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?

Options

A. `(Q//q)=(4//1)`
B. `(Q//q)=(2//1)`
C. `(Q//q)=(3//4)`
D. `(Q//q)=(5//1)`

Answer

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 the electrostatic force
The electrostatic force \( F \) between two charges \( q \) and \( Q - q \) separated by a distance \( d \) is given by Coulomb's law:
\[ F = k \frac{q (Q - q)}{d^2} \]
where \( k \) is Coulomb's constant.

### Step 2: Simplify the expression
We can rewrite the expression for the force:
\[ F = k \frac{Qq - q^2}{d^2} \]
Since \( k \) and \( d^2 \) are constants, we can focus on maximizing the expression \( Qq - q^2 \).

### 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:
\[ \frac{dF}{dq} = \frac{d}{dq}(Qq - q^2) = Q - 2q \]
Setting the derivative equal to zero gives:
\[ Q - 2q = 0 \]

### Step 4: Solve for \( q \)
From the equation \( Q - 2q = 0 \), we can solve for \( q \):
\[ 2q = Q \implies q = \frac{Q}{2} \]

### Step 5: Conclusion
Thus, to maximize the electrostatic repulsion between the two charges, the charge \( q \) must be:
\[ q = \frac{Q}{2} \]