A thin copper wire of length l increases in length by 1% when heated from `0^(@)C` to `100^(@)C`. If a then copper plate of area `2lxxl` is heated from `0^(@)C` to `100^(@)C`, the percentage increases in its area will be

Options

A. `1%`
B. `2%`
C. `3%`
D. `4%`

Explanation

To solve the problem, we need to find the percentage increase in the area of a thin copper plate when it is heated from \(0^\circ C\) to \(100^\circ C\). We know that the length of a thin copper wire increases by 1% under the same conditions. Let's break down the solution step by step.

### Step 1: Understanding Linear Expansion
The linear expansion of a material is given by the formula:
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]
where:
- \(\Delta L\) is the change in length,
- \(L\) is the original length,
- \(\alpha\) is the coefficient of linear expansion,
- \(\Delta T\) is the change in temperature.

### Step 2: Given Information
From the problem, we know that the percentage increase in length of the wire is 1%. This can be expressed as:
\[ \frac{\Delta L}{L} \cdot 100 = 1 \]
This implies:
\[ \Delta L = L \cdot \frac{1}{100} = 0.01L \]

### Step 3: Relating Change in Length to Coefficient of Linear Expansion
Substituting \(\Delta L\) into the linear expansion formula, we get:
\[ 0.01L = L \cdot \alpha \cdot 100 \]
Dividing both sides by \(L\) (assuming \(L \neq 0\)):
\[ 0.01 = \alpha \cdot 100 \]
Thus, we can solve for \(\alpha\):
\[ \alpha = \frac{0.01}{100} = 10^{-4} \, \text{per degree Celsius} \]

### Step 4: Coefficient of Area Expansion
The coefficient of area expansion \(\beta\) is related to the coefficient of linear expansion \(\alpha\) by the formula:
\[ \beta = 2\alpha \]
Substituting the value of \(\alpha\):
\[ \beta = 2 \cdot 10^{-4} = 2 \times 10^{-4} \, \text{per degree Celsius} \]

### Step 5: Calculating Percentage Increase in Area
The percentage increase in area can be calculated using the formula:
\[ \text{Percentage Increase in Area} = \frac{\Delta A}{A} \cdot 100 \]
Using the area expansion formula:
\[ \Delta A = A \cdot \beta \cdot \Delta T \]
Substituting this into the percentage formula, we have:
\[ \frac{\Delta A}{A} = \beta \cdot \Delta T \]
Thus,
\[ \text{Percentage Increase in Area} = \beta \cdot \Delta T \cdot 100 \]
Substituting \(\beta = 2 \times 10^{-4}\) and \(\Delta T = 100\):
\[ \text{Percentage Increase in Area} = (2 \times 10^{-4}) \cdot 100 \cdot 100 = 2\% \]

### Final Answer
The percentage increase in the area of the copper plate when heated from \(0^\circ C\) to \(100^\circ C\) is **2%**.

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