A cube of edge `(L)` and coefficient of linear expansion `(alpha)` is heated by `1^(0)C`. Its surface area increases by

Options

A. `6alphaL^(2)`
B. `8alphaL^(2)`
C. `12alphaL^(2)`
D. `2alphaL^(2)`

Explanation

To find the increase in the surface area of a cube when it is heated, we will follow these steps:

### Step 1: Understand the problem
We have a cube with edge length \( L \) and it is heated by \( 1^\circ C \). We need to find the increase in its surface area due to this temperature change.

### Step 2: Determine the initial surface area
The surface area \( A \) of a cube is given by the formula:
\[ A = 6L^2 \]

### Step 3: Identify the coefficient of areal expansion
The coefficient of linear expansion is given as \( \alpha \). The coefficient of areal expansion \( \beta \) is related to the coefficient of linear expansion by the formula:
\[ \beta = 2\alpha \]

### Step 4: Calculate the change in surface area
The change in surface area \( \Delta A \) can be calculated using the formula for areal expansion:
\[ \Delta A = A \cdot \beta \cdot \Delta T \]
where \( \Delta T \) is the change in temperature. Here, \( \Delta T = 1^\circ C \).

Substituting the values:
\[ \Delta A = A \cdot (2\alpha) \cdot (1) \]
\[ \Delta A = 6L^2 \cdot (2\alpha) \cdot 1 \]
\[ \Delta A = 12\alpha L^2 \]

### Step 5: Conclusion
The increase in surface area of the cube when heated by \( 1^\circ C \) is:
\[ \Delta A = 12\alpha L^2 \]

### Final Answer
The increase in surface area is \( 12\alpha L^2 \).

---