SI unit and CGS unit of certain quantity vary by `10^3` times.That quantity is

Options

A. Boltzmann constant
B. Gravitational constant
C. Planck's constant
D. Angular momentum

Answer

To solve the problem of identifying which physical quantity has SI and CGS units that vary by \(10^3\) times, we will analyze the given quantities: Boltzmann constant (kB), gravitational constant (G), Planck's constant (h), and angular momentum (L). We will derive their dimensional formulas and calculate the variation between their SI and CGS units.

### Step-by-Step Solution:

1. **Identify the quantities and their dimensions**:
- **Boltzmann constant (kB)**: The dimensional formula is \( [M^1 L^2 T^{-2} K^{-1}] \).
- **Gravitational constant (G)**: The dimensional formula is \( [M^{-1} L^3 T^{-2}] \).
- **Planck's constant (h)**: The dimensional formula is \( [M^1 L^2 T^{-1}] \).
- **Angular momentum (L)**: The dimensional formula is \( [M^1 L^2 T^{-1}] \).

2. **Identify the units in SI and CGS**:
- **Mass**:
- SI: kilogram (kg)
- CGS: gram (g)
- \(1 \text{ kg} = 10^3 \text{ g}\)
- **Length**:
- SI: meter (m)
- CGS: centimeter (cm)
- \(1 \text{ m} = 10^2 \text{ cm}\)
- **Time**:
- Both SI and CGS use seconds (s).
- **Temperature**:
- Both SI and CGS use Kelvin (K).

3. **Calculate the conversion factors for each quantity**:
- For **Boltzmann constant (kB)**:
\[ \text{SI unit} = kg \cdot m^2 \cdot K^{-1} \quad \text{and} \quad \text{CGS unit} = g \cdot cm^2 \cdot K^{-1} \]
\[ \text{Conversion} = 10^3 \cdot (10^2)^2 = 10^3 \cdot 10^4 = 10^7 \]

- For **Gravitational constant (G)**:
\[ \text{SI unit} = \frac{kg \cdot m^3}{s^2} \quad \text{and} \quad \text{CGS unit} = \frac{g \cdot cm^3}{s^2} \]
\[ \text{Conversion} = \frac{1}{10^3} \cdot (10^2)^3 = 10^{-3} \cdot 10^6 = 10^3 \]

- For **Planck's constant (h)**:
\[ \text{SI unit} = kg \cdot m^2 \cdot s^{-1} \quad \text{and} \quad \text{CGS unit} = g \cdot cm^2 \cdot s^{-1} \]
\[ \text{Conversion} = 10^3 \cdot (10^2)^2 = 10^3 \cdot 10^4 = 10^7 \]

- For **Angular momentum (L)**:
\[ \text{SI unit} = kg \cdot m^2 \cdot s^{-1} \quad \text{and} \quad \text{CGS unit} = g \cdot cm^2 \cdot s^{-1} \]
\[ \text{Conversion} = 10^3 \cdot (10^2)^2 = 10^3 \cdot 10^4 = 10^7 \]

4. **Conclusion**:
- The only quantity that has SI and CGS units varying by \(10^3\) times is the **gravitational constant (G)**.

### Final Answer:
The quantity whose SI unit and CGS unit vary by \(10^3\) times is the **gravitational constant (G)**.