The force between two charges separated by a distance 1m is 1.8N. The charges are in the ratio `1: 2` then the charges are

Options

A. `5mu C, 5mu C`
B. `5mu C, 10 mu C`
C. `1mu C, 10 mu C`
D. `10 mu C, 20 mu C`

Answer

To solve the problem, we need to find the values of two charges \( q_1 \) and \( q_2 \) given that they are in the ratio \( 1:2 \) and the force between them is \( 1.8 \, \text{N} \) when they are \( 1 \, \text{m} \) apart.

### Step-by-step Solution:

1. **Identify the Given Values:**
- Distance \( r = 1 \, \text{m} \)
- Force \( F = 1.8 \, \text{N} \)
- Ratio of charges \( q_1 : q_2 = 1 : 2 \)

2. **Express Charges in Terms of a Variable:**
- Let \( q_1 = q \)
- Then \( q_2 = 2q \)

3. **Use Coulomb's Law:**
- The formula for the force between two charges is given by:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
- Substituting the values we have:
\[ 1.8 = \frac{1}{4 \pi \epsilon_0} \frac{q \cdot 2q}{(1)^2} \]
- This simplifies to:
\[ 1.8 = \frac{1}{4 \pi \epsilon_0} \cdot 2q^2 \]

4. **Substitute the Value of \( \epsilon_0 \):**
- The value of \( \frac{1}{4 \pi \epsilon_0} \) is approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
- Thus, we have:
\[ 1.8 = 9 \times 10^9 \cdot 2q^2 \]

5. **Rearranging the Equation:**
- Rearranging gives:
\[ 2q^2 = \frac{1.8}{9 \times 10^9} \]
- Simplifying further:
\[ q^2 = \frac{1.8}{18 \times 10^9} = \frac{1}{10 \times 10^9} = \frac{1}{10^{10}} \]

6. **Calculate \( q \):**
- Taking the square root:
\[ q = \sqrt{\frac{1}{10^{10}}} = \frac{1}{10^5} = 10^{-5} \, \text{C} \]

7. **Convert to Microcoulombs:**
- Since \( 1 \, \text{C} = 10^6 \, \mu\text{C} \):
\[ q = 10^{-5} \, \text{C} = 10 \, \mu\text{C} \]

8. **Calculate \( q_2 \):**
- Since \( q_2 = 2q \):
\[ q_2 = 2 \cdot 10 \, \mu\text{C} = 20 \, \mu\text{C} \]

9. **Final Values of Charges:**
- Thus, the charges are:
- \( q_1 = 10 \, \mu\text{C} \)
- \( q_2 = 20 \, \mu\text{C} \)

### Conclusion:
The charges are \( 10 \, \mu\text{C} \) and \( 20 \, \mu\text{C} \).