A certain piece of copper is to be shaped into a conductor of minimum resistance . Its length and diameter should respectively be .

Options

A. L,D
B. `2L,(D)/(2)`
C. `(L)/(2),2D`
D. `L,(D)/(2)`

Answer

To solve the problem of shaping a piece of copper into a conductor of minimum resistance, we need to analyze the relationship between resistance, length, and diameter of the conductor.

### Step-by-Step Solution:

1. **Understand the Formula for Resistance**:
The resistance \( R \) of a conductor is given by the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (copper in this case),
- \( L \) is the length of the conductor,
- \( A \) is the cross-sectional area of the conductor.

2. **Express Area in Terms of Diameter**:
The cross-sectional area \( A \) of a cylindrical conductor can be expressed in terms of its diameter \( D \):
\[ A = \frac{\pi D^2}{4} \]

3. **Substitute Area into the Resistance Formula**:
Substituting the expression for \( A \) into the resistance formula gives:
\[ R = \frac{\rho L}{\frac{\pi D^2}{4}} = \frac{4\rho L}{\pi D^2} \]

4. **Minimizing Resistance**:
To minimize the resistance \( R \), we need to consider the factors affecting it:
- **Length \( L \)**: To minimize resistance, we should minimize \( L \).
- **Diameter \( D \)**: To minimize resistance, we should maximize \( D \).

5. **Choose Values for Length and Diameter**:
According to the problem, we need to determine the appropriate values for length and diameter. If we are given specific options, we should choose the one that provides the minimum length and maximum diameter.

6. **Evaluate Options**:
If the options provided are:
- Option 1: Length = \( L \), Diameter = \( D \)
- Option 2: Length = \( L/2 \), Diameter = \( D \)
- Option 3: Length = \( L/2 \), Diameter = \( 2D \)
- Option 4: Length = \( L \), Diameter = \( 2D \)

The best choice for minimum resistance would be:
- Minimum Length: \( L/2 \)
- Maximum Diameter: \( 2D \)

Thus, **Option 3** is the correct answer.

### Conclusion:
The conductor should have a length of \( L/2 \) and a diameter of \( 2D \) to achieve minimum resistance.