Understanding the Conversion: Erg to Joule

This question asks for the value of one erg when expressed in Joules. Both units measure energy or work, but they belong to different unit systems.

What are Ergs and Joules?

• Erg: This is the unit of energy in the CGS (centimeter-gram-second) system. It represents the work done when a force of one dyne acts over a distance of one centimeter.
• Joule: This is the standard unit of energy in the SI (International System of Units). It represents the work done when a force of one Newton acts over a distance of one meter.

Calculating the Conversion Factor

To find the relationship between these two units, we can express the Joule in terms of fundamental CGS units.

First, let's recall the definitions:

• $1 \, \text{erg} = 1 \, \text{dyne} \cdot \text{cm}$
• $1 \, \text{Joule} = 1 \, \text{N} \cdot \text{m}$

Now, let's use the conversion factors between the SI and CGS units:

• $1 \, \text{Newton} \, (\text{N}) = 10^5 \, \text{dyne}$
• $1 \, \text{meter} \, (\text{m}) = 100 \, \text{centimeters} \, (\text{cm})$

Substitute these into the definition of a Joule:

$1 \, \text{Joule} = 1 \, \text{N} \times 1 \, \text{m}$

$1 \, \text{Joule} = (10^5 \, \text{dyne}) \times (100 \, \text{cm})$

$1 \, \text{Joule} = 10^5 \times 10^2 \, \text{dyne} \cdot \text{cm}$

$1 \, \text{Joule} = 10^7 \, \text{dyne} \cdot \text{cm}$

Since $1 \, \text{erg} = 1 \, \text{dyne} \cdot \text{cm}$, we can see that:

$1 \, \text{Joule} = 10^7 \, \text{erg}$

Resulting Equivalence

The relationship shows that one Joule is equal to $10^7$ ergs.

To find out how many Joules are in one erg, we can rearrange this equation:

$1 \, \text{erg} = \frac{1}{10^7} \, \text{Joule}$

Using the rules of exponents, we can write $\frac{1}{10^7}$ as $10^{-7}$.

Therefore:

$1 \, \text{erg} = 10^{-7} \, \text{Joule}$

Below is a summary table showing the relationship:

So, one erg is equivalent to $10^{-7}$ Joules.