Block 10 - Field Patterns of key Geometries

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

• Warm-up (8–10 min)

Quick sketches: single charge, dipole, parallel plates. Poll for rules of field lines and equipotentials.

• Concept build & demonstrations (35–40 min)
  1. Rules for field lines: start at $+$, end at $-$, no intersections; density $\propto |\vec{E}|$; not particle trajectories.
  2. Homogeneous vs. inhomogeneous: parallel-plate region ($E=\frac{U}{d}$) vs. point/edge fields ($|\vec{E}|\sim 1/r^2$ near a point charge).
  3. Conductors in electrostatics: interior $E=0$, surface is an equipotential, $\vec{E}\perp$ surface; charge crowds near sharp curvature.
  4. Superposition: build dipole and two-like-charge patterns from single-charge fields.
• Guided simulations (20–25 min)

Move charges, toggle equipotentials, and compare line density to indicated $|\vec{E}|$; vary plate spacing $d$ and discuss $E=\frac{U}{d}$ (units: $\rm V/m$).

• Practice (10–15 min)

Mini-worksheet: sketch fields for two like charges and a dipole; mark where $|\vec{E}|$ is largest; short calc: $|\vec{E}|$ at $r$ from a charge.

• Wrap-up (5 min)

Summary map linking field lines ↔ equipotentials ↔ potential difference as bridge to capacitors and energy (next blocks).

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 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 figure 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 figure 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?

There are two different types of fields:

In the first thought experiment, a conductor (e.g., a metal plate) is charged, see figure 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.