Block 09 - Force on charges and electric field strength

Abb. 1: John Tra-Voltage

We had already considered the charge as the central quantity of electricity in the first chapter of the previous semester and recognized it as a multiple of the elementary charge. There was already a mutual force action (the Coulomb-force) derived. This will be more fully explained.

First, we shall define certain terms:

1. Electricity is a catch-all term for any occurrences involving moving and resting charges.
2. Electrostatics is the study of charges at rest and consequently electric fields that do not vary over time. As a result, the electrical quantities have no temporal dependence.
   For any function of the electric quantities, ${{{\rm d} f}\over{{\rm d} t}}=0$ holds mathematically.
3. Electrodynamics describes the behavior of moving charges. Hence, electrodynamics covers both changing electric fields and magnetic fields.
   For the time being, the simple explanation will be that magnetic fields are dependent on current or charge flow.
   It is no longer true in electrodynamics that the derivative is always necessary for any function of electric values.

Only electrostatics is discussed in this chapter. For the time being, magnetic fields are thus excluded. Furthermore, electrodynamics is not covered in this chapter and is provided in further detail in subsequent chapters.

Fields

The concept of a field will now be briefly discussed in more detail.

1. The introduction of the field distinguishes the cause from the effect.
   1. The field in space is caused by the charge $Q$.
   2. As a result of the field, the charge $q$ in space feels a force.
   3. This distinction is brought up again in this chapter.
      It is also fairly obvious in electrodynamics at high frequencies: the field corresponds to photons, i.e. to a transmission of effects with a finite (light)speed $c$.
2. There are different-dimensional fields, just like physical quantities:
   1. In a scalar field, each point in space is assigned a single number.
      For example,
      1. a temperature field $T(\vec{x})$ on a weather map or in an object
      2. a pressure field $p(\vec{x})$
   2. Each point in space in a vector field is assigned several numbers in the form of a vector. This reflects the action as it occurs along the spatial coordinates.
      As an example.
      1. gravitational field $\vec{g}(\vec{x})$ pointing to the object's center of mass.
      2. electric field $\vec{E}(\vec{x})$
      3. magnetic field $\vec{H}(\vec{x})$
3. A tensor field is one in which each point in space is associated with a two- or more-dimensional physical quantity - that is, a tensor. Tensor fields are useful in mechanics (for example, the stress tensor), but they are not required in electrical engineering.

Vector fields are defined as follows:

1. Effects along spatial axes $x$, $y$ and $z$ (Cartesian coordinate system).
2. Effect in magnitude and direction vector (polar coordinate system)

The electric Field

To determine the electric field, a measurement of its magnitude and direction is now required. The Coulomb force between two charges $Q_1$ and $Q_2$ is known from the first chapter of the previous semester:

\begin{align*} F_C = {{{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1 \cdot Q_2} \over {r^2}}} \end{align*}

The force on a (fictitious) sample charge $q$ is now considered to obtain a measure of the magnitude of the electric field.

\begin{align*} F_C &= {{{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1 \cdot q} \over {r^2}}} \\ &= \underbrace{{{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}}}_\text{=independent of q} \cdot q \\ \end{align*}

As a result, the left part is a measure of the magnitude of the field, independent of the size of the sample charge $q$. Thus, the magnitude of the electric field is given by

The result is therefore \begin{align*} \boxed{F_C = E \cdot q} \end{align*}

The direction of the electric field is switchable in Abbildung 2 via the „Electric Field“ option on the right.

Electric Field Lines

Electric field lines result from the (fictitious) path of a sample charge. Thus, also electric field lines of several charges can be determined. However, these also result from a superposition of the individual effects - i.e., electric field - at a measuring point $P$.

The superposition is sketched in Abbildung 4: Two charges $Q_1$ and $Q_2$ act on the test charge $q$ with the forces $F_1$ and $F_2$. Depending on the positions and charges, the forces vary, and so does the resulting force. The simulation also shows a single field line.

For a full picture of the field lines between charges, one has to start with a single charge. The in- and outgoing lines on this charge are drawn equidistant from the charge. This is also true for the situation with multiple charges. However, there, the lines are not necessarily run radially anymore. The test charge is influenced by all the single charges, and therefore, the field lines can get bent.

In Abbildung 6 the field lines are shown. The additional „equipotential lines“ will be discussed later and can be deactivated by clearing the checkmark Show Equipotentials. Try the following in the simulation:

• Get accustomed to the simulation. You can…
  • … move the charges by drag and drop.
  • … add another Charge with Add » Add Point Charge.
  • … delete components with a right click on them and delete
• Where is the density of the field lines higher?
• How does the field between two positive charges look? How does it look between two different charges?

Electric Charge and Coulomb Force (reloaded)

The electric charge and Coulomb force have already been described last semester. However, some points are to be caught up here.

Direction of the Coulomb force and Superposition

In the case of the force, only the direction has been considered so far, e.g., direction towards the sample charge. For future explanations, it is important to include the cause and effect in the naming. This is done by giving the correct labeling of the subscript of the force. In Abbildung 3 (a) and (b), the convention is shown: A force $\vec{F}_{21}$ acts on charge $Q_2$ and is caused by charge $Q_1$. As a mnemonic, you can remember „tip-to-tail“ (first the effect, then the cause).

Furthermore, several forces on a charge can be superimposed, resulting in a single, equivalent force.
Strictly speaking, it must hold that $\varepsilon$ is constant in the structure. For example, the resultant force in Abbildung 3 Fig. (c) on $Q_3$ becomes equal to: $\vec{F_3}= \vec{F_{31}}+\vec{F_{32}}$.

Common pitfalls

• Treating force and field as the same thing; forgetting $\vec{F}=q\,\vec{E}$ and the positive-test-charge convention.
• Mixing units (${\rm N}$, ${\rm C}$, ${\rm V}$, ${\rm m}$): not recognizing $1~{\rm N/C}=1~{\rm V/m}$.
• Drawing field lines as closed loops or allowing them to intersect (source field: start at $+$, end at $-$; no crossings).
• Ignoring vector addition in superposition (adding magnitudes instead of vectors).
• Assuming field exists only when a test charge is present; the field is a property of space due to sources.
• Using point-charge formulas too near extended objects; not identifying homogeneous vs. inhomogeneous regions.
• Forgetting conductor boundary facts: lines must be perpendicular to ideal conducting surfaces; interior $|\vec{E}|=0$ in electrostatics.

Exercises

Embedded resources