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. 4q
C. `-(q)/(2)`
D. `- 2 q`

Answer

To solve the problem, we need to find the value of charge \( Q \) such that the net force on charge \( q \) is zero. The charges are positioned as follows:

- Charge \( q \) is at position \( 0 \)
- Charge \( Q \) is at position \( \frac{L}{2} \)
- Charge \( -4q \) is at position \( L \)

### Step-by-Step Solution:

1. **Identify Forces Acting on Charge \( q \)**:
- The force on charge \( q \) due to charge \( Q \) (at \( \frac{L}{2} \)) is \( F_2 \).
- The force on charge \( q \) due to charge \( -4q \) (at \( L \)) is \( F_1 \).

2. **Calculate the Force \( F_1 \) due to Charge \( -4q \)**:
- The distance between charge \( q \) and charge \( -4q \) is \( L \).
- The force \( F_1 \) is given by Coulomb's law:
\[ F_1 = k \frac{|q \cdot (-4q)|}{L^2} = k \frac{4q^2}{L^2} \]
- Since \( -4q \) is negative, it attracts \( q \) towards it, so \( F_1 \) acts to the right.

3. **Calculate the Force \( F_2 \) due to Charge \( Q \)**:
- The distance between charge \( q \) and charge \( Q \) is \( \frac{L}{2} \).
- The force \( F_2 \) is also given by Coulomb's law:
\[ F_2 = k \frac{|q \cdot Q|}{\left(\frac{L}{2}\right)^2} = k \frac{4q|Q|}{L^2} \]
- The direction of \( F_2 \) depends on the sign of \( Q \). If \( Q \) is positive, \( F_2 \) acts to the right (repulsive). If \( Q \) is negative, \( F_2 \) acts to the left (attractive).

4. **Set the Forces Equal for Equilibrium**:
- For the net force on \( q \) to be zero, the magnitudes of the forces must be equal:
\[ F_1 = F_2 \]
\[ k \frac{4q^2}{L^2} = k \frac{4q|Q|}{L^2} \]

5. **Simplify the Equation**:
- Cancel \( k \) and \( L^2 \) from both sides:
\[ 4q^2 = 4q|Q| \]
- Divide both sides by \( 4q \) (assuming \( q \neq 0 \)):
\[ q = |Q| \]

6. **Determine the Value of Charge \( Q \)**:
- Since \( Q \) can be positive or negative, we conclude:
\[ Q = q \quad \text{or} \quad Q = -q \]

### Conclusion:
To make the net force on charge \( q \) zero, the charge \( Q \) must be equal to \( q \) (since we are looking for the magnitude).

### Final Answer:
\[ Q = q \]