A charges Q is placed at each of the two opposite corners of a square. A charge q is placed to each of the other two corners. If the resultant force on each charge q is zero, then

Options

A. `q=sqrt2Q`
B. `q=-sqrt2Q`
C. `q=2sqrt2Q`
D. `q=-2sqrt2Q`

Explanation

To solve the problem, we need to analyze the forces acting on the charges placed at the corners of a square. Let's break down the solution step by step.

### Step 1: Understand the Configuration
We have a square with side length \( l \). At two opposite corners, we place charges \( Q \), and at the other two opposite corners, we place charges \( q \).

### Step 2: Identify Forces on Charge \( q \)
For each charge \( q \), we need to consider the forces acting on it due to the two charges \( Q \) and the other charge \( q \).

1. The charge \( q \) at one corner experiences repulsive forces from the two charges \( Q \).
2. The charge \( q \) also experiences a repulsive force from the other charge \( q \) at the opposite corner.

### Step 3: Calculate the Forces
1. The force \( F \) between the charge \( Q \) and the charge \( q \) is given by Coulomb's law:
\[ F = k \frac{|Q \cdot q|}{l^2} \]
where \( k \) is Coulomb's constant.

2. The force between the two charges \( q \) (which are at a distance of \( \sqrt{2}l \) from each other) is:
\[ F' = k \frac{|q \cdot q|}{(\sqrt{2}l)^2} = k \frac{q^2}{2l^2} \]

### Step 4: Set Up the Conditions for Equilibrium
For the resultant force on each charge \( q \) to be zero, the forces acting on it must balance. Thus, we need to equate the forces:
\[ F + F' = 0 \]

### Step 5: Analyze the Directions of Forces
1. The forces due to \( Q \) are repulsive and directed away from \( Q \).
2. The force due to the other charge \( q \) is also repulsive.

### Step 6: Determine the Nature of Charge \( q \)
To achieve equilibrium, the charge \( q \) must be negative. If \( q \) is negative, the force due to \( Q \) will be attractive, which can balance the repulsive force from the other charge \( q \).

### Step 7: Equate Forces
From the previous calculations, we have:
\[ k \frac{|Q \cdot q|}{l^2} = k \frac{q^2}{2l^2} \]
Cancelling \( k \) and \( l^2 \) from both sides, we get:
\[ |Q \cdot q| = \frac{q^2}{2} \]

### Step 8: Solve for \( q \)
Assuming \( q \) is negative, we can write:
\[ |q| = -q \]
Substituting this into the equation gives:
\[ |Q \cdot (-q)| = \frac{(-q)^2}{2} \]
This simplifies to:
\[ |Q| \cdot |q| = \frac{q^2}{2} \]
Rearranging gives:
\[ |q| = \frac{2|Q|}{|q|} \]
Thus, we find:
\[ q = -2\sqrt{2}Q \]

### Conclusion
The charge \( q \) must be negative, and its magnitude must be equal to \( 2\sqrt{2}Q \).

### Final Answer
The charge \( q \) is:
\[ q = -2\sqrt{2}Q \]