25: Forces in a Three-Charge System

INTRO:
Coulomb's law for the magnitude of the force F between two particles with charges Q and Q′ separated by a distance d is |F|=K⋅|QQ′| / d²

where K=1/[4πϵ₀], and ϵ₀=8.854×10⁻¹²C²/(N⋅m²) is the permittivity of free space.

Consider two point charges located on the x axis: one charge, q₁ = -19.5 nC , is located at x₁ = -1.670 m ; the second charge, q₂ = 34.0 nC , is at the origin (x=0).

Part A:
What is the net force exerted by these two charges on a third charge q₃ = 52.5 nC placed between q₁ and q₂ at x₃ = -1.075 m ?

Your answer may be positive or negative, depending on the direction of the force.
Express your answer numerically in newtons to three significant figures.

SOLUTION:
givens:
q₁ = -19.5 nC
x₁ = -1.670 m
q₂ = 34.0 nC
x₂ = 0 m
q₃ = 52.5 nC
x₃ = -1.075 m

we begin by determining the two separate forces on q₃
F_1on3= K⋅|q₁⋅q₃| / [x₃-x₁]² = 1/[4πϵ₀]⋅|(-19.5 nC)⋅(52.5 nC)| / [(-1.075 m)-(-1.670 m)]² = 2.6×10^-5 N
F_2on3= K⋅|q₂⋅q₃| / [x₃-x₂]²= 1/[4πϵ₀]⋅|(34.0 nC)⋅(52.5 nC)| / [(-1.075 m)-(0 m)]² = 1.388×10^-5 N

however these are only the magnitudes of the forces, we also need to determine the directions so as to assign the forces positive or negative values
in other words, is each force pushing the charge in a positive x-direction or a negative x-direction?
⇐((-)q1) ((+)q3) ⇐((+)q2) 
F_1on3 is a negative charge pulling the positive charge in the negative x-direction
F_2on3 is a positive charge pushing the positive charge in the negative x-direction

SO, F_NETon3 = F_1on3(negative) + F_2on3(negative) =-2.6×10^-5 N + -1.388×10^-5 N = -3.99×10^-5 N