A charge Q is divided into two charge q and Q-q. The value of q such that the force between them is maximum is

Options

A. Q
B. `(3Q)/4`
C. `Q/2`
D. `Q/3`

Answer

To solve the problem of finding the value of charge \( q \) such that the force between two charges \( q \) and \( Q - q \) is maximized, we can follow these steps:

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

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

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

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

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

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

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

This gives us:

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

### Step 4: Set the derivative equal to zero
Setting the derivative equal to zero to find the critical points:

\[ k \frac{(Q - 2q)}{r^2} = 0 \]

Since \( k \) and \( r^2 \) are constants and cannot be zero, we can simplify this to:

\[ Q - 2q = 0 \]

### Step 5: Solve for \( q \)
Solving for \( q \):

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

### Conclusion
Thus, the value of charge \( q \) that maximizes the force between the two charges is:

\[ \boxed{\frac{Q}{2}} \]