v_d\)), electric field (\(E\)), and the average time between collisions (\(\tau\)) can be expressed with the formula:

\[ v_d = \frac{eE\tau}{m} \]

Here, \(e\) represents the charge of the electron, \(m\) is the mass of the electron, and \(\tau\) is the mean free time, which is the average time an electron travels between collisions. The mean free time is crucial for understanding how often electrons interact with the atomic structure of the conductor.

Current (\(I\)) can be calculated using the drift velocity and the free electron density (\(n\)), which is the number of free electrons per cubic meter. The equation for current in this microscopic context is:

\[ I = n e^2 E \frac{\tau}{m} A \]

Where \(A\) is the cross-sectional area of the conductor. The current density (\(J\)), which is the current per unit area, simplifies to:

\[ J = \frac{I}{A} = n e^2 \frac{\tau}{m} E \]

In a practical example, if a conductor has \(1 \times 10^{20}\) electrons per cubic meter with 1% being free electrons, an electric field of \(5000 \, \text{N/C}\), and a mean free time of \(5 \, \mu s\), the free electron density would be \(1 \times 10^{18} \, \text{m}^{-3}\). Plugging these values into the current density equation yields approximately \(7 \times 10^{8} \, \text{A/m}^2\).

Resistivity (\(\rho\)) is another important property of conductors, defined as the inherent resistance to the flow of current. It can be derived from the microscopic view using the equation:

\[ \rho = \frac{m}{n e^2 \tau} \]

Conductivity (\(\sigma\)), the reciprocal of resistivity, indicates how well a conductor can carry an electric current. It is given by:

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

For instance, if copper has a conductivity of \(5.8 \times 10^{7} \, \text{S/m}\) and a free electron density of \(5 \times 10^{17} \, \text{m}^{-3}\), the mean free time can be calculated by rearranging the conductivity formula:

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

Substituting the known values results in a mean free time of approximately \(4.13 \, \text{ms}\), indicating the average time between collisions for electrons in copper.

This exploration of the microscopic view of current in conductors highlights the intricate relationship between electric fields, electron movement, and material properties, providing a deeper understanding of electrical conduction.

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.