Around the year 1785, Charles Augustin de Coulomb discovered that electrically charged bodies attract each other more strongly the closer they are. Analogous to the point mass in mechanics, in electricity one uses the concept of a point charge, which itself has no spatial extension. This assumption is perfectly valid for electrons, since they are currently considered point particles, and approximately valid for protons.
If closed surfaces are drawn around a charge, all field lines of the charge pass through these surfaces. Multiplying the field strength at each point by the corresponding surface element and summing over all results yields a value proportional to the enclosed charge. Mathematically, this is expressed by the closed integral:
$$
\oint \vec{E}\cdot d\vec{A} = \frac{Q}{\varepsilon_0}
$$
This relation is called Gauss’s law of electrostatics. The reciprocal of $\varepsilon_0$ is the proportionality constant. The value of this electric constant $\varepsilon_0$ is:
$$
\varepsilon_0 = 8.8541878128 \cdot 10^{-12}\,\frac{\mathrm{A\,s}}{\mathrm{V\,m}}
$$
Here, the units ampere (A) and volt (V) will be introduced later. In the case of spherical surfaces and point- or spherical charges, the electric field is perpendicular to the surface and points in the direction of its normal vector. The surface integral on the left then reduces to the magnitude of the field strength multiplied by the surface area of the sphere, leading to:
$$
4\pi r^2 E = \frac{Q}{\varepsilon_0}
$$
Rearranging for $E$ and multiplying by the charge $q$ to calculate the force yields Coulomb’s law for two point charges $q_1$ and $q_2$:
$$
\boxed{\vec{F}_C = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}\hat{\vec{r}}}
$$
The unit vector $\hat{\vec{r}}$ indicates that the force is radially directed, just as in the case of gravitation. Thus, Coulomb’s law strongly resembles Newton’s law of gravitation, with charges replacing masses. The only difference lies in the constants: the gravitational constant $G$ for gravity, and the electric constant $\varepsilon_0$ for electrostatics.
If the charges are placed in a dielectric, the force is reduced due to the lower field strength. To describe this, the dimensionless relative permittivity or dielectric constant $\varepsilon_r$ is introduced. For the ideal vacuum, this value is exactly 1. For low-density gases, such as air, it can also be approximated as 1.
Below is an overview of the dielectric constants of some materials:
| Material | Dielectric Constant $\varepsilon_r$ |
|---|---|
| Ceramic | 2 – 6 |
| Glass | 6 – 8 |
| Tantalum pentoxide | 27 |
| Water | 88 |
| Barium titanate | $10^3$ – $10^4$ |
Often, $\varepsilon_r$ is multiplied directly with the electric constant $\varepsilon_0$. The resulting value:
$$
\varepsilon = \varepsilon_0 \varepsilon_r
$$
is called the permittivity constant and is therefore material dependent. Coulomb’s law (in magnitude form) can then be written as:
$$
\boxed{F_C = \frac{1}{4\pi\varepsilon_0\varepsilon_r}\frac{q_1q_2}{r^2}}
$$
The biggest differences between gravity and electric forces are the strength of the interaction and the fact that there are both negative and positive charges, but only positive masses. Therefore, it is impossible to shield masses from each other or to create regions of vanishing gravity, whereas the world around us is largely free of strong electric fields.