Question

Unlike charges attract
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 = -2.00µC is at the origin. Where must a positive charge q3 placed on the x-axis so that the resultant force on it is zero?

Answer 1

Let q1 = +10.0 μC at x1 = 1.00 m and q2 = −2.00 μC at x2 = 0. Place q3 = +q at x (unknown). For equilibrium the magnitudes of the forces from q1 and q2 on q3 must be equal and in opposite directions.

For x < 0 the forces are opposite (q1 repels q3 to the left, q2 attracts q3 to the right), so set magnitudes equal:
k q1 / (x − 1)^2 = k |q2| / x^2
Cancel k and substitute q1/|q2| = 10/2 = 5:
(x − 1)^2 / x^2 = 5
(x − 1)^2 = 5x^2
4x^2 + 2x − 1 = 0

Solve:
x = [−2 ± √(4 + 16)] / 8 = (−1 ± √5)/4

The physically acceptable root (x < 0) is
x = (−1 − √5)/4 ≈ −0.809 m.

Thus q3 must be placed at x ≈ −0.809 m (0.809 m to the left of the origin).