Solar constant is defined as energy received by Earth per `cm^(2)` per minute. Find the dimensions of solar constant.

Options

A. `[M^0L^0T^0]`
B. `[MLT^(-2)]`
C. `[ML^2T^(-2)]`
D. `[M T^(-3)]`

Answer

To find the dimensions of the solar constant, we need to analyze its definition and break it down step by step.

### Step 1: Understand the Definition
The solar constant is defined as the energy received by the Earth per square centimeter per minute.

### Step 2: Identify the Units
1. **Energy (E)**: The SI unit of energy is Joules (J).
2. **Area**: The area is given in square centimeters (cm²). We will convert this to meters for our calculations.
3. **Time**: The time is given in minutes. We will convert this to seconds.

### Step 3: Convert Units
1. **Area**:
- 1 cm² = \( (0.01 \, m)² = 0.0001 \, m² = 10^{-4} \, m² \)

2. **Time**:
- 1 minute = 60 seconds

### Step 4: Write the Expression for Solar Constant
The solar constant can be expressed as:
\[ \text{Solar Constant} = \frac{\text{Energy}}{\text{Area} \times \text{Time}} = \frac{E}{\text{Area} \times \text{Time}} = \frac{J}{cm² \times min} \]

### Step 5: Substitute the Units
Substituting the units we have:
\[ \text{Solar Constant} = \frac{J}{10^{-4} \, m² \times 60 \, s} \]

### Step 6: Dimensional Formula of Joule
The dimensional formula for Joules (J) is:
\[ [J] = [M][L²][T^{-2}] \]
Where:
- \( [M] \) = Mass
- \( [L] \) = Length
- \( [T] \) = Time

### Step 7: Combine the Dimensions
Now substituting the dimensions into the expression for solar constant:
\[ \text{Solar Constant} = \frac{[M][L²][T^{-2}]}{[L²][T]} = [M][L²][T^{-2}] \cdot [L^{-2}][T^{-1}] \]

### Step 8: Simplify the Dimensions
When we simplify this expression:
\[ \text{Solar Constant} = [M][L^{2-2}][T^{-2-1}] = [M][T^{-3}] \]

### Final Result
Thus, the dimensions of the solar constant are:
\[ \text{Dimensions of Solar Constant} = [M][T^{-3}] \]