The work \( W \) required to move a charge \( Q \) from infinity to the square's center is equivalent to the system's potential energy.
The potential \( V \) at the square's center due to a single charge \( Q \) located at a distance \( r \) from the center is calculated as:
\[ V = \frac{Q}{4 \pi \varepsilon_0 r} \]
For four charges at the square's corners, the distance \( r \) from each corner to the center is \( r = \frac{l}{\sqrt{2}} \).
Therefore, the potential at the center due to one charge is:
\[ V = \frac{Q}{4 \pi \varepsilon_0 \frac{l}{\sqrt{2}}} = \frac{\sqrt{2} Q}{4 \pi \varepsilon_0 l} \]
The total potential at the center from all four charges is:
\[ V_{\text{total}} = 4 \times \frac{\sqrt{2} Q}{4 \pi \varepsilon_0 l} = \frac{\sqrt{2} Q}{\pi \varepsilon_0 l} \]
The work done in bringing charge \( Q \) from infinity to the center is:
\[ W = Q \times V_{\text{total}} = Q \times \frac{\sqrt{2} Q}{\pi \varepsilon_0 l} \]
\[ W = \frac{\sqrt{2} Q^2}{\pi \varepsilon_0 l} \]
Consequently, the work done to bring the charge \( Q \) to the center of the square is:
\[ W = \frac{\sqrt{2} Q^2}{\pi \varepsilon_0 l} \]