A certain charge Q is divided into two parts q and `Q-q`, wheich are then separated by a cetain 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 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, \( k = \frac{1}{4 \pi \epsilon_0} \).

### Step 2: Rewrite the force equation
Substituting \( k \) into the equation, we have:

\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q (Q - q)}{r^2} \]

### Step 3: Differentiate the force with respect to \( q \)
To find the value of \( q \) that maximizes the force, we need to differentiate \( F \) with respect to \( q \) and set the derivative equal to zero:

\[ \frac{dF}{dq} = \frac{1}{4 \pi \epsilon_0 r^2} \left( Q - 2q \right) = 0 \]

### Step 4: Solve for \( q \)
Setting the derivative equal to zero gives:

\[ Q - 2q = 0 \]

Solving for \( q \):

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

### Step 5: Conclusion
Thus, to maximize the electrostatic repulsion between the two charges, the value of \( q \) must be:

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