Superposition of Forces

The superposition principle (also known as superposition property) states that: for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. For Coulomb's law, the stimuli are forces. Therefore, the principle suggests that total force is a vector sum of individual forces.

Coulomb Force

The scalar form of Coulomb's Law relates the magnitude and sign of the electrostatic force F, acting simultaneously on two point charges q₁ and q₂:

$|\boldsymbol{F}|= \frac{1}{4\pi ar\epsilon_0}\frac{|q_1q_2|}{r^2}$,

where r is the separation distance and ε₀ is electric permittivity. If the product q₁q₂ is positive, the force between them is repulsive; if q₁q₂ is negative, the force between them is attractive. The principle of linear superposition allows the extension of Coulomb's law to include any number of point charges—in order to derive the force on any one point charge by a vector addition of these individual forces acting alone on that point charge. The resulting force vector happens to be parallel to the electric field vector at that point, with that point charge removed.

To calculate the force on a small test charge q at position $r$, due to a system of N discrete charges:

$\begin{aligned} \boldsymbol{F(r)} &= \frac{q}{4 \pi ar \epsilon_0} \sum_{i=1}^N q_i \frac{\boldsymbol{r-r_i}}{|\boldsymbol{r-r_i}|^3} \ &= \frac{q}{r 4 \pi ar \epsilon_0} \sum_{i=1}^Nq_i\frac{\boldsymbol{\widehat{R_i}}}{|\boldsymbol{R_i}|^2} \end{aligned}$,

where q_i and r_i are the magnitude and position vector of the i-th charge, respectively, and $\boldsymbol{\widehat{R_i}}$ is a unit vector in the direction of $\boldsymbol{R}_{i} = \boldsymbol{r} - \boldsymbol{r}_i$ (a vector pointing from charges q_i to q. )

Of course, our discussion of superposition of forces applies to any types (or combinations) of forces. For example, when a charge is moving in the presence of a magnetic field as well as an electric field, the charge will feel both electrostatic and magnetic forces. Total force, affecting the motion of the charge, will be the vector sum of the two forces. (In this particular example of the moving charge, the force due to the presence of electromagnetic field is collectively called Lorentz force (see ).

Lorentz Force on a Moving Particle

Lorentz force f on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time.