Understanding Drift Velocity and Current Density

When a voltage is applied across a conductor, free electrons (or other charge carriers) inside the conductor don't move randomly anymore. They acquire a small average velocity in a direction opposite to the electric field (if the charge carriers are negative, like electrons). This small average velocity is called the drift velocity (\(v_d\)).

Current density (\(J\)) is a vector quantity that describes the electric current flowing through a unit cross-sectional area of the conductor. It tells us how concentrated the current is at a particular point.

Relating Current, Drift Velocity, and Charge Carriers

Let's consider a conductor with a uniform cross-sectional area \(A\).

• Let \(n\) be the number of charge carriers per unit volume (number density).
• Let \(e\) be the magnitude of the charge of each carrier (e.g., for electrons, \(e = 1.6 \times 10^{-19}\) C).
• Let \(v_d\) be the average drift velocity of the charge carriers.

In a time interval \(\Delta t\), the charge carriers that pass through a cross-section of the conductor are those initially contained in a cylinder of length \(v_d \Delta t\) and area \(A\). The volume of this cylinder is \(A \times (v_d \Delta t)\).

The number of charge carriers in this volume is \(n \times (A v_d \Delta t)\).

The total charge (\(\Delta Q\)) passing through the cross-section in time \(\Delta t\) is the number of carriers multiplied by the charge of each carrier:

\(\Delta Q = (n A v_d \Delta t) \times e\)

The electric current (\(I\)) is the rate of flow of charge:

\(I = \frac{\Delta Q}{\Delta t} = \frac{n A v_d e \Delta t}{\Delta t} = n A v_d e\)

So, the relation between current and drift velocity is \(I = n A v_d e\).

Deriving the Relation between Drift Velocity and Current Density

Current density \(J\) is defined as the current per unit area:

\(J = \frac{I}{A}\)

Now, substitute the expression for \(I\) (\(I = n A v_d e\)) into the definition of \(J\):

\(J = \frac{n A v_d e}{A}\)

The area \(A\) cancels out:

\(J = n v_d e\)

or \(J = n e v_d\)

We need to find the relation for drift velocity \(v_d\). We can rearrange this equation to solve for \(v_d\):

\(v_d = \frac{J}{ne}\)

This equation shows the relationship between drift velocity (\(v_d\)) and current density (\(J\)), number density of charge carriers (\(n\)), and elementary charge (\(e\)).

Analyzing the Options

Let's compare our derived relation with the given options:

• Option 1: \(v_d = Jne\). This is incorrect.
• Option 2: \(v_d = \frac{J}{ne}\). This matches our derived relation.
• Option 3: \(v_d = \frac{J}{neA}\). This includes the area \(A\), which is not present in the current density definition, so it's incorrect. \(J\) already accounts for the area.
• Option 4: \(v_d = \frac{J}{ne^2A}\). This includes \(A\) and \(e^2\), which is also incorrect.

Therefore, the correct relation is \(v_d = \frac{J}{ne}\).

Revision Table: Key Formulas

Additional Information: Drift Velocity and Mobility

The drift velocity is often related to the electric field \(E\) applied across the conductor. The relation is \(v_d = \mu E\), where \(\mu\) is the mobility of the charge carriers. Mobility is a measure of how easily charge carriers move through a material under the influence of an electric field.

Substituting this into the current density equation \(J = ne v_d\), we get:

\(J = ne (\mu E) = (ne\mu) E\)

We know that Ohm's Law can be written in terms of current density and electric field as \(J = \sigma E\), where \(\sigma\) is the conductivity of the material. Comparing this with \(J = (ne\mu) E\), we find that the conductivity of a material is related to the number density and mobility of charge carriers:

\(\sigma = ne\mu\)

These relationships highlight how microscopic properties (like \(n\), \(e\), \(v_d\), \(\mu\)) are connected to macroscopic properties (like \(I\), \(J\), \(E\), \(\sigma\)).