Consider a square of side length a = 9.4 cm with charged particles at each of the corners.Three of the particles carry +9 nC of charge while the fourth particle carries -9 nC.In volts, what is the electric potential at the center of the square?
Hint: Choose the potential to be zero for a position very far from the square.
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.
To find the electric potential at the center of the square, we need to calculate the contributions of each charged particle and then sum them up.
The electric potential at a point in space due to a single charged particle can be calculated using the formula:
V = k * (q / r)
where V is the electric potential, k is Coulomb's constant (9 * 10^9 Nm²/C²), q is the charge, and r is the distance from the point to the charged particle.
Since three particles carry +9 nC of charge each and the fourth carries -9 nC, we can consider the potential at the center of the square as the total sum of the potentials contributed by each particle.
Let's name the distances from the center of the square to each corner as r1, r2, r3, and r4. In a square, the diagonals bisect each other at 90 degrees, so we can divide the square into four congruent right triangles. Each triangle will have legs that are half the side length of the square (a/2) and the hypotenuse will be the distance from the center to one corner (r).
Using the Pythagorean theorem, we can calculate the distance from the center of the square to each corner:
r1 = r2 = r3 = r4 = √((a/2)² + (a/2)²) = √(a²/4 + a²/4) = √(2a²/4) = √(a²/2) = √(9.4²/2) = √(88.36/2) = √(44.18) = 6.64 cm
Now, let's calculate the electric potential due to each charged particle at the center:
V1 = k * (q / r1) = (9 * 10^9 Nm²/C²) * (9 * 10^-9 C / 6.64 cm) = (9/6.64) * 10^9 Nm²/C * 10^-9 C / cm = 1.354 Nm/C
V2 = k * (q / r2) = V1
V3 = k * (q / r3) = V1
V4 = k * (q / r4) = (9 * 10^9 Nm²/C²) * (-9 * 10^-9 C / 6.64 cm) = (-9/6.64) * 10^9 Nm²/C * 10^-9 C / cm = -1.354 Nm/C
Finally, to find the electric potential at the center of the square, we add up the potentials contributed by each particle:
Vtotal = V1 + V2 + V3 + V4 = (1.354 Nm/C) + (1.354 Nm/C) + (1.354 Nm/C) + (-1.354 Nm/C) = 2.708 Nm/C - 1.354 Nm/C = 1.354 Nm/C
Therefore, the electric potential at the center of the square is 1.354 Nm/C.