Deduce Ohm’s law using the concept of drift velocity

To deduce Ohm's law using the concept of drift velocity, we need to understand how the flow of electric current in a conductor is related to the electric field and the properties of the material. Let's go through the derivation step by step:

### 1. **Understanding Drift Velocity**:

When a potential difference (voltage) is applied across a conductor, an electric field (\(E\)) is established within the conductor. Free electrons in the conductor experience a force due to this electric field, which causes them to accelerate. However, due to frequent collisions with the ions in the conductor, they don't accelerate indefinitely. Instead, they attain a steady average velocity known as the **drift velocity** (\(v_d\)).

The drift velocity can be given by:

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

where:
- \( e \) is the charge of an electron (\( \approx 1.6 \times 10^{-19} \) coulombs),
- \( E \) is the electric field strength (volts per meter, V/m),
- \( \tau \) is the average relaxation time between collisions,
- \( m \) is the mass of an electron (\( \approx 9.1 \times 10^{-31} \) kg).

### 2. **Current Density (\(J\))**:

The current density \(J\) is the current (\(I\)) per unit area (\(A\)) through which the charge flows. It can be expressed in terms of drift velocity as:

\[
J = n e v_d
\]

where:
- \( n \) is the number density of free electrons (number of free electrons per unit volume),
- \( e \) is the charge of an electron,
- \( v_d \) is the drift velocity.

### 3. **Relating Current Density to Electric Field**:

Substituting the expression for drift velocity into the current density equation, we get:

\[
J = n e \left(\frac{eE\tau}{m}\right)
\]

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

### 4. **Ohm’s Law Derivation**:

Ohm's law states that the current \(I\) through a conductor between two points is directly proportional to the voltage \(V\) across the two points, and inversely proportional to the resistance \(R\):

\[
V = IR
\]

Using the relationship \(J = \sigma E\) (where \(\sigma\) is the electrical conductivity), we can identify:

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

Thus:

\[
J = \sigma E
\]

\[
J = \frac{I}{A} \quad \text{and} \quad E = \frac{V}{L}
\]

where \(A\) is the cross-sectional area of the conductor and \(L\) is the length of the conductor. Substituting these into \(J = \sigma E\), we get:

\[
\frac{I}{A} = \sigma \frac{V}{L}
\]

Rearranging to solve for \(V\), we find:

\[
V = \frac{I L}{\sigma A}
\]

Comparing this with Ohm's law \(V = IR\), the resistance \(R\) can be identified as:

\[
R = \frac{L}{\sigma A}
\]

### Conclusion:

- **Ohm's law**: \( V = IR \), where \( R = \frac{L}{\sigma A} \).
- The resistance \( R \) depends on the material's properties (\(\sigma\)) and the dimensions of the conductor (\(L\) and \(A\)).

This derivation shows that the electric current in a conductor is proportional to the applied voltage, thus verifying Ohm's law using the concept of drift velocity.