A wire stretched to increase its length by`5%`. Calculate percentage charge in its resistance.

Explanation

To solve the problem of calculating the percentage change in resistance when a wire is stretched to increase its length by 5%, we will follow these steps:

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

### Step 2: Calculate the new length of the wire
If the original length of the wire is \( L \), and it is stretched by 5%, the new length \( L' \) can be calculated as:
\[ L' = L + 0.05L = 1.05L \]

### Step 3: Determine the change in cross-sectional area
When the wire is stretched, its volume remains constant. Therefore, the volume before stretching \( V \) is equal to the volume after stretching \( V' \):
\[ V = A \cdot L = A' \cdot L' \]
Substituting \( L' \):
\[ A \cdot L = A' \cdot (1.05L) \]
From this, we can express the new area \( A' \):
\[ A' = \frac{A}{1.05} \]

### Step 4: Calculate the new resistance
Now we can find the new resistance \( R' \) using the new length and area:
\[ R' = \frac{\rho L'}{A'} = \frac{\rho (1.05L)}{\frac{A}{1.05}} = \frac{\rho (1.05L) \cdot 1.05}{A} = \frac{\rho L \cdot 1.1025}{A} = 1.1025 R \]

### Step 5: Calculate the percentage change in resistance
The percentage change in resistance can be calculated using the formula:
\[ \text{Percentage Change} = \frac{R' - R}{R} \times 100 \]
Substituting \( R' \):
\[ \text{Percentage Change} = \frac{1.1025R - R}{R} \times 100 = \frac{0.1025R}{R} \times 100 = 10.25\% \]

### Final Answer
The percentage change in resistance when the wire is stretched by 5% is **10.25%**.

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