A point charge +6.7 uC is placed at the origin, and a second charge –3.6 uC is placed in the X - Y plane at (0.40 m , 0.55 m ). Where should a third charge be placed in the plane so that the net force acting on it is zero?

tanα=0.55/0.4 =1.375.

α=54⁰
The distance between the charges
d = sqrt(0.4²+0.55²) = 0.68 m
The electric field at the disired point is zero =>
k•q₁/0.68+r)²=k•q₂/r²,
0.86r²-1.36r-0.46=0,
r ={1.36±sqrt(1.85+1.58)}/1.72 = >
r₁=1.867 m , r₂= - 0.28 m (extraneous root)

d+r=0.68+1.867=2.547 m
x=2.547•cos54=1.5 m
y=2.547•sin54=2.06 m.

To determine the position of the third charge where the net force acting on it is zero, we need to calculate the vector sum of the forces exerted by the first and second charges at that point.

First, let's calculate the force exerted on the third charge by the positive charge at the origin (charge 1):

1. Find the distance (r1) between the first charge and the third charge. Since the first charge is at the origin, the distance is simply the magnitude of the position vector of the third charge.

r1 = √(x1² + y1²) = √(0.40² + 0.55²)

Now, using Coulomb's Law, we can calculate the magnitude of the force (F1) exerted by charge 1 on the third charge:

F1 = (k * |q1| * |q3|) / r1²

where k is the electrostatic constant (9.0 x 10^9 N * m² / C²), |q1| is the magnitude of charge 1 (6.7 μC = 6.7 x 10^-6 C), and |q3| is the magnitude of charge 3.

Next, let's calculate the force exerted on the third charge by the negative charge in the X-Y plane (charge 2):

2. Find the distance (r2) between the second charge and the third charge. Since the second charge is located at (0.40 m, 0.55 m), the distance is the magnitude of the position vector difference between the two charges.

r2 = √((x3 - x2)² + (y3 - y2)²) = √((x3 - 0.40)² + (y3 - 0.55)²)

Using Coulomb's Law, we can calculate the magnitude of the force (F2) exerted by charge 2 on the third charge:

F2 = (k * |q2| * |q3|) / r2²

where |q2| is the magnitude of charge 2 (-3.6 μC = -3.6 x 10^-6 C) and |q3| is the magnitude of charge 3.

Finally, to make the net force on the third charge zero, the forces F1 and F2 must have equal magnitudes but opposite directions. This means F1 = F2.

Now you can set up the equation:

(k * |q1| * |q3|) / r1² = (k * |q2| * |q3|) / r2²

Cancel out k and |q3| on both sides of the equation:

|q1| / r1² = |q2| / r2²

Now solve for |q3|:

|q3| = (|q1| * r2²) / r1²

Plug in the given values and calculate:

|q3| = (6.7 x 10^-6 C * ((x3 - 0.40)² + (y3 - 0.55)²)) / (0.40² + 0.55²)