To determine the electric potential at the center of a square arrangement of charges, we start by recalling the formula for electric potential, which is given by:

$V^{=}\frac{kq}{r}$

Here, k is Coulomb's constant, approximately equal to \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\), q is the charge, and r is the distance from the charge to the point of interest.

In this scenario, we need to find the total potential at the center of the square formed by the charges. The first step is to calculate the distance from the center to each corner of the square. If the side length of the square is \(5 \, \text{mm}\), then the distance to the center can be determined using the Pythagorean theorem. By dividing the square into two right triangles, we find that:

$r=\; \backslash sqrt\{(2.5\; \backslash ,\; \backslash text\{mm\})^2\; +\; (2.5\; \backslash ,\; \backslash text\{mm\})^2\}\; =\; 3.54\; \backslash ,\; \backslash text\{mm\}\; =\; 3.54\; \backslash times\; 10^\{-3\}\; \backslash ,\; \backslash text\{m\}$

Next, we can calculate the total potential at the center by summing the potentials contributed by each charge. Since electric potential is a scalar quantity, we can simply add the potentials from each charge:

$V\_\{\backslash text\{total\}\}\; =\; \backslash sum\; \backslash frac\{k\; \backslash cdot\; q\_i\}\{r\}$

In this case, the charges are \(2 \, \text{nC}\), \(-3 \, \text{nC}\), \(1 \, \text{nC}\), and \(-1.5 \, \text{nC}\). The total potential can be simplified by factoring out \(k/r\):

$V\_\{\backslash text\{total\}\}\; =\; \backslash frac\{k\}\{r\}\; \backslash sum\; q\_i$

Calculating the sum of the charges:

$2\; \backslash ,\; \backslash text\{nC\}\; -\; 3\; \backslash ,\; \backslash text\{nC\}\; +\; 1\; \backslash ,\; \backslash text\{nC\}\; -\; 1.5\; \backslash ,\; \backslash text\{nC\}\; =\; -1.5\; \backslash ,\; \backslash text\{nC\}$

Now substituting the values into the potential formula:

$V\_\{\backslash text\{total\}\}\; =\; \backslash frac\{8.99\; \backslash times\; 10^9\; \backslash ,\; \backslash text\{N\; m\}^2/\backslash text\{C\}^2\}\{3.54\; \backslash times\; 10^\{-3\}\; \backslash ,\; \backslash text\{m\}\}\; \backslash cdot\; (-1.5\; \backslash times\; 10^\{-9\}\; \backslash ,\; \backslash text\{C\})$

Calculating this gives:

$V\_\{\backslash text\{total\}\}\; \backslash approx\; -3.81\; \backslash times\; 10^3\; \backslash ,\; \backslash text\{V\}$

This negative value indicates that the potential at the center is lower than the reference potential, which is an important aspect to consider in electrostatics. Understanding how to simplify the calculations by factoring out common terms can significantly streamline the process of finding electric potentials in complex charge arrangements.