A -9.00 μC point charge and +7.00 μC point charge are placed along the x-axis at x = 0.000 m and x = 0.4 m, respectively. Where must a third charge be placed along the x-axis (relative to the origin) so that it does not experience any net electric force due to the other two charges? The position on the x-axis is in meters and use three significant digits.

x₃ = 0.397 m

Explanation:

For this exercise we use the vector sum of the forces, where the force is electric given by the Coulomb equation

           F =

We also use that charges of the same sign repel and charges of a different sign attract.

   In this case the force between load 1 and 3 has one direction and the force between 2 and 3 has the opposite direction, in the exercise they ask that the force on load 3 be zero

           ∑ F = F₁₃ - F₂₃ = 0

            F₁₃ = F₂₃

we write the expressions of the forces, the distances are

         r₁₃ = (x₁-x₃) ²

         r₂₃ = (x₂ - x₃) ²

x₁ = 0 m

         q₁ (x₂ - x₃) ² = q₂ x₃²

          (x₂ -x₃) = x₃

we substitute the values

           (0.4 -x₃) = x₃

         x₃ (1 + ) =  0.4

         x₃ (1.13389) = 0.453557

         x₃ = 0.453557 / 1.13389

         x₃ = 0.397 m