The ratio of S.I. to C.G.S units for Stefan's constant is.

Options

A. `1/100`
B. `1/1000`
C. `100`
D. `1000`

Explanation

To find the ratio of the S.I. units to the C.G.S. units for Stefan's constant, we can follow these steps:

### Step 1: Understand the formula for Stefan's constant
The Stefan-Boltzmann law states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature. The formula can be expressed as:
\[ P = A \cdot \sigma \cdot T^4 \]
Where:
- \( P \) is the power,
- \( A \) is the area,
- \( \sigma \) is Stefan's constant,
- \( T \) is the temperature in Kelvin.

From this, we can derive Stefan's constant:
\[ \sigma = \frac{P}{A \cdot T^4} \]

### Step 2: Determine the dimensions of Stefan's constant
- Power (\( P \)) has the dimensions of \( [M^1 L^2 T^{-3}] \) (mass, length, time).
- Area (\( A \)) has the dimensions of \( [L^2] \).
- Temperature (\( T \)) has the dimensions of \( [K] \).

Thus, the dimensions of Stefan's constant can be calculated as follows:
\[ \sigma = \frac{[M^1 L^2 T^{-3}]}{[L^2] \cdot [K^4]} \]
This simplifies to:
\[ \sigma = [M^1 T^{-3} K^{-4}] \]

### Step 3: Convert dimensions to S.I. and C.G.S. units
- In S.I. units:
- Mass = kg
- Time = s
- Temperature = K

Thus, in S.I. units, Stefan's constant is expressed as:
\[ \sigma_{SI} = \frac{kg}{s^3 \cdot K^4} \]

- In C.G.S. units:
- Mass = g (1 kg = 1000 g)
- Time = s
- Temperature = K

Thus, in C.G.S. units, Stefan's constant is expressed as:
\[ \sigma_{CGS} = \frac{g}{s^3 \cdot K^4} \]

### Step 4: Find the ratio of S.I. to C.G.S. units
To find the ratio of S.I. to C.G.S. units for Stefan's constant:
\[ \text{Ratio} = \frac{\sigma_{SI}}{\sigma_{CGS}} = \frac{\frac{kg}{s^3 \cdot K^4}}{\frac{g}{s^3 \cdot K^4}} \]

Since \( K^4 \) and \( s^3 \) cancel out, we have:
\[ \text{Ratio} = \frac{kg}{g} \]

Now, substituting the conversion factor \( 1 kg = 1000 g \):
\[ \text{Ratio} = \frac{1000 g}{g} = 1000 \]

### Final Answer
The ratio of S.I. to C.G.S. units for Stefan's constant is **1000**.

---