Two free charges +q and +4q are a distance l apart. A third charge is placed so that the entire system is in equilibrium. Find the location, magnitude and sign of the third charge.

the value of the charge is   q₃ = - 4/9 q, in the position  x= l/3

Explanation:

The forces in this system are given by Coulomb's law

           F =

As the forces are vector, we must add them as a vector, also let us take that charges of the same sign repel and charges of the opposite sign attract, therefore for the system to be in equilibrium the third charge must be of the opposite sign, that is, NEGATIVE .

Let's analyze the situation for card charge, let's use the indexes 1 for the charge q₁ = + q located at the origin (x₁ = 0), index 2 for the face q₂ = + 4q located at x₂ = l and the index 3 for the third charge

Let's find the location of charge 3 so that it is in balance

               ∑ F = F₁₃ - F₂₃ = 0

               F₁₃ = F₂₃

we seek every force

              F₁₃ = k

              F₂₃ = k

the distances are

              x₁₃ = x-0 = x

              x₂₃ = l -x

we substitute

              k \frac{q_1q_3 }{x^2} = k \frac{q_2q_3 }{(l-x)^2}

we solve

             (l-x)² =   x²

              l² - 2lx + x² = \frac{4q}{q} x²

              3x² + 2l x - l² = 0

we solve the quadratic equation

              x = [-2l ± / (2 3)

              x = [-2l + 2l ] / 6 = 2l [-1 ± 2] / 6

              x₁ = -l

              x₂ = l / 3

as charge 3 must be between the two charges the correct answer is

              x = l / 3

with charge 3 in this location it remains in equilibrium regardless of the value of its NEGATIVE charge.

now let's analyze the force on the other charges

charge 1 so that it is in balance

           ∑F = F₁₃ - F₁₂ = 0

           F₁₃ = F₁₂

           l² q₃ = q₂ (l/3)²

           q₃ = q₂ / 9

            q₃ = 4q / 9

            q₃ =

in summary the value of the charge is   q₃ = - 4/9 q

in the position  x= l/3

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