If n,e,A and `v_(d)` are free electron density inside conductor , charge of electron , area of cross-section of conductor and drift velocity of free electrons inside conductor, then current l through the conductor is

Options

A. `n e A v_(d)`
B. `n e^(2) Av_(d)`
C. `n ev_(d)`
D. `n^(2) e Abar(v) _(d)`

Answer

To find the current \( I \) through a conductor in terms of the free electron density \( n \), charge of an electron \( e \), area of cross-section \( A \), and drift velocity \( v_d \), we can follow these steps:

### Step 1: Understand the parameters
- **Free electron density (n)**: This is the number of free electrons per unit volume in the conductor.
- **Charge of an electron (e)**: This is the fundamental charge of an electron, approximately \( 1.6 \times 10^{-19} \) coulombs.
- **Area of cross-section (A)**: This is the area through which the current flows.
- **Drift velocity (v_d)**: This is the average velocity that a free electron attains due to an electric field.

### Step 2: Define the volume of a small element
Consider a small segment of the conductor with a length \( dx \) and area \( A \). The volume \( dV \) of this small element can be expressed as:
\[ dV = A \cdot dx \]

### Step 3: Calculate the number of electrons in the small volume
The number of free electrons \( dN \) in the small volume \( dV \) can be calculated as:
\[ dN = n \cdot dV = n \cdot (A \cdot dx) \]

### Step 4: Calculate the total charge in the small volume
The total charge \( dq \) in this small volume due to the free electrons can be expressed as:
\[ dq = dN \cdot e = (n \cdot A \cdot dx) \cdot e = n \cdot e \cdot A \cdot dx \]

### Step 5: Relate charge to current
The current \( I \) is defined as the rate of flow of charge. Therefore, we can express the current as:
\[ I = \frac{dq}{dt} \]

### Step 6: Substitute \( dq \) into the current equation
To find the current, we need to relate \( dx \) to time \( dt \). The distance \( dx \) covered by the electrons in a small time \( dt \) can be expressed as:
\[ dx = v_d \cdot dt \]
Substituting this into the equation for \( dq \):
\[ dq = n \cdot e \cdot A \cdot (v_d \cdot dt) \]

### Step 7: Substitute \( dq \) into the current equation
Now substituting \( dq \) into the current equation gives:
\[ I = \frac{dq}{dt} = \frac{n \cdot e \cdot A \cdot (v_d \cdot dt)}{dt} = n \cdot e \cdot A \cdot v_d \]

### Final Result
Thus, the current \( I \) through the conductor can be expressed as:
\[ I = n \cdot e \cdot A \cdot v_d \]