Four equal charges `Q` are placed at the four corners of a square of each side is 'a'. Work done in removing a charge `-Q` from its centre to infinity is

Options

A. `0`
B. `(sqrt(2)Q^(2))/(4pi epsilon_(0)a)`
C. `(sqrt(2)Q^(2))/(pi epsilon_(0)a)`
D. `(Q^(2))/(2pi epsilon_(0)a)`

Explanation

To find the work done in removing a charge \(-Q\) from the center of a square (formed by four equal charges \(Q\) at its corners) to infinity, we can follow these steps:

### Step 1: Understand the Configuration
We have four equal charges \(Q\) placed at the corners of a square. The side length of the square is \(a\). The center of the square is equidistant from all four charges.

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

### Step 3: Calculate the Electric Potential at the Center
The electric potential \(V\) at a point due to a point charge is given by:
\[ V = \frac{kQ}{r} \]
where \(k\) is Coulomb's constant. Since there are four charges \(Q\) at the corners, the total electric potential \(V\) at the center due to all four charges is:
\[ V_{\text{total}} = 4 \cdot \frac{kQ}{\frac{a}{\sqrt{2}}} = \frac{4kQ\sqrt{2}}{a} \]

### Step 4: Calculate the Work Done
The work done \(W\) in moving a charge \(q\) from a point where the potential is \(V\) to a point at infinity (where the potential is zero) is given by:
\[ W = q(V_{\text{final}} - V_{\text{initial}}) \]
In this case, the initial potential at the center is \(V_{\text{total}}\) and the final potential at infinity is \(0\). Thus, the work done in moving the charge \(-Q\) from the center to infinity is:
\[ W = -Q \left(0 - \frac{4kQ\sqrt{2}}{a}\right) = \frac{4kQ^2\sqrt{2}}{a} \]

### Final Answer
The work done in removing the charge \(-Q\) from the center of the square to infinity is:
\[ W = \frac{4kQ^2\sqrt{2}}{a} \]