Electrostatics: Coulomb's law

1. Coulomb's law

The electric force on the two stationary charges, qa and qb, distant by r, is given by the Coulomb's law: $$\Large\bf\color{brown}{ F = k \frac{q_a q_b}{r^2}\hat r } \Large\bf\color{green}{ \text { }\\ \text { }\\ \text { }\\ k = \frac{1}{4\pi\epsilon_0} }$$ The constant εo is called the permittivity of free space or vaccum. The forces in the figure are oriented according to Newton's third law. Here the charges are of like sign.

2. Example: Three points charges

We want to calculate the Coulomb forces at the point c of charge - q_c.

The charge on the point a is positive and the charge on the point c is negative, so the related forces are attractive.

Fac = Fca = k qa qc/n².

Fac has no components on the y-axis.

The charge on the point b is negative and the charge on the point c is negative, so the related forces are repulsive.

Fbc = Fcb = k qb qc/(n² + m²)

Fbc has two components: one on the y-axis, and the other on the z-axis.

The components of Fbc:

Fbc-y = Fbc cos(π - θ) = - Fbc cos θ
Fbc-z = + Fbc sin θ

Finally, the components of the net force F on c are:

Fy = - Fbc cos θ
Fz = - Fac + Fbc sin θ

cos θ = m/[m² + n²]^1/2
sin θ = n/[m² + n²]^1/2

F = [Fy ² + Fz²]^1/2
tg (β) = Fz/Fy