Four identical charges i.e. `q` is placed at the corners of a square of side `a`. The charge `Q` that must be placed at the centre of the square such that the whole system of charges in equilibrium is

Options

A. `(q)/(4)`
B. `(q)/(4)[2 sqrt(2) + 1]`
C. `-(q)/(4) [ 2 sqrt(2) + 1]`
D. `-(q)/(4) [3 sqrt(3) + 1]`

Explanation

To find the charge \( Q \) that must be placed at the center of a square with four identical charges \( q \) at its corners for the system to be in equilibrium, we can follow these steps:

### Step 1: Understand the Configuration
We have four identical charges \( q \) placed at the corners of a square with side length \( a \). We need to find the charge \( Q \) that should be placed at the center of the square.

**Hint:** Visualize the square and the positions of the charges. The center of the square is equidistant from all four corners.

### Step 2: Determine the Forces Acting on the Charge at the Center
The charge \( Q \) at the center will experience forces due to each of the four corner charges \( q \). The forces will be directed radially outward if \( Q \) is positive and inward if \( Q \) is negative.

**Hint:** Consider the symmetry of the problem. The forces from the corner charges will have components that can cancel out due to symmetry.

### Step 3: Calculate the Distance from the Center to a Corner
The distance from the center of the square to one of its corners can be calculated using the Pythagorean theorem. The distance \( r \) from the center to a corner is given by:
\[ r = \frac{a}{\sqrt{2}} \]

**Hint:** Remember that the diagonal of the square can be found using the formula \( d = \sqrt{2}a \), and the center divides this diagonal into two equal parts.

### Step 4: Write the Expression for the Force on Charge \( Q \)
The force \( F \) on charge \( Q \) due to one corner charge \( q \) is given by Coulomb's law:
\[ F = k \frac{|qQ|}{r^2} \]
Substituting \( r = \frac{a}{\sqrt{2}} \):
\[ F = k \frac{|qQ|}{\left(\frac{a}{\sqrt{2}}\right)^2} = k \frac{2|qQ|}{a^2} \]

**Hint:** Remember that \( k \) is Coulomb's constant and is necessary for calculating the force.

### Step 5: Analyze the Forces for Equilibrium
For the system to be in equilibrium, the net force acting on charge \( Q \) must be zero. Since there are four identical charges \( q \), the forces due to these charges will be equal in magnitude but opposite in direction.

The total force due to the four corner charges can be expressed as:
\[ F_{\text{net}} = 4F \cos(45^\circ) = 4 \left(k \frac{2|qQ|}{a^2}\right) \frac{1}{\sqrt{2}} = \frac{4\sqrt{2}k|qQ|}{a^2} \]

**Hint:** Use the angle to resolve the forces into components. The angle is \( 45^\circ \) because of the symmetry of the square.

### Step 6: Set the Forces Equal to Zero
For equilibrium, we need the force due to \( Q \) to balance the forces from the corner charges:
\[ F_{\text{net}} = 0 \implies \frac{4\sqrt{2}k|qQ|}{a^2} = 0 \]

This implies that \( Q \) must be negative to counteract the repulsive forces from the positive charges \( q \).

### Step 7: Solve for Charge \( Q \)
From the equilibrium condition, we can derive that:
\[ Q = -\frac{q}{4(2\sqrt{2} + 1)} \]

**Hint:** Make sure to keep track of the signs. Since \( Q \) must be negative, we include the negative sign in the final expression.

### Final Result
Thus, the charge \( Q \) that must be placed at the center of the square for the system to be in equilibrium is:
\[ Q = -\frac{q}{4(2\sqrt{2} + 1)} \]