A total charge of `20 muC` is divided into two parts and placed at some distance apart. If the charges experience maximum coulombian repulsion, the charge should be :

Options

A. `5 muC, 15 muC`
B. `10 muC, 10 muC`
C. `12 muC, 8 muC`
D. `40/3 muC, 20/3 muC`

Answer

To solve the problem of dividing a total charge of `20 µC` into two parts such that the charges experience maximum Coulombian repulsion, we can follow these steps:

### Step 1: Define the Charges
Let the total charge `Q_total = 20 µC` be divided into two parts:
- Charge 1: `Q`
- Charge 2: `Q' = 20 µC - Q`

### Step 2: Write the Expression for Coulomb's Force
The force of repulsion between the two charges can be expressed using Coulomb's Law:
\[ F = k \frac{Q \cdot Q'}{r^2} \]
Where:
- \( k \) is Coulomb's constant,
- \( r \) is the distance between the charges.

Substituting \( Q' \):
\[ F = k \frac{Q \cdot (20 - Q)}{r^2} \]

### Step 3: Differentiate the Force with Respect to Q
To find the value of \( Q \) that maximizes the force, we differentiate \( F \) with respect to \( Q \):
\[ \frac{dF}{dQ} = k \frac{(20 - Q) - Q}{r^2} \]
\[ \frac{dF}{dQ} = k \frac{20 - 2Q}{r^2} \]

### Step 4: Set the Derivative Equal to Zero
To find the maximum force, set the derivative equal to zero:
\[ 20 - 2Q = 0 \]

### Step 5: Solve for Q
Solving for \( Q \):
\[ 2Q = 20 \]
\[ Q = 10 \, \mu C \]

### Step 6: Find the Other Charge
Now, substituting back to find the other charge:
\[ Q' = 20 \, \mu C - Q = 20 \, \mu C - 10 \, \mu C = 10 \, \mu C \]

### Conclusion
Thus, the charges should be divided into:
- Charge 1: `10 µC`
- Charge 2: `10 µC`

The final answer is that the charges are `10 µC` and `10 µC`.