The coefficient of linear exopansion of a metal is `1 xx 10^(-5//0)C`. The percentage increase in area of a square plate of that metal when it is heated through `100^(0)C` is

Options

A. `0.02%`
B. `0.1%`
C. `0.001%`
D. `0.2%`

Explanation

To solve the problem of finding the percentage increase in the area of a square plate of metal when heated, we can follow these steps:

### Step 1: Understand the Coefficient of Linear Expansion
The coefficient of linear expansion (α) is given as \(1 \times 10^{-5} \, \text{°C}^{-1}\). This means that for every degree Celsius increase in temperature, the length of the metal expands by \(1 \times 10^{-5}\) times its original length.

### Step 2: Relate Linear Expansion to Area Expansion
For a two-dimensional object (like a square plate), the area expansion coefficient (β) is related to the linear expansion coefficient (α) by the formula:
\[ \beta = 2\alpha \]
Substituting the given value of α:
\[ \beta = 2 \times (1 \times 10^{-5}) = 2 \times 10^{-5} \, \text{°C}^{-1} \]

### Step 3: Calculate the Change in Area
The change in area (ΔA) can be calculated using the formula:
\[ \Delta A = A \cdot \beta \cdot \Delta T \]
Where:
- A is the original area
- ΔT is the change in temperature

Since we are interested in the percentage increase in area, we can express it as:
\[ \frac{\Delta A}{A} \times 100 = \beta \cdot \Delta T \times 100 \]

### Step 4: Substitute the Values
Given that the temperature change (ΔT) is \(100 \, \text{°C}\), we can substitute the values into the equation:
\[ \frac{\Delta A}{A} \times 100 = (2 \times 10^{-5}) \cdot (100) \cdot 100 \]

### Step 5: Simplify the Expression
Calculating the right-hand side:
\[ \frac{\Delta A}{A} \times 100 = (2 \times 10^{-5}) \cdot 10000 = 2 \times 10^{-1} = 0.2 \]

### Step 6: Final Result
Thus, the percentage increase in area is:
\[ \text{Percentage Increase} = 0.2\% \]

### Conclusion
The percentage increase in the area of the square plate when heated through \(100 \, \text{°C}\) is \(0.2\%\).

---