What is the relationship between current density and drift velocity?

Options

A. `J = n e v_(d)`
B. `J= e v_(d//n)`
C. `J = n //e v_(d)`
D. `J= n v_(d)`

Explanation

To find the relationship between current density (j) and drift velocity (vd), we can start from the fundamental definitions and equations in current electricity.

### Step-by-Step Solution:

1. **Understanding Current Density (j)**:
- Current density (j) is defined as the current (I) flowing per unit area (A) through a conductor. Mathematically, it is expressed as:
\[ j = \frac{I}{A} \]

2. **Defining Drift Velocity (vd)**:
- Drift velocity (vd) is the average velocity that a charge carrier, such as an electron, attains due to an electric field. It is a measure of how quickly the charge carriers move through the conductor.

3. **Relating Current, Charge, and Drift Velocity**:
- The current (I) can also be expressed in terms of charge (Q) and time (t):
\[ I = \frac{Q}{t} \]
- For a conductor with n free charge carriers (electrons) per unit volume, the total charge (Q) that passes through a cross-sectional area in time t can be given by:
\[ Q = n \cdot A \cdot vd \cdot t \]
- Here, \(n\) is the number of charge carriers per unit volume, \(A\) is the cross-sectional area, and \(vd\) is the drift velocity.

4. **Substituting into the Current Equation**:
- Substituting this expression for Q into the equation for current (I):
\[ I = \frac{n \cdot A \cdot vd \cdot t}{t} = n \cdot A \cdot vd \]

5. **Substituting into Current Density Equation**:
- Now substituting this expression for I back into the equation for current density (j):
\[ j = \frac{I}{A} = \frac{n \cdot A \cdot vd}{A} = n \cdot vd \]

6. **Final Relationship**:
- Therefore, we arrive at the relationship between current density and drift velocity:
\[ j = n \cdot e \cdot vd \]
- Here, \(e\) is the charge of an electron. Thus, the relationship can also be expressed as:
\[ j = n \cdot e \cdot vd \]

### Conclusion:
The correct relationship between current density (j) and drift velocity (vd) is given by:
\[ j = n \cdot e \cdot vd \]