A square with side length s = 2a has four charges at its corners: three charges of +Q and one charge of -Q. What is the electric potential at the center of the square?

To find the electric potential at point P, the center of the square, we need to sum the potentials due to each of the four point charges. Step 1: Determine the distance from each charge to point P. The side length of the square is s = 2a. Point P is the center of the square. The distance from any corner to the center of a square is half the length of its diagonal. The length of the diagonal d of a square with side length s is d = ssqrt(2). In this case, d = (2a)sqrt(2). The distance r from each charge to point P is half the diagonal: r = (d)/(2) = (2a)sqrt(2)2 = asqrt(2) Step 2: Calculate the electric potential due to each charge at point P. The electric potential V due to a point charge q at a distance r is given by the formula: V = (1)/(4_0) (q)/(r) There are three charges of +Q and one charge of -Q. All are at the same distance r = asqrt(2) from point P. Let V_1, V_2, V_3 be the potentials due to the three +Q charges, and V_4 be the potential due to the -Q charge. V_1 = (1)/(4_0) (+Q)/(asqrt(2)) V_2 = (1)/(4_0) (+Q)/(asqrt(2)) V_3 = (1)/(4_0) (+Q)/(asqrt(2)) V_4 = (1)/(4_0) (-Q)/(asqrt(2)) Step 3: Sum the potentials to find the total electric potential at P. The total potential V_total at P is the algebraic sum of the individual potentials: V_total = V_1 + V_2 + V_3 + V_4 V_total = (1)/(4_0) (Q)/(asqrt(2)) + (1)/(4_0) (Q)/(asqrt(2)) + (1)/(4_0) (Q)/(asqrt(2)) + (1)/(4_0) (-Q)/(asqrt(2)) Factor out the common term (1)/(4_0 asqrt(2)): V_total = (1)/(4_0 asqrt(2)) (Q + Q + Q - Q) V_total = (1)/(4_0 asqrt(2)) (2Q) V_total = (2Q)/(4_0 asqrt(2)) Step 4: Simplify the expression. V_total = (Q)/(2_0 asqrt(2)) This can also be written as: V_total = (Q)/(2sqrt(2)_0 a) Comparing this result with the given options: A) (Q)/(2sqrt(2)_0 a) B) (Q)/(sqrt(2)_0 a) C) (Q)/(2_0 a) D) (Q)/(4_0 a) The calculated total potential matches option A. The final answer is A