A thin wire of length `l` when heated to a certain temperature increases its length by `1%`. A sheet of the same material of area `2lxxl` is heated to the same temperature then percentage increase in area will be :-

Options

A. `4%`
B. `2.5%`
C. `2%`
D. `1.5%`

Answer

To solve the problem, we need to determine the percentage increase in the area of a sheet of material when it is heated, given that a wire of the same material increases in length by 1% when heated to the same temperature.

### Step-by-Step Solution:

1. **Understand the Linear Expansion of the Wire:**
- The problem states that a thin wire of length \( l \) increases its length by \( 1\% \) when heated.
- This can be expressed mathematically as:
\[ \frac{\Delta L}{L_0} = \frac{1}{100} \]
- Here, \( \Delta L \) is the change in length, and \( L_0 \) is the original length.

2. **Relate Linear Expansion to Temperature Change:**
- The change in length due to temperature change is given by:
\[ \Delta L = L_0 \alpha \Delta \theta \]
- Where \( \alpha \) is the coefficient of linear expansion and \( \Delta \theta \) is the change in temperature.
- From the previous step, we can equate:
\[ \alpha \Delta \theta = \frac{1}{100} \]

3. **Determine the Area of the Sheet:**
- The area of the sheet is given as \( A_0 = 2l \times l = 2l^2 \).

4. **Use the Area Expansion Formula:**
- The percentage increase in area due to temperature change is given by:
\[ \frac{\Delta A}{A_0} = \beta \Delta \theta \]
- Where \( \beta \) is the coefficient of area expansion. For a material, the coefficient of area expansion is related to the coefficient of linear expansion by:
\[ \beta = 2\alpha \]

5. **Substitute the Values:**
- Substitute \( \beta \) into the area expansion formula:
\[ \frac{\Delta A}{A_0} = 2\alpha \Delta \theta \]
- Now, substitute \( \alpha \Delta \theta \) from step 2:
\[ \frac{\Delta A}{A_0} = 2 \left(\frac{1}{100}\right) = \frac{2}{100} = 0.02 \]

6. **Calculate Percentage Increase in Area:**
- To find the percentage increase in area:
\[ \text{Percentage Increase in Area} = \frac{\Delta A}{A_0} \times 100 = 0.02 \times 100 = 2\% \]

### Final Answer:
The percentage increase in area when the sheet is heated to the same temperature is \( 2\% \).

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