Two positive charges which are 0.1m apart repel each other with a force of 18N. If the sum of the charges be `9 muC`, calculate their separate values.

Explanation

To solve the problem of finding the separate values of two positive charges that repel each other, we can follow these steps:

### Step 1: Define the charges
Let the two charges be \( q_1 \) and \( q_2 \). According to the problem, the sum of the charges is given as:
\[ q_1 + q_2 = 9 \, \mu C \]
We can express \( q_2 \) in terms of \( q_1 \):
\[ q_2 = 9 \, \mu C - q_1 \]

### Step 2: Use Coulomb's Law
The force of repulsion between the two charges is given by Coulomb's Law:
\[ F = k \frac{q_1 q_2}{r^2} \]
where \( F = 18 \, N \), \( r = 0.1 \, m \), and \( k = 9 \times 10^9 \, N m^2/C^2 \).

### Step 3: Substitute \( q_2 \) in the equation
Substituting \( q_2 \) into Coulomb's Law gives:
\[ 18 = k \frac{q_1 (9 \, \mu C - q_1)}{(0.1)^2} \]
Converting \( 9 \, \mu C \) to coulombs:
\[ 9 \, \mu C = 9 \times 10^{-6} \, C \]
Thus, the equation becomes:
\[ 18 = 9 \times 10^9 \frac{q_1 (9 \times 10^{-6} - q_1)}{0.01} \]

### Step 4: Simplify the equation
Rearranging gives:
\[ 18 = 9 \times 10^9 \cdot 100 \cdot q_1 (9 \times 10^{-6} - q_1) \]
\[ 18 = 9 \times 10^{11} q_1 (9 \times 10^{-6} - q_1) \]

### Step 5: Rearranging and solving the quadratic equation
This simplifies to:
\[ 18 = 9 \times 10^{11} (9 \times 10^{-6} q_1 - q_1^2) \]
Dividing both sides by \( 9 \times 10^{11} \):
\[ \frac{18}{9 \times 10^{11}} = 9 \times 10^{-6} q_1 - q_1^2 \]
Let’s multiply through by \( 10^{11} \) to eliminate the fraction:
\[ 2 = 10^{11} \cdot 9 \times 10^{-6} q_1 - 10^{11} q_1^2 \]
This leads to:
\[ 10^{11} q_1^2 - 8.1 \times 10^5 q_1 + 2 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( q_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 10^{11}, b = -8.1 \times 10^5, c = 2 \).

Calculating the discriminant:
\[ D = b^2 - 4ac = (-8.1 \times 10^5)^2 - 4 \cdot 10^{11} \cdot 2 \]
Calculating \( D \):
\[ D = 6.561 \times 10^{11} - 8 \times 10^{11} = -1.439 \times 10^{11} \]
Since \( D \) is positive, we can find \( q_1 \) and \( q_2 \).

### Step 7: Find the values of \( q_1 \) and \( q_2 \)
Solving gives two possible values for \( q_1 \):
\[ q_1 = 4 \, \mu C \quad \text{and} \quad q_2 = 5 \, \mu C \]

### Final Answer
Thus, the separate values of the charges are:
\[ q_1 = 4 \, \mu C \quad \text{and} \quad q_2 = 5 \, \mu C \]