To find the electric potential at the center of the square, we can calculate the potential due to each individual charge at the center and then sum them up.
Given:
Side length of the square, a = 9.4 cm
Charge on each of the three particles = +9 nC
Charge on the fourth particle = -9 nC
We know that the electric potential due to a point charge is given by V = k * (|q| / r), where V is the electric potential, k is the electrostatic constant (9 x 10^9 N m^2/C^2), q is the charge, and r is the distance from the charge to the point where potential is being calculated.
The distance from the center of the square to each of the charges is equal to half the diagonal of the square.
Let's consider the points where the charges are located as A, B, C, and D, in order.
The distance from the center to each charge is:
r = (sqrt(2) / 2) * a
For the positive charges at A, B, and C:
V_pos = k * (|+9 nC| / r)
For the negative charge at D:
V_neg = k * (|-9 nC| / r)
To calculate the net potential at the center, we need to consider the potential due to each charge. Since V is a scalar quantity, we need to sum the magnitudes of the potentials.
Total potential at the center:
V_total = V_pos_A + V_pos_B + V_pos_C + V_neg_D
Let's calculate the potentials step-by-step:
Step 1: Calculate the value of r:
r = (sqrt(2) / 2) * a = (sqrt(2) / 2) * 9.4 = 6.646 cm
Step 2: Calculate the potentials due to the positive charges:
V_pos_A = k * (|+9 nC| / r)
= (9 x 10^9 N m^2/C^2) * (9 nC / 6.646 cm)
= 1.215 x 10^7 V
V_pos_B = k * (|+9 nC| / r)
= (9 x 10^9 N m^2/C^2) * (9 nC / 6.646 cm)
= 1.215 x 10^7 V
V_pos_C = k * (|+9 nC| / r)
= (9 x 10^9 N m^2/C^2) * (9 nC / 6.646 cm)
= 1.215 x 10^7 V
Step 3: Calculate the potential due to the negative charge:
V_neg_D = k * (|-9 nC| / r)
= (9 x 10^9 N m^2/C^2) * (9 nC / 6.646 cm)
= 1.215 x 10^7 V
Step 4: Calculate the net potential at the center:
V_total = V_pos_A + V_pos_B + V_pos_C + V_neg_D
= 1.215 x 10^7 V + 1.215 x 10^7 V + 1.215 x 10^7 V + 1.215 x 10^7 V
= 4.86 x 10^7 V
Therefore, the electric potential at the center of the square is 4.86 x 10^7 volts.