For a given volume, the resistance of a conductor is proportional to its

Options

A. length
B. `"(length)"^2`
C. `(1)/("length")`
D. `(1)/("( length)"^2)`

Explanation

To solve the question, we need to determine how the resistance of a conductor relates to its dimensions when the volume is kept constant. Here’s a step-by-step breakdown of the solution:

### Step-by-Step Solution:

1. **Understand the relationship between resistance, length, and area**:
The resistance \( R \) of a conductor is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length of the conductor, and \( A \) is the cross-sectional area.

2. **Express volume in terms of length and area**:
The volume \( V \) of the conductor can be expressed as:
\[ V = A \times L \]
Rearranging this gives:
\[ A = \frac{V}{L} \]

3. **Substitute the expression for area into the resistance formula**:
Now, substitute \( A \) in the resistance formula:
\[ R = \frac{\rho L}{\frac{V}{L}} = \frac{\rho L^2}{V} \]

4. **Identify the relationship between resistance and length**:
From the equation \( R = \frac{\rho L^2}{V} \), we can see that for a constant volume \( V \) and constant resistivity \( \rho \), the resistance \( R \) is proportional to the square of the length \( L \):
\[ R \propto L^2 \]

5. **Conclusion**:
Therefore, for a given volume, the resistance of a conductor is proportional to the square of its length.

### Final Answer:
The resistance of a conductor is proportional to the square of its length when the volume is kept constant.

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