My question was a little long, so it is in the description! PHYSCIS 2 QUESTION

Question

Two charges, q₁ = +4.0 μC and q₂ = -9.0 μC, lie along the x axis in a vacuum. Find the position of the third charge so that the net force on it is zero, if the two charges are 20 cm apart? Assume that q₁ is at x = 0.

Answer 1

Force between two charges is kq₁q₂/r₁₂²

Thus force between charge 1 and 3 and that between charge 2 and 3 is kq₁q₃/r₁₃² and kq₂q₃/r₂₃² respectively.

We know that r₁₂ = 20 cm = .02 m. r₁₃ + r₂₃ = 20 cm by default.

Without loss of generality, let us set r₁₃ = x, so that r₂₃ = .02 - x (x is in meters).

At net force = 0, the force between charge 1 and 3 and charges 2 and 3 are equal. Hence:

kq₁q₃/r₁₃ = kq₂q₃/r₂₃, or substituting above, and eliminating q3 from both sides,

kq₁/x² = kq₂/(20-x)^2, q1 = 4 uC, and q2 = 9 uC.

Solving for x gives you a quadratic equation (I won't include all of the math, since you can use the quadratic formula or a calculator to determine this. You will get a positive solution, which is the true answer, and a negative one, which is extraneous and can be discarded). For me, I get .008 m, or 8 mm.

Best,

Robert