To find the electric potential at the center of a square system of four point charges, each with charge q, located at the corners of a square of side length a, we can use the principle of superposition. The electric potential at a point due to multiple point charges is the algebraic sum of the potentials due to each individual charge.
Let's denote the distance from the center of the square to each charge as R. Since the square has side length a, the diagonal distance from the center to a corner is R = a/√2.
The electric potential at the center of the square due to a single charge q is given by the equation:
V = k * q / R,
where k is the electrostatic constant (k = 9 × 10^9 N m^2/C^2) and R is the distance between the charge and the center of the square.
Since all four charges are identical and equidistant from the center, the potential due to each charge is the same. Therefore, the total electric potential at the center is the sum of the potentials due to the four charges:
V_total = V + V + V + V = 4V.
Substituting the values into the equation, we have:
V_total = 4 * (k * q / R) = 4 * (9 × 10^9 N m^2/C^2) * q / (a/√2).
Simplifying further, we get:
V_total = (36 × 10^9 N m^2/C^2) * q / (a/√2).
So, the electric potential at the center of the square system of four point charges is (36 × 10^9 N m^2/C^2) times the charge q divided by the distance a/√2.