A cylindrical wire is stretched to increase its length by 10%. Calculate the percentage increase in resistance.

Answer

To solve the problem of calculating the percentage increase in resistance when a cylindrical wire is stretched to increase its length by 10%, we can follow these steps:

### Step 1: Understand the relationship between resistance, length, and area
The resistance \( R \) of a cylindrical 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,
- \( A \) is the cross-sectional area of the wire.

### Step 2: Relate the change in length to the change in area
When the wire is stretched, its length increases by 10%. Therefore, if the original length is \( L \), the new length \( L' \) is:
\[ L' = 1.1L \]
Since the volume of the wire remains constant during stretching, we can express the volume \( V \) as:
\[ V = A \cdot L = A' \cdot L' \]
where \( A' \) is the new cross-sectional area after stretching. Since the volume is constant:
\[ A \cdot L = A' \cdot (1.1L) \]
From this, we can solve for the new area \( A' \):
\[ A' = \frac{A}{1.1} \]

### Step 3: Calculate the initial and final resistance
Now we can calculate the initial resistance \( R \) and the final resistance \( R' \):
- Initial resistance:
\[ R = \frac{\rho L}{A} \]
- Final resistance after stretching:
\[ R' = \frac{\rho L'}{A'} = \frac{\rho (1.1L)}{(A/1.1)} = \frac{\rho (1.1L) \cdot 1.1}{A} = \frac{\rho (1.21L)}{A} \]
Thus, the final resistance becomes:
\[ R' = 1.21 \cdot \frac{\rho L}{A} = 1.21R \]

### Step 4: Calculate the percentage increase in resistance
The percentage increase in resistance can be calculated using the formula:
\[ \text{Percentage Increase} = \frac{R' - R}{R} \times 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \frac{1.21R - R}{R} \times 100 = \frac{0.21R}{R} \times 100 = 21\% \]

### Final Answer
The percentage increase in resistance when the cylindrical wire is stretched to increase its length by 10% is **21%**.

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