The electric field

1 Characterisation of the electric field

As shown, electric charges have a direct influence on other charges. This occurs because they alter the properties of the space around them. These changes can be described by a field model. he concept of a field is of fundamental importance in physics. Examples of this are the temperature field, in which the temperature is described by a scalar quantity depending on its location.

In addition to such scalar fields, there are also vector fields in which each location is assigned not only a quantity but also a direction in which it acts. Examples of this are the gravitational field or a flow field, which describes, for example, the direction and velocity of water particles.

The electric field is described by magnitude (in newtons) and the direction of the force \(\vec {F_\mathrm {E}}\) on a positive test charge \(Q_2\) and characterised by field lines. This arrangement is shown in Figure 1.

The electric field strength is therefore:

\begin {equation} \vec {E_1} = \frac {\vec {F_2}}{Q_2} \left [ \frac {\mathrm {N}}{\mathrm {C}} = \frac {\frac {\mathrm {kg \cdot m}}{\mathrm {s}}}{\mathrm {A \cdot s}} = \frac {\mathrm {V}}{\mathrm {m}} \right ] \label {eq:e_feld} \end {equation}

At first glance, a disadvantage of this definition appears to be that the electric field strength seems to depend on the test charge \(Q_2\). However, by inserting Coulomb’s law (equation ??) into equation 1, it can be shown that the force \(F_2\) always increases proportionally to the test charge \(Q_2\). Their ratio therefore always yields the same value for the electric field strength, which, as shown in Figure 2, becomes independent of the test charge:

\begin {equation} \vec {E_1} = \frac {\vec {F_2}}{Q_2} =\frac {1}{4 \pi \varepsilon _0} \cdot \frac {Q_1 \cdot Q_2}{r^2} \cdot \frac {1}{Q_2} = \frac {1}{4 \pi \varepsilon _0} \cdot \frac {Q_1}{r^2} \label {e_feld2} \end {equation}

2 Electrical conductors and electrostatic fields I

The previous chapter examines the effect of point charges on the electric field. However, since electric fields in everyday life often originate from charged electrical conductors (such as metals), they are listed separately here. If there is an excess of electrons, the positive test charge \(Q\) is attracted towards the plate surfaces, as shown in Figure ??. is attracted towards the plate surfaces. The field lines \(\vec {E}\) of the electric field also run in the direction of the conductor surface. In the case of an electron deficiency, the test charge is repelled and the field lines also lead away from the positive conductor.

Since electrons are freely mobile within a conductor, in the case of an electric field within a conductor, they would be attracted (or repelled) by this field until the field within the conductor has balanced out (see Figure 4). Consequently, in this case, no electric field can exist within a conductor.

This leads to the realisation that the electric field lines must necessarily be perpendicular to the surface of conductors (see Figure 5). Otherwise, an oblique field line could be vectorially decomposed into a component running perpendicular and a component running horizontally on the conductor surface. Since this horizontal component is instantly balanced by the moving charge carriers, it cannot exist. This effect occurs exclusively in electrically conductive materials; in non-conductive materials (so-called insulators), there are no moving charge carriers that can balance the horizontal component.

3 Examples of electric fields

The simplest electric field is the homogeneous field. In this field, both the magnitude and the direction of the electric field strength \(\vec {E}\) are constant at every point. A test charge located in this field therefore experiences the same force at every point. Such a field can be generated, as shown in Figure 6, for example between two parallel metal plates that are charged with different charges \(Q\). The field lines generally run from the positively charged to the negatively charged metal plate. The density with which the field lines are drawn is often used as a measure of the strength of the electric field. The denser the field lines are drawn, the stronger the electric field is at that point.

The electric field of a point charge is also referred to as the radial field (see Figure 7). Under the idealised assumption that there is a shell with opposite charge infinitely far away, the field lines propagate in a straight line. In their symmetrical arrangement, they point in the direction of the negative shell for positive point charges, and in the direction of the point charge for negative point charges.

The electric field strength \(\vec {E}\) is directly proportional to the charge \(Q\), but decreases quadratically with increasing distance \(r\) from it. Its magnitude can be calculated using:

\begin {equation} E = \frac {1}{4 \pi \varepsilon _0} \frac {Q}{r^2} \end {equation}

The electrical field strength, which weakens with increasing distance \(r\) from the point charge, is evident not only in the mathematical relationship but also in the increasing distance between the field lines.

More complicated electric fields arise when a second point charge is added. In the case of two charges of equal magnitude but opposite sign (Figure 8), an almost uniform field exists between the charges, while it becomes significantly weaker further out.

The case with two identical point charges shows a special feature of the electric field, as illustrated in Figure 9. While the field lines in the rest of the space point away from the positive charges, as expected, there is a free space between the point charges without any field lines. This is because the opposing field lines in this space between the charges cancel each other out. There is no electric field strength at the point between the two charges; on the plane between the charges, the respective horizontal components neutralise each other, and there is only an electric field in the respective vertical direction.