When a steady current flows through a metal conductor of non-uniform cross-section , then drift velocity is

Options

A. Independent of area of cross-section
B. Directly proportional to the area of cross-section
C. Inversely proportional to the area of cross-section
D. Inversely proportional to the square of area of cross-section.

Answer

To solve the problem, we need to analyze the relationship between current, drift velocity, and the cross-sectional area of a metal conductor. Here’s a step-by-step solution:

### Step 1: Understand the relationship between current, drift velocity, and cross-sectional area.
The current \( I \) flowing through a conductor can be expressed by the formula:
\[ I = n \cdot e \cdot A \cdot V_d \]
where:
- \( I \) = current (in Amperes)
- \( n \) = number of charge carriers per unit volume (in m\(^{-3}\))
- \( e \) = charge of an electron (approximately \(1.6 \times 10^{-19}\) Coulombs)
- \( A \) = cross-sectional area of the conductor (in m\(^2\))
- \( V_d \) = drift velocity of the charge carriers (in m/s)

### Step 2: Rearranging the formula to find drift velocity.
From the equation above, we can rearrange it to express drift velocity \( V_d \):
\[ V_d = \frac{I}{n \cdot e \cdot A} \]

### Step 3: Analyze the conditions given in the problem.
We know that:
- The current \( I \) is steady (constant).
- The number of charge carriers per unit volume \( n \) and the charge \( e \) are also constants for a given material.

### Step 4: Determine the relationship between drift velocity and cross-sectional area.
Since \( I \) is constant and both \( n \) and \( e \) are constants, we can conclude that:
\[ V_d \propto \frac{1}{A} \]
This means that drift velocity \( V_d \) is inversely proportional to the cross-sectional area \( A \).

### Step 5: Conclusion based on the analysis.
Since we have established that drift velocity is inversely proportional to the area of the cross-section, we can conclude that:

**Drift velocity is inversely proportional to the area of cross-section.**

### Step 6: Check the options.
1. **Independent of area of cross-section** - False
2. **Directly proportional to area of cross-section** - False
3. **Inversely proportional to area of cross-section** - True

Thus, the correct answer is that the drift velocity is inversely proportional to the area of the cross-section.

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