Three charge `q , Q` and `-4q` are placed in a straight line , line of length `L`at points distant `0 ,L//2` and `L` respectively from one end. In order to make the net force on `q` zero, the charge `Q` must be equal to

Options

A. `-q`
B. `-2q`
C. `-(q)/(2)`
D. `q`

Explanation

To find the value of charge \( Q \) such that the net force on charge \( q \) is zero, we can follow these steps:

### Step 1: Identify the positions of the charges
- Charge \( q \) is located at position \( 0 \).
- Charge \( Q \) is located at position \( \frac{L}{2} \).
- Charge \( -4q \) is located at position \( L \).

### Step 2: Determine the forces acting on charge \( q \)
- The force on charge \( q \) due to charge \( Q \) (located at \( \frac{L}{2} \)) will be attractive if \( Q \) is negative and repulsive if \( Q \) is positive.
- The force on charge \( q \) due to charge \( -4q \) (located at \( L \)) will always be attractive since it is negative.

### Step 3: Calculate the force between \( q \) and \( Q \)
The force \( F_{qQ} \) between charges \( q \) and \( Q \) is given by Coulomb's law:
\[ F_{qQ} = \frac{1}{4 \pi \epsilon_0} \frac{|q \cdot Q|}{\left(\frac{L}{2}\right)^2} = \frac{1}{4 \pi \epsilon_0} \frac{4|q \cdot Q|}{L^2} \]

### Step 4: Calculate the force between \( q \) and \( -4q \)
The force \( F_{q(-4q)} \) between charges \( q \) and \( -4q \) is:
\[ F_{q(-4q)} = \frac{1}{4 \pi \epsilon_0} \frac{|q \cdot (-4q)|}{(L - 0)^2} = \frac{1}{4 \pi \epsilon_0} \frac{4q^2}{L^2} \]

### Step 5: Set the forces equal for equilibrium
For the net force on charge \( q \) to be zero, the magnitudes of the forces must be equal:
\[ F_{qQ} = F_{q(-4q)} \]
Substituting the expressions we derived:
\[ \frac{4|q \cdot Q|}{4 \pi \epsilon_0 L^2} = \frac{4q^2}{4 \pi \epsilon_0 L^2} \]
This simplifies to:
\[ |Q| = q \]

### Step 6: Determine the sign of \( Q \)
Since \( F_{qQ} \) must be attractive (to counteract the attractive force from \( -4q \)), \( Q \) must be negative:
\[ Q = -q \]

### Conclusion
Thus, the charge \( Q \) must be equal to \( -q \) in order to make the net force on \( q \) zero.