The ratio of SI units to CGS units of energy

Options

A. `10^(5)`
B. `10 ^(6)`
C. `10^(7)`
D. `10^(-7)`

Answer

To find the ratio of SI units to CGS units of energy, we will follow these steps:

### Step 1: Identify the SI and CGS units of energy
- The SI unit of energy is the Joule (J).
- The CGS unit of energy is the Erg.

### Step 2: Express the SI unit in base units
- 1 Joule (J) is defined as 1 Newton (N) × 1 meter (m).
- 1 Newton is defined as 1 kg × m/s².
- Therefore, 1 Joule can be expressed as:
\[ 1 J = 1 \, \text{kg} \cdot \text{m}^2/\text{s}^2 \]

### Step 3: Express the CGS unit in base units
- 1 Erg is defined as 1 dyne × 1 centimeter (cm).
- 1 dyne is defined as 1 g × cm/s².
- Therefore, 1 Erg can be expressed as:
\[ 1 \, \text{Erg} = 1 \, \text{g} \cdot \text{cm}^2/\text{s}^2 \]

### Step 4: Write the ratio of SI unit to CGS unit
- Now we write the ratio of Joules to Ergs:
\[ \text{Ratio} = \frac{1 \, \text{J}}{1 \, \text{Erg}} = \frac{1 \, \text{kg} \cdot \text{m}^2/\text{s}^2}{1 \, \text{g} \cdot \text{cm}^2/\text{s}^2} \]

### Step 5: Convert units to the same system
- We need to convert kg to grams and meters to centimeters:
- 1 kg = 1000 g
- 1 m = 100 cm
- Substituting these values into the ratio gives:
\[ \text{Ratio} = \frac{1000 \, \text{g} \cdot (100 \, \text{cm})^2}{1 \, \text{g} \cdot \text{cm}^2} \]

### Step 6: Simplify the ratio
- Simplifying the expression:
\[ \text{Ratio} = \frac{1000 \, \text{g} \cdot 10000 \, \text{cm}^2}{1 \, \text{g} \cdot \text{cm}^2} = \frac{1000 \cdot 10000}{1} = 10^3 \cdot 10^4 = 10^7 \]

### Conclusion
- Therefore, the ratio of SI units to CGS units of energy is:
\[ \text{Ratio} = 10^7 \]

### Final Answer
- The correct option is 10 to the power 7.

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