Find the electric potential energy for the following array of charges: charge q1 = +3.0 µC is located at (x, y) = (0.0, 0.0) m; charge q2 = +3.0 µC is located at (4.0, 3.0) m; and charge q3 = −1.0 µC is located at (0.0, 3.0) m.
Three point charges are arranged along the x-axis. Charge q1 = +4.25 µC is at the origin, and charge q2 = -6.25 µC is at x = 0.160 m. Charge q3 = -8.75 µC. Where is q3 located if the net force on q1 is 6.50 N in the −x-direction?
x =
m
To find the electric potential energy for this array of charges, we need to calculate the potential energy between each pair of charges and then sum them up.
The potential energy between two charges can be calculated using the formula:
U = k * q1 * q2 / r
Where:
U is the electric potential energy (in joules)
k is the Coulomb's constant (approximately 8.99 × 10^9 Nm^2/C^2)
q1 and q2 are the magnitudes of the charges (in coulombs)
r is the distance between the charges (in meters)
Let's calculate the potential energy for each pair of charges:
1. Potential energy between q1 and q2:
q1 = +3.0 µC,
q2 = +3.0 µC,
r = distance between (x, y) = (0.0, 0.0) m and (4.0, 3.0) m.
Using the distance formula, we get:
r = sqrt((4.0 - 0.0)^2 + (3.0 - 0.0)^2) = sqrt(16 + 9) = sqrt(25) = 5.0 m.
U1-2 = (8.99 × 10^9 Nm^2/C^2) * (+3.0 µC) * (+3.0 µC) / (5.0 m) = 4.79 × 10^-7 J
2. Potential energy between q1 and q3:
q1 = +3.0 µC,
q3 = -1.0 µC,
r = distance between (x, y) = (0.0, 0.0) m and (0.0, 3.0) m.
Using the distance formula, we get:
r = sqrt((0.0 - 0.0)^2 + (3.0 - 0.0)^2) = sqrt(0 + 9) = sqrt(9) = 3.0 m.
U1-3 = (8.99 × 10^9 Nm^2/C^2) * (+3.0 µC) * (-1.0 µC) / (3.0 m) = -8.99 × 10^-7 J
3. Potential energy between q2 and q3:
q2 = +3.0 µC,
q3 = -1.0 µC,
r = distance between (x, y) = (4.0, 3.0) m and (0.0, 3.0) m.
Using the distance formula, we get:
r = sqrt((4.0 - 0.0)^2 + (3.0 - 3.0)^2) = sqrt(16 + 0) = sqrt(16) = 4.0 m.
U2-3 = (8.99 × 10^9 Nm^2/C^2) * (+3.0 µC) * (-1.0 µC) / (4.0 m) = -6.74 × 10^-7 J
Now, let's sum up the potential energies:
U_total = U1-2 + U1-3 + U2-3
U_total = (4.79 × 10^-7 J) + (-8.99 × 10^-7 J) + (-6.74 × 10^-7 J)
U_total = -1.94 × 10^-7 J
Therefore, the electric potential energy for the given array of charges is approximately -1.94 × 10^-7 Joules.