State the relation between the erg and the joule.

Answer

To find the relation between the erg and the joule, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Joule**:
- The joule (J) is the SI unit of energy. It is defined as the energy transferred when a force of one newton is applied over a distance of one meter.
- Mathematically, 1 Joule = 1 Newton × 1 meter.

2. **Understand the Definition of Erg**:
- The erg is a unit of energy in the centimeter-gram-second (CGS) system. It is defined as the energy transferred when a force of one dyne is applied over a distance of one centimeter.
- Mathematically, 1 Erg = 1 dyne × 1 centimeter.

3. **Convert Newton to Dyne**:
- We know that 1 Newton is equal to \(10^5\) dynes.
- Therefore, we can express 1 Joule in terms of dynes:
\[ 1 \text{ Joule} = (10^5 \text{ dynes}) \times (1 \text{ meter}) \]

4. **Convert Meter to Centimeter**:
- We also know that 1 meter is equal to \(10^2\) centimeters.
- Substituting this into the equation gives:
\[ 1 \text{ Joule} = (10^5 \text{ dynes}) \times (10^2 \text{ centimeters}) \]

5. **Combine the Units**:
- Now, we can multiply the two values:
\[ 1 \text{ Joule} = 10^5 \times 10^2 \text{ dyne-centimeters} = 10^7 \text{ dyne-centimeters} \]

6. **Relate Joules to Ergs**:
- Since 1 Erg is defined as 1 dyne-centimeter, we can conclude:
\[ 1 \text{ Joule} = 10^7 \text{ Ergs} \]

### Final Relation:
Thus, the relation between erg and joule is:
\[ 1 \text{ Joule} = 10^7 \text{ Ergs} \]