Superposition of Forces

Net force on a charge q due to n other charges is the vector sum of individual Coulomb forces. Each pair interacts independently.

The principle

The superposition principle is an empirical fact: the force on a charge $q$ due to a collection of charges $q_1, q_2, \ldots, q_n$ is the vector sum of the individual Coulomb forces. The presence of other charges does not alter the interaction between any given pair.

Derivation

Let charge $q$ be at position $\vec{r}$, and charge $q_i$ at position $\vec{r}_i$. Define $\vec{r}_{i} = \vec{r} - \vec{r}_i$ as the vector from $q_i$ to $q$, with $r_i = |\vec{r} - \vec{r}_i|$.

Force on $q$ due to $q_i$ alone:

$$\vec{F}_i = \frac{1}{4\pi\varepsilon_0}\frac{q\, q_i}{r_i^2}\hat{r}_i$$

By superposition, the net force:

$$\vec{F} = \sum_{i=1}^{n} \vec{F}_i = \frac{q}{4\pi\varepsilon_0}\sum_{i=1}^{n} \frac{q_i}{r_i^2}\hat{r}_i$$ $$\boxed{\vec{F} = k q \sum_{i=1}^{n} \frac{q_i}{r_i^2}\hat{r}_i}$$

Continuous distribution

When charge is distributed continuously with volume density $\rho(\vec{r}')$:

$$\vec{F} = \frac{q}{4\pi\varepsilon_0}\int \frac{\rho(\vec{r}')(\vec{r}-\vec{r}')}{|\vec{r}-\vec{r}'|^3}dV'$$