Question

3. Three charges lie along the x-axis. A positive charge q1 =
10.0µC is at x = 1.00 m, and a negative charge q2 = -
Charging and discharging
2.00µC is at the origin. Where must a positive charge q3 be

Answer 1

It looks like your problem text was cut off. Which of these do you mean?

A) “Where must a positive charge q3 be placed so that the net force on q3 is zero?”
B) “Where must q3 be placed so that the net force on q1 (or on q2) is zero?”

If you meant (A) (the common textbook version), here is the solution.

Given q1 = +10.0 μC at x1 = 1.00 m and q2 = −2.00 μC at x2 = 0, place q3 at x so the forces from q1 and q2 on q3 cancel. For x < 0 the two forces are opposite, so set their magnitudes equal:

k q1 q3/(1 − x)^2 = k |q2| q3/x^2.

q3 and k cancel, and with q1 = 10, |q2| = 2 (in μC units):

10/(1 − x)^2 = 2/x^2 ⇒ 5 x^2 = (1 − x)^2
⇒ 4 x^2 + 2 x − 1 = 0.

Solve: x = [−2 ± sqrt(4 + 16)]/8 = [−2 ± sqrt(20)]/8. The physically appropriate root with x < 0 is

x ≈ −0.809 m.

So q3 must be placed at x ≈ −0.809 m (about 0.809 m to the left of the origin) for the net force on q3 to be zero.

If you meant a different equilibrium (net force on q1 or q2 = 0), tell me which and I’ll solve that case.