A steady current is set up in a metalic wire of non- uniform cross-section. How is the rate of flow free electrons related

Options

A. `R prop A^(-1)`
B. `R prop A`
C. `R prop A^2`
D. R is independent of A

Answer

To solve the question regarding the relationship between the rate of flow of free electrons and the cross-sectional area of a metallic wire with a non-uniform cross-section, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Current**:
- The rate of flow of free electrons in a conductor is defined as the electric current (I). Current is the amount of charge (Q) passing through a cross-section of the wire per unit time (t).
- Mathematically, \( I = \frac{Q}{t} \).

2. **Current Density**:
- Current density (J) is defined as the current per unit area of cross-section. It can be expressed as:
\[ J = \frac{I}{A} \]
where \( A \) is the cross-sectional area.

3. **Drift Velocity**:
- The drift velocity (v_d) of free electrons is related to current density and charge density (n) as follows:
\[ J = n e v_d \]
where \( e \) is the charge of an electron.

4. **Relation to Cross-Sectional Area**:
- From the equations above, we can see that if the area \( A \) changes, the current density \( J \) and drift velocity \( v_d \) will also change. Specifically, if the area decreases, the current density increases, assuming the current remains constant.

5. **Independence of Current**:
- The total current \( I \) remains constant regardless of the cross-sectional area of the wire. This means that while \( J \) and \( v_d \) are dependent on the area, the total current \( I \) is independent of the area.

6. **Conclusion**:
- Therefore, the rate of flow of free electrons (current) is independent of the area of cross-section in a metallic wire with a steady current.

### Final Answer:
The rate of flow of free electrons (current) is independent of the cross-sectional area of the wire.

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