A certain charge Q is divided into two parts q and `Q-q`, which are then separated by a certain distance. What must q be in terms of Q to maximum the electrostatic repulsion between the two charges?

Explanation

To solve the problem of maximizing the electrostatic repulsion between two charges \( q \) and \( Q - q \), we can follow these steps:

### Step 1: Write the expression for electrostatic force
The electrostatic force \( F \) between two point charges \( q \) and \( Q - q \) separated by a distance \( R \) is given by Coulomb's law:

\[ F = k \frac{q (Q - q)}{R^2} \]

where \( k \) is Coulomb's constant.

### Step 2: Expand the expression
We can expand the expression for \( F \):

\[ F = k \frac{(qQ - q^2)}{R^2} \]

### 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 \( q \):

\[ \frac{dF}{dq} = k \frac{d}{dq} \left( \frac{qQ - q^2}{R^2} \right) \]

Using the quotient rule, we get:

\[ \frac{dF}{dq} = k \frac{(Q - 2q)}{R^2} \]

### Step 4: Set the derivative equal to zero
To find the maximum, we set the derivative equal to zero:

\[ Q - 2q = 0 \]

### Step 5: Solve for \( q \)
Now, we can solve for \( q \):

\[ 2q = Q \implies q = \frac{Q}{2} \]

### Conclusion
Thus, to maximize the electrostatic repulsion between the two charges, the charge \( q \) must be:

\[ q = \frac{Q}{2} \]