Four equal charges each `16 muC` are placed on four corners of a square of side 0.4m. Calculate force on q is zero, how are Q and q related ?

Explanation

To solve the problem, we need to analyze the forces acting on the charges placed at the corners of the square.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have four equal charges \( Q = 16 \, \mu C \) placed at the corners of a square with side length \( a = 0.4 \, m \).

2. **Identifying the Forces**:
- Each charge experiences a force due to the other three charges. We need to find the net force acting on one of the charges (let's say charge at corner A).

3. **Calculating the Forces**:
- The force between two point charges is given by Coulomb's law:
\[ F = k \frac{|Q_1 Q_2|}{r^2} \]
where \( k = 8.99 \times 10^9 \, N m^2/C^2 \) is Coulomb's constant, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

4. **Distance Between Charges**:
- The distance between adjacent charges (e.g., A and B) is \( a = 0.4 \, m \).
- The distance between diagonal charges (e.g., A and C) can be calculated using the Pythagorean theorem:
\[ r_{AC} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} = 0.4\sqrt{2} \, m \]

5. **Calculating the Forces on Charge A**:
- The force due to charge B (adjacent):
\[ F_{AB} = k \frac{Q^2}{a^2} = 8.99 \times 10^9 \frac{(16 \times 10^{-6})^2}{(0.4)^2} \]
- The force due to charge D (adjacent):
\[ F_{AD} = k \frac{Q^2}{a^2} = F_{AB} \]
- The force due to charge C (diagonal):
\[ F_{AC} = k \frac{Q^2}{(0.4\sqrt{2})^2} = k \frac{Q^2}{0.32} \]

6. **Direction of Forces**:
- The forces \( F_{AB} \) and \( F_{AD} \) will act along the sides of the square, while \( F_{AC} \) will act diagonally towards charge C.

7. **Setting Up the Equations**:
- For the net force on charge A to be zero, the horizontal and vertical components of the forces must balance out.
- The horizontal components of \( F_{AB} \) and \( F_{AD} \) must equal the horizontal component of \( F_{AC} \).

8. **Finding the Relationship**:
- By symmetry and the requirement for equilibrium, we can derive that for the net force to be zero, the charge \( q \) placed at the center of the square must satisfy:
\[ F_{AB} + F_{AD} = 2 F_{AC} \]
- This leads to a relationship between \( Q \) and \( q \).

9. **Conclusion**:
- The relationship can be derived from the balance of forces, leading to the conclusion that \( q \) must be such that it counteracts the resultant force from the four corner charges.