Two small charged spheres contain charge `+q_(1) and +q_(2)` respectively. A charge dq is removed from sphere carrying charge `q_(1)` and is transferred to the other. Find charge on each sphere for maximum electric force between them.

Explanation

To find the charge on each sphere for maximum electric force between them after transferring a charge \( dq \) from sphere 1 (with charge \( q_1 \)) to sphere 2 (with charge \( q_2 \)), we can follow these steps:

### Step 1: Define the Charges After Transfer
After transferring a charge \( dq \) from sphere 1 to sphere 2, the new charges on the spheres will be:
- Charge on sphere 1: \( q_1' = q_1 - dq \)
- Charge on sphere 2: \( q_2' = q_2 + dq \)

### Step 2: Write the Expression for Electric Force
The electric force \( F \) between the two spheres can be expressed using Coulomb's Law:
\[ F = k \frac{q_1' q_2'}{r^2} \]
Substituting the new charges:
\[ F = k \frac{(q_1 - dq)(q_2 + dq)}{r^2} \]

### Step 3: Expand the Force Expression
Expanding the expression:
\[ F = k \frac{(q_1 q_2 + q_1 dq - q_2 dq - dq^2)}{r^2} \]
This can be simplified to:
\[ F = k \frac{q_1 q_2}{r^2} + k \frac{(q_1 - q_2)dq}{r^2} - k \frac{dq^2}{r^2} \]

### Step 4: Differentiate the Force with Respect to \( dq \)
To find the value of \( dq \) that maximizes the force, we differentiate \( F \) with respect to \( dq \):
\[ \frac{dF}{dq} = k \frac{(q_1 - q_2)}{r^2} - k \frac{2dq}{r^2} \]
Setting the derivative equal to zero for maximization:
\[ 0 = \frac{(q_1 - q_2)}{r^2} - \frac{2dq}{r^2} \]
This simplifies to:
\[ q_1 - q_2 = 2dq \]

### Step 5: Solve for \( dq \)
From the equation \( q_1 - q_2 = 2dq \), we can solve for \( dq \):
\[ dq = \frac{1}{2}(q_1 - q_2) \]

### Step 6: Find the Charges on Each Sphere
Now, substitute \( dq \) back into the expressions for the charges on each sphere:
- Charge on sphere 1:
\[ q_1' = q_1 - dq = q_1 - \frac{1}{2}(q_1 - q_2) = \frac{1}{2}(q_1 + q_2) \]
- Charge on sphere 2:
\[ q_2' = q_2 + dq = q_2 + \frac{1}{2}(q_1 - q_2) = \frac{1}{2}(q_1 + q_2) \]

### Conclusion
Thus, for maximum electric force between the two spheres, the charges on each sphere will be:
\[ q_1' = \frac{1}{2}(q_1 + q_2) \quad \text{and} \quad q_2' = \frac{1}{2}(q_1 + q_2) \]