To find the position of the third particle with a positive charge that would result in the net electric field at the origin being zero, we need to consider the electric forces and fields due to the two existing charges.
Let's denote the charge of the third particle as q (in coulombs), and its position as x (in meters).
The electric field due to the particle at the origin (charge -2.82 nC) at the origin (x = 0) is:
E1 = k * q1 / r^2,
where k is the electrostatic constant (approximately 8.99 x 10^9 N m²/C²), q1 is the charge (-2.82 nC = -2.82 x 10^(-9) C), and r is the distance from the origin to the particle at the origin (which is 0 since it's at the origin).
The electric field due to the particle at x = 51.0 cm = 0.51 m is:
E2 = k * q / (0.51 m)^2.
Now, for the net electric field to be zero at the origin, the electric field due to the third particle must exactly cancel out the electric field due to the other two particles.
So, we have:
E1 + E2 = 0.
Substituting the expressions for E1 and E2:
k * q1 / r^2 + k * q / (0.51 m)^2 = 0.
Now, we can solve for q:
k * q1 / r^2 = -k * q / (0.51 m)^2
q1 / r^2 = -q / (0.51 m)^2
q = -q1 * (0.51 m)^2 / r^2
q = -(2.82 x 10^(-9) C) * (0.51 m)^2 / r^2
Now, you would need to specify the value of r (the distance from the origin to the third particle) to calculate the charge q of the third particle in this setup. The third particle with a positive charge will be located at a distance r from the origin.