### 5 Vectors and electrostatics

Electricity is important in several branches of engineering - not only in electrical or electronic engineering. For example the design of the electrostatic precipitator plates for cleaning the solid fuel power stations involves both mechanical engineering (structures and mechanical rapping systems for cleaning the plates) and electrostatics (to determine the electrical forces between solid particles and plates).

The following example and tasks relate to the electrostatic forces between particles. Electric charge is measured in coulombs (C). Charges can be either positive or negative.

The force between two charges

Let ${q}_{1}$ and ${q}_{2}$ be two charges in free space located at points ${P}_{1}$ and ${P}_{2}$ . Then ${q}_{1}$ will experience a force due to the presence of ${q}_{2}$ and directed from ${P}_{2}$ towards ${P}_{1}$ .

This force is of magnitude $K\phantom{\rule{1em}{0ex}}\frac{{q}_{1}{q}_{2}}{{r}^{2}}$ where $r$ is the distance between ${P}_{1}$ and ${P}_{2}$ and $K$ is a constant.

In vector notation this coulomb force (measured in newtons) can then be expressed as $\underset{̲}{F}=K\frac{{q}_{1}{q}_{2}}{{r}^{2}}\phantom{\rule{1em}{0ex}}\stackrel{̂}{\underset{̲}{r}}$ where $\stackrel{̂}{\underset{̲}{r}}$ is a unit vector directed from ${P}_{2}$ towards ${P}_{1}$ .

The constant $K$ is known to be $\frac{1}{4\pi {\epsilon }_{0}}$ where ${\epsilon }_{0}=8.854×1{0}^{-12}\text{F}{\text{m}}^{-1}$ (farads per metre).

The electric field

A unit charge located at a general point $G$ will then experience a force $\frac{K{q}_{1}}{{r}_{1}^{2}}{\stackrel{̂}{\underset{̲}{r}}}_{1}$ (where ${\stackrel{̂}{\underset{̲}{r}}}_{1}$ is the unit vector directed from ${P}_{1}$ towards $G$ ) due to a charge ${q}_{1}$ located at ${P}_{1}$ . This is the electric field $\underset{̲}{E}$ newtons per coulomb ( $\text{N}{\text{C}}^{-1}$ or alternatively $\text{V}{\text{m}}^{-1}$ ) at $G$ due to the presence of ${q}_{1}$ .

For several point charges ${q}_{1}$ at ${P}_{1},\phantom{\rule{1em}{0ex}}{q}_{2}$ at ${P}_{2}$ etc., the total electric field $\underset{̲}{E}$ at $G$ is given by

$\phantom{\rule{2em}{0ex}}\underset{̲}{E}=\frac{K{q}_{1}}{{r}_{1}^{2}}\phantom{\rule{1em}{0ex}}{\stackrel{̂}{\underset{̲}{r}}}_{1}+\frac{K{q}_{2}}{{r}_{2}^{2}}\phantom{\rule{1em}{0ex}}{\stackrel{̂}{\underset{̲}{r}}}_{2}+\dots$

where ${\stackrel{̂}{\underset{̲}{r}}}_{i}$ is the unit vector directed from point ${P}_{i}$ towards $G$ .

From the definition of a unit vector we see that

$\phantom{\rule{2em}{0ex}}\underset{̲}{E}=\frac{K{q}_{1}}{{r}_{1}^{2}}\phantom{\rule{1em}{0ex}}\frac{{\underset{̲}{r}}_{1}}{\left|{\underset{̲}{r}}_{1}\right|}+\frac{K{q}_{2}}{{r}_{2}^{2}}\phantom{\rule{1em}{0ex}}\frac{{\underset{̲}{r}}_{2}}{\left|{\underset{̲}{r}}_{2}\right|}+\dots =\frac{K{q}_{1}}{{\left|{r}_{1}\right|}^{3}}{\underset{̲}{r}}_{1}+\frac{K{q}_{2}}{{\left|{\underset{̲}{r}}_{2}\right|}^{3}}{\underset{̲}{r}}_{2}+\dots =\frac{1}{4\pi {\epsilon }_{0}}\left[\frac{{q}_{1}}{{\left|{\underset{̲}{r}}_{1}\right|}^{3}}{\underset{̲}{r}}_{1}+\frac{{q}_{2}}{{\left|{\underset{̲}{r}}_{2}\right|}^{3}}{\underset{̲}{r}}_{2}+\dots \right]$

where ${\underset{̲}{r}}_{i}$ is the vector directed from point ${P}_{i}$ towards $G,$ so that ${\underset{̲}{r}}_{1}=\underset{̲}{OG}-{\underset{̲}{OP}}_{1}$ etc., where $\underset{̲}{OG}$ and ${\underset{̲}{OP}}_{1}$ are the position vectors of $G$ and ${P}_{1}$ (see Figure 34).

Figure 34

$\phantom{\rule{2em}{0ex}}\underset{̲}{O{P}_{1}}+\underset{̲}{{P}_{1}G}=\underset{̲}{OG}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\underset{̲}{{P}_{1}G}=\underset{̲}{OG}-\underset{̲}{O{P}_{1}}$

The work done

The work done $W$ (energy expended) in moving a charge $q$ through a distance $Ṣ$ , in a direction given by the unit vector $\underset{̲}{S}∕\left|\underset{̲}{S}\right|$ , in an electric field $\underset{̲}{E}$ is (defined by)

$\phantom{\rule{2em}{0ex}}W=-q\underset{̲}{E}.\underset{̣}{\underset{̲}{S}}$ (4)

where $W$ is in joules.