A piece of wire is cut into four equal parts and the pieces are bundled together side by side to from a thicker wire. Compared with that of the original wire, the resistance of the bundle is

Options

A. The same
B. `(1)/(16)` as much
C. `(1)/(8)` as much
D. `(1)/(4) ` as much

Answer

To solve the problem, we need to analyze the resistance of the original wire and the resistance of the bundle formed by cutting the wire into four equal parts.

### Step-by-Step Solution:

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

2. **Define the Original Wire**:
Let the original wire have a length \( L \) and a cross-sectional area \( A \). The resistance of the original wire \( R \) can be expressed as:
\[ R = \frac{\rho L}{A} \]

3. **Cut the Wire into Four Equal Parts**:
When the wire is cut into four equal parts, each piece will have a length of:
\[ L' = \frac{L}{4} \]
The cross-sectional area of each piece remains \( A \).

4. **Calculate the Resistance of Each Piece**:
The resistance \( R' \) of each piece can be calculated using the resistance formula:
\[ R' = \frac{\rho L'}{A} = \frac{\rho \left(\frac{L}{4}\right)}{A} = \frac{\rho L}{4A} = \frac{R}{4} \]
Thus, each piece has a resistance of \( \frac{R}{4} \).

5. **Combine the Four Pieces in Parallel**:
When the four pieces are bundled together side by side, they are effectively in parallel. The total resistance \( R_{total} \) of resistors in parallel is given by:
\[ \frac{1}{R_{total}} = \frac{1}{R'} + \frac{1}{R'} + \frac{1}{R'} + \frac{1}{R'} = 4 \cdot \frac{1}{R'} \]
Substituting \( R' = \frac{R}{4} \):
\[ \frac{1}{R_{total}} = 4 \cdot \frac{1}{\frac{R}{4}} = \frac{4 \cdot 4}{R} = \frac{16}{R} \]
Therefore, the total resistance \( R_{total} \) is:
\[ R_{total} = \frac{R}{16} \]

6. **Conclusion**:
The resistance of the bundle compared to the original wire is:
\[ R_{total} = \frac{R}{16} \]
This means the resistance of the bundle is \( \frac{1}{16} \) of the original wire's resistance.

### Final Answer:
The resistance of the bundle is \( \frac{1}{16} \) of the original wire's resistance.

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