If a wire is stretched to double its length, find the new resistance if the original resistance of the wire was R.

Options

A. R
B. 2R
C. 3R
D. 4R

Explanation

To find the new resistance of a wire when it is stretched to double its length, we can follow these steps:

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

### Step 2: Analyze the change in length
If the wire is stretched to double its original length, then the new length \( L' \) becomes:
\[ L' = 2L \]

### Step 3: Consider the volume of the wire
The volume \( V \) of the wire remains constant when it is stretched. The volume is given by:
\[ V = A \cdot L \]
When the length is doubled, the new volume can also be expressed as:
\[ V = A' \cdot L' \]
where \( A' \) is the new cross-sectional area. Since the volume remains constant, we have:
\[ A \cdot L = A' \cdot (2L) \]
From this, we can solve for the new area \( A' \):
\[ A' = \frac{A}{2} \]

### Step 4: Substitute the new values into the resistance formula
Now we can substitute the new length \( L' \) and the new area \( A' \) into the resistance formula:
\[ R' = \frac{\rho L'}{A'} \]
Substituting \( L' = 2L \) and \( A' = \frac{A}{2} \):
\[ R' = \frac{\rho (2L)}{\frac{A}{2}} = \frac{2\rho L}{\frac{A}{2}} = \frac{2\rho L \cdot 2}{A} = \frac{4\rho L}{A} \]

### Step 5: Relate the new resistance to the original resistance
We know that the original resistance \( R \) is:
\[ R = \frac{\rho L}{A} \]
Thus, we can express the new resistance \( R' \) in terms of the original resistance \( R \):
\[ R' = 4 \cdot \frac{\rho L}{A} = 4R \]

### Conclusion
The new resistance \( R' \) when the wire is stretched to double its length is:
\[ R' = 4R \]