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Block 10 - Field patterns of key geometries

Learning objectives

Preparation at Home

And again:

• Please read through the following chapter.
• Also here, there are some clips for more clarification under 'Embedded resources'.

For checking your understanding please do the following exercise:

• 1.1.2
• 1.2.5

90-minute plan

1. Warm-up (8–10 min):
2. Concept build & demonstrations (35–40 min):
   1. Field lines: definition, drawing rules, sources/sinks, no intersections; relate density to magnitude.
   2. Homogeneous vs. inhomogeneous fields; conductor boundary facts (perpendicular $\vec{E}$, interior field-free).
3. Guided simulations (20–25 min)
4. Practice (10–15 min):
   1. Short worksheet: sketch field lines for two like charges and a dipole; compute $|\vec{E}|$ at a marked point.
5. Wrap-up (5 min):
   1. Summary map: link to equipotentials and energy (next block).

Conceptual overview

Core content

Geometric Distribution of Charges

In previous chapters, only single charges (e.g., $Q_1$, $Q_2$) were considered.

• The charge $Q$ was previously reduced to a point charge.
  This can be used, for example, for the elementary charge or for extended charged objects from a large distance. The distance is sufficiently large if the ratio between the largest object extent and the distance to the measurement point $P$ is small.
• If the charges are lined up along a line, this is referred to as a line charge.
  Examples of this are a straight trace on a circuit board or a piece of wire. Furthermore, this also applies to an extended charged object, which has exactly an extension that is no longer small in relation to the distance. For this purpose, the charge $Q$ is considered to be distributed over the line. Thus, a (line) charge density $\rho_l$ can be determined:

  $\rho_l = {{Q}\over{l}}$

  or, in the case of different charge densities on subsections:

  $\rho_l = {{\Delta Q}\over{\Delta l}} \rightarrow \rho_l(l)={{\rm d}\over{{\rm d}l}} Q(l)$

• It is spoken of as an area charge when the charge is distributed over an area.
  Examples of this are the floor or the plate of a capacitor. Again, an extended charged object can be considered when two dimensions are no longer small in relation to the distance (e.g. surface of the earth). Again, a (surface) charge density $\rho_A$ can be determined:

  $\rho_A = {{Q}\over{A}}$

  or if there are different charge densities on partial surfaces:

  $\rho_A = {{\Delta Q}\over{\Delta A}} \rightarrow \rho_A(A) ={{\rm d}\over{{\rm d}A}} Q(A)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}} Q(A)$

• Finally, a space charge is the term for charges that span a volume.
  Here, examples are plasmas or charges in extended objects (e.g., the doped volumes in a semiconductor). As with the other charge distributions, a (space) charge density $\rho_V$ can be calculated here:

  $\rho_V = {{Q}\over{V}}$

  or for different charge density in partial volumes:

  $\rho_V = {{\Delta Q}\over{\Delta V}} \rightarrow \rho_V(V) ={{\rm d}\over{{\rm d}V}} Q(V)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}}{{\rm d}\over{{\rm d}z}} Q(V)$

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 3 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?

Types of Fields depending on the Charge Distribution

There are two different types of fields:

Stationary Situation of a charged conducting Object (without an external Field)

In the first thought experiment, a conductor (e.g., a metal plate) is charged, see Abbildung 7. The additional charges create an electric field. Thus, a resultant force acts on each charge. The causes of this force are the electric fields of the surrounding electric charges. So the charges repel and move apart.