Particle A of charge 2.76 10-4 C is at the origin, particle B of charge -6.54 10-4 C is at (4.00 m, 0), and particle C of charge 1.02 10-4 C is at (0, 3.00 m). We wish to find the net electric force on C.

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To find the net electric force on particle C, we need to calculate the electric force exerted on particle C by both particle A and particle B, and then add these forces vectorially.

The electric force between two charged particles can be calculated using Coulomb's law:

F = k * q1 * q2 / r^2

where F is the electric force, k is the electrostatic constant (9.0 x 10^9 N m^2/C^2), q1 and q2 are the charges of the particles, and r is the distance between them.

Let's start by calculating the electric force exerted by particle A on particle C. The distance between particle A at the origin and particle C at (0, 3.00 m) is 3.00 m. Given that the charge of particle A is 2.76 x 10^(-4) C, the formula becomes:

F_A = k * q_A * q_C / r_AC^2

where F_A is the electric force exerted by particle A on particle C, q_A is the charge of particle A, q_C is the charge of particle C, and r_AC is the distance between particle A and particle C.

Substituting the values, we get:

F_A = (9.0 x 10^9 N m^2/C^2) * (2.76 x 10^(-4) C) * (1.02 x 10^(-4) C) / (3.00 m)^2

Now, let's calculate the electric force exerted by particle B on particle C. The distance between particle B at (4.00 m, 0) and particle C at (0, 3.00 m) can be found using the Pythagorean theorem:

r_BC = sqrt((4.00 m)^2 + (3.00 m)^2)

Given that the charge of particle B is -6.54 x 10^(-4) C, the formula becomes:

F_B = k * q_B * q_C / r_BC^2

where F_B is the electric force exerted by particle B on particle C, q_B is the charge of particle B, q_C is the charge of particle C, and r_BC is the distance between particle B and particle C.

Substituting the values, we get:

F_B = (9.0 x 10^9 N m^2/C^2) * (-6.54 x 10^(-4) C) * (1.02 x 10^(-4) C) / r_BC^2

Finally, to find the net electric force on particle C, we need to add the forces vectorially. Since the force vectors F_A and F_B are in different directions, we need to consider their magnitudes and directions correctly. To do this, we can break down the forces into their x-components and y-components using trigonometry.

Let's calculate the x-components and y-components of the electric forces F_A and F_B:

F_Ax = F_A * cos(theta_A)
F_Ay = F_A * sin(theta_A)

F_Bx = F_B * cos(theta_B)
F_By = F_B * sin(theta_B)

where theta_A and theta_B are the angles that the forces F_A and F_B make with the x-axis, respectively.

After calculating the x-components and y-components of the forces, we can then add them up:

F_net_x = F_Ax + F_Bx
F_net_y = F_Ay + F_By

Finally, the magnitude and direction of the net electric force on particle C can be found using the Pythagorean theorem and trigonometry:

F_net = sqrt(F_net_x^2 + F_net_y^2)
theta_net = atan(F_net_y / F_net_x)

These calculations will give you the net electric force on particle C.