Electricity

Coulomb's law - formula in LaTeX

The force between two point charges in vacuum; it falls off with the square of the distance. \varepsilon gives the rounded epsilon and \epsilon the other, lunate shape of the same letter. Both are correct, as long as you do not mix them in one document.

F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}

LaTeX code

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$$
F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}
$$

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What the symbols mean

$F$: force between the charges (N)
$q_1, q_2$: charges (C)
$r$: distance between the charges (m)
$\varepsilon_0$: vacuum permittivity, 8.85·10⁻¹² F/m

An electricity and magnetism formula - the basis of circuit problems and electrical lab reports.

Worked example

Two 1 μC charges 3 m apart attract or repel each other with a force of 1 mN. Powers of ten are written 10^{-6}: a negative exponent needs braces, or only the minus sign is raised.

F = 9\cdot 10^{9}\,\dfrac{\mathrm{N\,m^2}}{\mathrm{C^2}} \cdot \dfrac{(10^{-6}\,\mathrm{C})^2}{(3\,\mathrm{m})^2} = 10^{-3}\,\mathrm{N}

Other ways to write it

With the constant k instead of 1/(4πε₀) and an absolute value, when only the magnitude matters.
$F = k\,\dfrac{|q_1 q_2|}{r^2}$
```
F = k\,\dfrac{|q_1 q_2|}{r^2}
```
Vector form; \hat puts a hat over the unit vector.
$\vec{F}_{12} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}\,\hat{r}_{12}$
```
\vec{F}_{12} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}\,\hat{r}_{12}
```
In a medium with relative permittivity εᵣ the force is εᵣ times weaker than in vacuum.
$F = \dfrac{1}{4\pi\varepsilon_0\varepsilon_r}\dfrac{q_1 q_2}{r^2}$
```
F = \dfrac{1}{4\pi\varepsilon_0\varepsilon_r}\dfrac{q_1 q_2}{r^2}
```

LaTeX commands in this formula

\dfrac: fraction (display)
\pi: Greek letter pi
\varepsilon: Greek letter varepsilon
x_{0}: subscript; wrap several characters in braces
x^{2}: superscript (power); wrap several characters in braces

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