Find the two possible values x3, r and x3,ℓ for the position of charge 3.

Given:

q1 = q

q2 = 4q

q3 = q

d = 2.00 cm

xq1 = 0 cm

xq2 = 2.00 cm

q = 2.00 nC

Let us determine the forces:

F 1on3 = K ⋅ |q1 | ⋅ |q 3| / r2 = K ⋅ |q| ⋅ |q| / (d 1-3) 2 = K ⋅ q 2 / (d1-3) 2

F 2on3 = K ⋅ |q2 | ⋅ |q 3| / r2 = K ⋅ |4q| ⋅ |q| / (d 3-2) 2 = K ⋅ 4q 2 / (d3-2) 2

We're able to solve for the variables in both cases since the forces are supposed to equal each other in each case.

F 1on3 = F2on3

K ⋅ q 2 / x 2 = K ⋅ 4q2 / (2.00 cm - x) 2

1/x2 = 4 / [4.00 cm2 - (4.00 cm⋅x) + x 2]

[4.00 cm2 - (4.00 cm ⋅ x) + x 2] / x2 = 4

4.00 cm 2 / x2 - (4.00 cm ⋅ x) / x 2 + x 2 / x2 - 4 = 0

4.00 cm 2 / x2 - (4.00 cm) / x + 1 - 4 = 0

4.00 cm 2 / x2 - (4.00 cm) / x - 3 = 0

3 = 4.00 cm 2 / x2 - (4.00 cm) / x

3x 2 = 4.00 cm - 4.00 cm ⋅ x

3x2 + 4.00 cm ⋅ x - 4.00 cm = 0

After we plug that into the quadratic equation, we get:

x = - 2 & x = 2/3

x3, r, x3,ℓ = 0.667,-2.00 cm, cm