The electrical charge

1 The atomic structure

Many physical processes cannot be explained solely by classical phenomena from mechanics. As early as around 600 BC, the Greek philosopher Thales of Miletus discovered that amber, when rubbed with a fur, attracts light objects such as feathers. In further experiments, it was proven much later that, in addition to the force of gravity, there is another force, known as the electric force exists in addition to the force of gravity. This force can have both an attractive and a repulsive effect. Electric charges are postulated as the cause of these forces and are defined (arbitrarily, from a historical perspective) as positive and negative. Charges of the same name repel each other, while charges of different names attract each other.

In 1913, Nils Bohr developed a simple model based on the observed effects. According to this atomic model, shown in Figure 1, each atom consists of an atomic nucleus and an electron shell. The atomic nucleus, in turn, consists of densely packed protons (positive charge carriers) and neutrons (electrically neutral). The shell consists of electrons (negative charge carriers) that orbit the atomic nucleus in concentric paths with different radii.

The Load \(Q\) of a proton and an electron is equal in magnitude and is referred to as the elementary charge \(e\). Its experimentally determined value is \(e = 1.6 \cdot 10^{-19} \, \mathrm {C}\). The proton has a positive charge and the electron has a negative charge. An atom usually has the same number of protons as electrons and is therefore electrically neutral when viewed from the outside.

The coulomb is the unit of electric charge and is described as follows:

\begin {equation*} \mathit {[Q]} = \rm { 1 \, Coulomb = 1 \, C = 1 \, As} \end {equation*}

The number of electrons in an atom can change through the supply of energy or through interaction with other particles (for example, through friction between a plastic rod and a piece of fabric, friction between clothing and a plastic slide, or UV radiation).

If an atom gains one or more additional electrons, i.e. if there is an Elektronenüberschuss, it becomes negatively charged. If it loses electrons, i.e. if there is a Elektronenmangel, it becomes positively charged. In both cases, it is referred to as an ion.

Since only whole electrons can be added to or removed from the ion, the corresponding charge \(Q\) of an ion (and thus also of any other object) is always a multiple of the elementary charge \(e\).

The mass of the protons and neutrons in the nucleus is many orders of magnitude higher than that of the electrons. Consequently, the electrons released from the atomic structure are of great importance to electrical engineering as the smallest, lightest and most mobile charge carriers.

2 Charge densities

Not only the number, but also the distribution of electrical charge carriers plays an important role in electrical engineering. For example, it is crucial for the design and functioning of electrical components (such as resistors, capacitors or semiconductor devices) to know how the charges are distributed along lines, on surfaces or within volumes. This module assumes a uniform distribution of charge carriers.

The point charge serves as an idealised model of a charge distribution at a point without spatial extension. If the actual spatial distribution at the corresponding location is not relevant, charges are usually considered as point charges.

The line charge density \(\lambda \) (lambda), given in C/m, describes the distribution of the electric charge along a line, as occurs, for example, in a very thin wire, which is extended in only one dimension (see Figure 3).

Since the charge carriers and their atomic structures are negligible compared to the structures used in components, a continuous distribution can be assumed instead of a discrete one. Consequently, the line charge density can be described as the derivative of the charge per line length.

\begin {equation*} \lambda = \lim _{l \to 0} \frac {\Delta Q}{\Delta l} = \frac {\mathrm {d}Q}{\mathrm {d}l} \end {equation*}

Similarly, the charge \(Q\) is the integral of the line charge density \(\lambda \) over the length \(l\).

\begin {equation*} Q = \int _l \lambda \, \mathrm {d}l \end {equation*}

The Surface charge density \(\sigma \) (sigma) in \(\mathrm {C/m^2}\) describes the distribution of the charge \(Q\), which is distributed per unit area \(A\) (see Figure 4). It is required, for example, when calculating capacitors, which will be introduced at a later stage.

Similar to the line charge density, \(\sigma \) can also be described using a limit analysis and thus a derivative:

\begin {equation*} \sigma = \lim _{A \to 0} \frac {\Delta Q}{\Delta A} = \frac {\mathrm {d}Q}{\mathrm {d}A} \end {equation*}

\begin {equation*} Q = \iint _A \sigma \, \mathrm {d}A \end {equation*}

The Space charge density \(\rho \) (rho) in \(\mathrm {C/m^3}\) indicates the number of free charge carriers in the volume under consideration. These can be, for example, electrons in a conductor or charged ions in a gas mixture.

\begin {equation*} \rho = \lim _{V \to \infty } \frac {\Delta Q}{\Delta V} = \frac {\mathrm {d}Q}{\mathrm {d}V} \end {equation*} \begin {equation*} Q = \iiint _V \rho \, \mathrm {d}V \end {equation*}

3 Electrical conductors - metals

This distribution of charge carriers can be clearly seen in metals, for example. Many metals have a special atomic structure. The atoms are arranged in a regular Lattice structure. As a result, the outermost electrons of each atom within the metal body are almost free to move. These free electrons give metals their electrically conductive properties. Figure 6 illustrates this structure. Metals are therefore often referred to in electrical engineering as (electrical) conductors. In general, conductors are electrically neutral, since in each atom the sum of all electrons corresponds to the sum of all protons.

The addition of electrons creates an excess of electrons, causing the conductor to become negatively charged. f electrons are removed, an electron deficiency occurs and the conductor becomes positively charged. (Abbildung 7).

4 Coulomb’s law

Charged particles influence each other, with charges of the same sign repelling each other and charges of different signs attracting each other. As early as 1785, Charles Augustin de Coulomb was able to experimentally prove that the magnitude of the force is proportional to each of the spherically symmetric charges \(Q_1\) and \(Q_2\), but inversely proportional to the square of the distance \(r\) between the two charges.

The proportional interaction between the charges \(Q_1\) and \(Q_2\) and the square of the distance \(r^2\) leads to the force between the charges with the proportionality constant \(\frac {1}{4 \pi \varepsilon _0}\). This proportional relationship is shown in Figure 8.

By introducing a proportionality term, the resulting force \(F\) can be calculated directly from the previously determined proportional relationship. The resulting equation 1 is called Coulomb’s law.

\begin {equation} F \sim \frac {Q_1 Q_2}{r^2} \rightarrow F = \frac {1}{4 \pi \varepsilon _0} \frac {Q_1 Q_2}{r^2} \label {eq:coulomb} \end {equation}

The factor \(\varepsilon _0\) is referred to as the electric field constant (also known as the dielectric constant of vacuum), while the 4 \(\pi \) originates from geometric considerations of the arrangement.

The value of the electric field constant is approximately:

\begin {equation*} \varepsilon _0 = 8,854 \cdot 10^{-12} \rm {As/Vm} \end {equation*}

If there are more than two charges, the resulting force on each of the individual charges can be calculated by adding all the individual forces resulting from each combination of two charges in which the target charge is involved. This effect is known as the superposition principle, which is also used in Module 5 Advanced DC Networks to calculate resulting voltages.

In practice, this effect is used, among other things, in the electrostatic tensioning of paper on plotters, in laser printers to transfer toner powder to the paper, in touch screens to determine the point of contact, and in plasma screens for the controlled gas discharge of ions to generate light.