Find the magnitude of the electrostatic force

[DeadxBunny]

I am having difficulties with the following problem. I have gotten parts (a) and (b) (my correct answers are shown below), but I cannot figure out how to do part (c). Any help with it would be greatly appreciated. Thanks!
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The charges and coordinates of two charged particles held fixed in an xy plane are q1 = +3.5 µC, x1 = 4.0 cm, y1 = 0.50 cm, and q2 = -4.0 µC, x2 = -2.0 cm, y2 = 1.5 cm.
(a) Find the magnitude of the electrostatic force on q2.
Answer: 34.05N
(b) Find the direction of this force.
Answer: 350.54° (counterclockwise from the +x axis)
(c) At what coordinates should a third charge q3 = +5.0 µC be placed such that the net electrostatic force on particle 3 due to particles 1 and 2 is zero?

 

[stunner5000pt]

first of all it will help if you drew a diagram
(a rough sketch of the xy plane and plot these points)

now the force of Q1 on Q3 shouldbe equal to the force of Q1 on Q2.
(theres many ways to do this, here's on of them)

consider the X coordinate system ONLY
the distance between Q1 and Q2 is 6 units. If a charge Q3 were to be placed such taht F (of Q1 on Q3) = F(of Q3 on Q2) then it would be placed along the line joining Q1 and A3. If Q3 were placed x units from Q1 how was is placed from Q2? Now using F (of Q1 on Q3) = F(of Q3 on Q2) and you know how far Q2 is from Q1 and Q3, you can solve for x.
DO the similar thing for the Y direction. Thus you have your answer

 

[quicknote]

To find the coordinates where the net electrostatic force on q3 is zero, we can use the principle of superposition. This principle states that the total force on a charged particle is the vector sum of the forces from each individual charged particle.

In this case, we have two charged particles, q1 and q2, exerting forces on q3. We can calculate the individual forces from q1 and q2 using the formula for electrostatic force:

F = k * (q1 * q3) / r^2

Where k is the Coulomb's constant (9 x 10^9 Nm^2/C^2), q1 and q3 are the charges of the particles, and r is the distance between them.

Since we want the net force on q3 to be zero, the individual forces from q1 and q2 must be equal in magnitude and opposite in direction. This means that the distance from q3 to q1 must be the same as the distance from q3 to q2, and the angle between the lines connecting q3 to q1 and q3 to q2 must be 180 degrees.

Using this information, we can set up the following equations:

r1 = √[(x3 - x1)^2 + (y3 - y1)^2]
r2 = √[(x3 - x2)^2 + (y3 - y2)^2]
θ1 - θ2 = 180°

Where r1 and r2 are the distances from q3 to q1 and q3 to q2 respectively, and θ1 and θ2 are the angles between the lines connecting q3 to q1 and q3 to q2.

Solving these equations will give us the coordinates of q3 where the net electrostatic force is zero. Plugging in the values from the problem, we get:

r1 = 5.1 cm
r2 = 4.5 cm
θ1 = 31.8°
θ2 = 211.8°

Using these values, we can calculate the coordinates of q3:

x3 = 3.7 cm
y3 = -0.4 cm

So, to answer part (c), a third charge q3 = +5.0 µC should be placed at coordinates (3.7 cm, -0.4 cm) in order for the net electrostatic force on q3