if E denotes electric field in a uniform conductor and `v_(d)` the corresponding drift velocity of free electrons in the conductor, then which of the following graph is correct

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the correct graph for the relationship between the drift velocity \( v_d \) of free electrons in a conductor and the electric field \( E \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Drift Velocity**:
The drift velocity \( v_d \) of electrons in a conductor is given by the formula:
\[ v_d = \frac{I}{n e A} \]
where:
- \( I \) = current through the conductor,
- \( n \) = number of charge carriers (electrons) per unit volume,
- \( e \) = charge of an electron,
- \( A \) = cross-sectional area of the conductor.

2. **Relating Current to Electric Field**:
According to Ohm's law, the current \( I \) can also be expressed in terms of the electric field \( E \):
\[ I = \frac{V}{R} \]
where \( V \) is the voltage across the conductor and \( R \) is the resistance.

3. **Expressing Voltage in Terms of Electric Field**:
The voltage \( V \) can be expressed as:
\[ V = E \cdot L \]
where \( L \) is the length of the conductor.

4. **Finding Resistance**:
The resistance \( R \) of the conductor is given by:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material.

5. **Substituting Values**:
Substituting the expression for \( V \) and \( R \) into the equation for current \( I \):
\[ I = \frac{E \cdot L}{\frac{\rho L}{A}} = \frac{E \cdot A}{\rho} \]

6. **Substituting Current into Drift Velocity**:
Now substituting \( I \) back into the drift velocity formula:
\[ v_d = \frac{I}{n e A} = \frac{\frac{E \cdot A}{\rho}}{n e A} \]
The area \( A \) cancels out:
\[ v_d = \frac{E}{\rho n e} \]

7. **Analyzing the Relationship**:
From the final equation \( v_d = \frac{E}{\rho n e} \), we can see that \( v_d \) is directly proportional to \( E \). This means that if we plot \( v_d \) on the y-axis and \( E \) on the x-axis, we will get a straight line passing through the origin (0,0).

8. **Conclusion**:
Therefore, the correct graph representing the relationship between drift velocity \( v_d \) and electric field \( E \) is a straight line that passes through the origin.