To find the energy equivalent of a mass of 10 micrograms, we will use Einstein's mass-energy equivalence formula, which is given by:
\[ E = mc^2 \]
Where:
- \( E \) is the energy equivalent,
- \( m \) is the mass in kilograms,
- \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).
### Step-by-Step Solution:
**Step 1: Convert mass from micrograms to kilograms.**
- We know that \( 1 \, \text{microgram} = 10^{-6} \, \text{grams} \).
- Therefore, \( 10 \, \text{micrograms} = 10 \times 10^{-6} \, \text{grams} = 10^{-5} \, \text{grams} \).
- Now, convert grams to kilograms:
\[ 10^{-5} \, \text{grams} = 10^{-5} \times 10^{-3} \, \text{kg} = 10^{-8} \, \text{kg} \]
**Step 2: Use the speed of light in the equation.**
- The speed of light \( c \) is \( 3 \times 10^8 \, \text{m/s} \).
**Step 3: Substitute the values into the equation.**
- Now we can substitute \( m = 10^{-8} \, \text{kg} \) and \( c = 3 \times 10^8 \, \text{m/s} \) into the equation:
\[ E = (10^{-8} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2 \]
**Step 4: Calculate \( c^2 \).**
- First, calculate \( (3 \times 10^8)^2 \):
\[ (3 \times 10^8)^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \]
**Step 5: Calculate the energy equivalent.**
- Now substitute \( c^2 \) back into the equation:
\[ E = 10^{-8} \times 9 \times 10^{16} \]
\[ E = 9 \times 10^{8} \, \text{Joules} \]
### Final Answer:
The energy equivalent to a mass of 10 micrograms is \( 9 \times 10^{8} \, \text{Joules} \).
---
