If a wire is stretched to four times its length, then the specific resistance of the wire will

Options

A. become 4 times
B. become 1/4 times
C. become 16 times
D. remain the same

Answer

To determine how the specific resistance of a wire changes when it is stretched to four times its length, we can follow these steps:

### Step-by-Step Solution:

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

2. **Effect of Stretching the Wire**:
When the wire is stretched to four times its original length, we can denote the new length as:
\[ L' = 4L \]
The volume of the wire remains constant when it is stretched. Therefore, if the original cross-sectional area is \( A \), the new cross-sectional area \( A' \) can be found using the relationship:
\[ L \times A = L' \times A' \implies A' = \frac{A}{4} \]

3. **Calculate New Resistance**:
Now, substituting the new length and area into the resistance formula:
\[ R' = \rho \frac{L'}{A'} = \rho \frac{4L}{A/4} = \rho \frac{4L \cdot 4}{A} = 16 \frac{\rho L}{A} = 16R \]
This shows that the resistance increases by a factor of 16.

4. **Understanding Specific Resistance**:
Specific resistance \( \rho \) is a material property and is defined as:
\[ \rho = R \frac{A}{L} \]
Since \( R \) has changed but the specific resistance \( \rho \) depends only on the material properties (number density of electrons and relaxation time), it does not change with the dimensions of the wire.

5. **Conclusion**:
Therefore, even though the resistance of the wire increases, the specific resistance remains the same when the wire is stretched to four times its length.

### Final Answer:
The specific resistance of the wire will **remain the same**.

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