The ratio of SI unit of CGS unit of a plysical constant is `10^(7)`. That costant is

Options

A. Universal gas constant
B. Universal gravitational constant
C. Magnetic induction c
D. Impulse

Answer

To solve the problem, we need to find a physical constant whose ratio of SI unit to CGS unit is \(10^7\). Let's analyze the options provided one by one.

### Step 1: Analyze the Universal Gas Constant (R)

1. **SI Unit of R**: The SI unit of the universal gas constant \(R\) is Joules per mole per Kelvin, which can be expressed as:
\[ \text{SI unit of } R = \frac{\text{J}}{\text{mol} \cdot \text{K}} \]

2. **Convert Joules to CGS**: In CGS, the unit of energy is erg. The conversion is:
\[ 1 \text{ Joule} = 10^7 \text{ erg} \]

3. **CGS Unit of R**: The CGS unit of the gas constant can be expressed as:
\[ \text{CGS unit of } R = \frac{10^7 \text{ erg}}{\text{mol} \cdot \text{°C}} \]
(Note: 1 Kelvin = 1 °C in terms of temperature difference)

4. **Ratio of SI to CGS**:
\[ \text{Ratio} = \frac{\frac{\text{J}}{\text{mol} \cdot \text{K}}}{\frac{10^7 \text{ erg}}{\text{mol} \cdot \text{°C}}} = \frac{1}{10^7} \]
This simplifies to:
\[ \text{Ratio} = 10^7 \]

### Step 2: Analyze the Universal Gravitational Constant (G)

1. **SI Unit of G**: The SI unit of the universal gravitational constant \(G\) is:
\[ \text{SI unit of } G = \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \]

2. **Convert Newtons to CGS**: In CGS, the unit of force is dyne. The conversion is:
\[ 1 \text{ N} = 10^5 \text{ dyne} \]

3. **Convert Meters to Centimeters**: The conversion is:
\[ 1 \text{ m} = 10^2 \text{ cm} \]

4. **Convert Kilograms to Grams**: The conversion is:
\[ 1 \text{ kg} = 10^3 \text{ g} \]

5. **CGS Unit of G**: Therefore, the CGS unit of \(G\) can be expressed as:
\[ \text{CGS unit of } G = \frac{10^5 \text{ dyne} \cdot (10^2 \text{ cm})^2}{(10^3 \text{ g})^2} = \frac{10^5 \cdot 10^4 \text{ dyne} \cdot \text{cm}^2}{10^6 \text{ g}^2} = \frac{10^9 \text{ dyne} \cdot \text{cm}^2}{10^6 \text{ g}^2} = 10^3 \frac{\text{dyne} \cdot \text{cm}^2}{\text{g}^2} \]

6. **Ratio of SI to CGS**:
\[ \text{Ratio} = \frac{\frac{\text{N} \cdot \text{m}^2}{\text{kg}^2}}{\frac{10^3 \text{ dyne} \cdot \text{cm}^2}{\text{g}^2}} \neq 10^7 \]

### Step 3: Analyze Magnetic Induction (B)

1. **SI Unit of B**: The SI unit of magnetic induction \(B\) is Tesla.

2. **Convert Tesla to CGS**: The CGS unit of magnetic induction is Gauss. The conversion is:
\[ 1 \text{ T} = 10^4 \text{ G} \]

3. **Ratio of SI to CGS**:
\[ \text{Ratio} = \frac{1 \text{ T}}{10^4 \text{ G}} = 10^{-4} \]
This does not equal \(10^7\).

### Step 4: Analyze Impulse (J)

1. **SI Unit of J**: The SI unit of impulse \(J\) is Newton-second (N·s).

2. **Convert Newtons to CGS**: As before, \(1 \text{ N} = 10^5 \text{ dyne}\).

3. **CGS Unit of J**: The CGS unit of impulse can be expressed as:
\[ \text{CGS unit of } J = 10^5 \text{ dyne} \cdot \text{s} \]

4. **Ratio of SI to CGS**:
\[ \text{Ratio} = \frac{1 \text{ N·s}}{10^5 \text{ dyne·s}} = 10^{-5} \]
This does not equal \(10^7\).

### Conclusion

After analyzing all the options, we find that the universal gas constant \(R\) is the only physical constant whose ratio of SI unit to CGS unit is \(10^7\).

**Final Answer**: The physical constant is the **Universal Gas Constant (R)**.

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