To practice Problem-Solving Strategy 22.1 for electric force problems. Two charged particles, with charges q1=qq1=q and q2=4qq2=4q, are located on the x axis separated by a distance of 2.00cm2.00cm . A third charged particle, with charge q3=qq3=q, is placed on the x axis such that the magnitude of the force that charge 1 exerts on charge 3 is equal to the force that charge 2 exerts on charge 3.Find the position of charge 3 when qqq = 2.00 nCnC .

x₂ = 0.01336

Explanation:

In this exercise we use the translational equilibrium equation

          F₁₃ - F₂₃ = 0

          F₁₃ = F₂₃

at the point where charge 3 is placed the two electric forces have the same magnitude

let us use the expression of Coulomb's law for the electric force

         F₁₃ = ka q₁ q₃ / r₁₃²

         F₂₃ = ka q₂ q₃ / r₂₃²

we substitute

         k q₁ q₃ / r₁₃² = k q₂ q₃ / r₂₃²

now imprescriptibility suppose that particle 1 is at the origin of the coordinate system, particle 2 is at a distance d = 2.00cm = 2 10-2 m, therefore let's call the distance from particle 1 to particle 3 as x

            r₁₃ = x

            R₂₃ = d-x

In the exercise we are given the charges for the particle1 q1 = q, for the particle 2 the charge is q2 = 4q the distance between them is d = 2.00cm = 0.0200 m, the value of q = 2.00 nC = 2.00 10⁻⁹ C

let's substitute these values

              q₁ / x₂ = q₂ / (d-x)²

let's clear x

              (d-x)² = q₂ / q₁    x²

              d² - 2dx + x₂ = q₂ /q₁   x²

               x² (1-q₂ / q₁) - 2d x + d² = 0

let's substitute the values and solve the quadratic equation

              x² (1 - 4q / q) - 2 0.02 x + 0.02² = 0

              -3 x² - 0.04 x + 0.0004 = 0

                x² + 0.0133 x - 0.0001333 = 0

               x = [-0.0133 ±√(0.0133² + 4 0.00013333)] / 2

                x = [-0.0133 + - 0.026664] /2

              x₁ = -0.01998 m

              x₂ = 0.01336 m

Since load 3 must be between charged 1 and 2 the correct answer is

x₂ = 0.01336