Across a metallic conductor of non-uniform cross-section a constant potential difference is applied. The quantity which remains constant is

Options

A. current density
B. current
C. drift velocity
D. electric field

Answer

To solve the problem, we need to analyze the behavior of a metallic conductor with a non-uniform cross-section when a constant potential difference is applied across it. We will identify the quantity that remains constant under these conditions.

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have a metallic conductor with a non-uniform cross-section connected to a constant potential difference \( V \). The cross-sectional areas at two points can be denoted as \( A_1 \) and \( A_2 \).

**Hint**: Visualize the conductor and label the areas clearly.

2. **Current Flow**:
The current \( I \) flowing through the conductor is defined as the total charge flowing per unit time. According to the principle of conservation of charge, the current must remain constant throughout the conductor, regardless of the changes in cross-sectional area.

**Hint**: Remember that current is conserved in a closed circuit.

3. **Current Density**:
The current density \( J \) is defined as the current per unit area. For different cross-sections, we can express the current density as:
\[ J_1 = \frac{I}{A_1} \quad \text{and} \quad J_2 = \frac{I}{A_2} \]
Since \( A_1 \) and \( A_2 \) are different, the current densities \( J_1 \) and \( J_2 \) will not be equal, indicating that current density varies with the cross-section.

**Hint**: Current density depends on the area; larger area means lower current density if current remains constant.

4. **Drift Velocity**:
The drift velocity \( v_d \) can be expressed as:
\[ v_{d1} = \frac{I}{n e A_1} \quad \text{and} \quad v_{d2} = \frac{I}{n e A_2} \]
Since \( A_1 \neq A_2 \), it follows that \( v_{d1} \) and \( v_{d2} \) will also differ, indicating that drift velocity changes with the cross-section.

**Hint**: Drift velocity is influenced by both current and area.

5. **Electric Field**:
The electric field \( E \) in the conductor is related to the drift velocity. Since drift velocity changes with the cross-section, the electric field will also vary:
\[ E \propto v_d \]
Therefore, the electric field is not constant along the conductor.

**Hint**: Recall the relationship between electric field and drift velocity.

6. **Conclusion**:
The only quantity that remains constant across the metallic conductor with a non-uniform cross-section when a constant potential difference is applied is the **current \( I \)**.

**Final Answer**: The quantity which remains constant is the current \( I \).