electrical_engineering_and_electronics_1:block10 [MEXLE Wiki]

Diese Seite ist nicht editierbar. Sie können den Quelltext sehen, jedoch nicht verändern. Kontaktieren Sie den Administrator, wenn Sie glauben, dass hier ein Fehler vorliegt.
====== Block 10 - Field Patterns of key Geometries ======

~~PAGEBREAK~~ ~~CLEARFIX~~
===== Learning objectives =====
<callout>
By the end of this section, you will be able to:
  * Explain and sketch **electric field lines** for single and multiple charges; state that line **direction** follows the force on a positive test charge and line **density** indicates $|\vec{E}|$.
  * Distinguish **homogeneous** fields (e.g. ideal parallel plates) from **inhomogeneous** fields (e.g. point charge, edges) and relate $E=\frac{U}{d}$ in plate geometries.
  * State conductor boundary facts in electrostatics: $\vec{E}$ is **perpendicular** to conducting surfaces and the **interior is field-free**; surfaces are **equipotentials**.
  * Use the **superposition principle** to construct field patterns.
  * Compute $|\vec{E}|$ for a **point charge** with $\varepsilon=\varepsilon_0\varepsilon_r$: $\displaystyle |\vec{E}|=\frac{1}{4\pi\varepsilon}\frac{|Q|}{r^2}$.
</callout>

~~PAGEBREAK~~ ~~CLEARFIX~~
===== 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

~~PAGEBREAK~~ ~~CLEARFIX~~
===== 90-minute plan =====
  * **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)**  
    - Rules for **field lines**: start at $+$, end at $-$, no intersections; density $\propto |\vec{E}|$; not particle trajectories.  
    - **Homogeneous vs. inhomogeneous**: parallel-plate region ($E=\frac{U}{d}$) vs. point/edge fields ($|\vec{E}|\sim 1/r^2$ near a point charge).  
    - **Conductors in electrostatics**: interior $E=0$, surface is an **equipotential**, $\vec{E}\perp$ surface; charge crowds near sharp curvature.  
    - **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).

===== Conceptual overview =====
<callout icon="fa fa-lightbulb-o" color="blue">
  - **Field lines** visualize $\vec{E}$: start at $+$, end at $-$, never intersect; higher line density ⇔ larger $|\vec{E}|$; lines are **not** particle trajectories.
  - **Homogeneous fields** (ideal between large parallel plates): parallel, equally spaced lines; **inhomogeneous fields** elsewhere (e.g., point charges, edges).
  - **Conductors (electrostatics)**: $\vec{E}$ is perpendicular to the surface; interior is field-free; surface charge arranges to enforce these conditions.

  * **What field lines mean:** visual aid for $\vec{E}$. \\ they start on positive charge and end on negative charge; their **density** reflects the **magnitude** $|\vec{E}|$; arrows show the **force direction on a positive test charge**. Lines never intersect.
  * **Homogeneous vs. inhomogeneous:** between large, parallel plates the field is approximately uniform with $E=\frac{U}{d}$; \\ around localized or curved conductors and point charges the field varies with position (e.g. $|\vec{E}|\propto 1/r^2$ for a point charge).
  * **Conductors (electrostatics):** inside an ideal conductor $E=0$; surfaces are equipotentials; \\ $\vec{E}$ meets the surface **perpendicularly**; surface charge re-arranges to enforce these conditions and concentrates at sharp edges.
  * **Superposition:** total field is the vector sum of contributions from all charges; use it to construct patterns for dipoles and multi-charge systems.

</callout>

~~PAGEBREAK~~ ~~CLEARFIX~~

===== 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: <WRAP centeralign>$\rho_l = {{Q}\over{l}}$</WRAP> or, in the case of different charge densities on subsections: <WRAP centeralign>$\rho_l = {{\Delta Q}\over{\Delta l}} \rightarrow \rho_l(l)={{\rm d}\over{{\rm d}l}} Q(l)$</WRAP>
  * 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: <WRAP centeralign>$\rho_A = {{Q}\over{A}}$</WRAP> or if there are different charge densities on partial surfaces: <WRAP centeralign>$\rho_A = {{\Delta Q}\over{\Delta A}} \rightarrow \rho_A(A) ={{\rm d}\over{{\rm d}A}} Q(A)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}} Q(A)$</WRAP>
  * 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: <WRAP centeralign>$\rho_V = {{Q}\over{V}}$</WRAP> or for different charge density in partial volumes: <WRAP centeralign>$\rho_V = {{\Delta Q}\over{\Delta V}} \rightarrow \rho_V(V) ={{\rm d}\over{{\rm d}V}} Q(V)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}}{{\rm d}\over{{\rm d}z}} Q(V)$</WRAP>

==== 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 <imgref ImgNr032>: 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. 

<WRAP>
<imgcaption ImgNr039 | examples of field lines>
</imgcaption> <WRAP>
{{url>https://www.geogebra.org/material/iframe/id/qIXZJKqj/width/500/height/500/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/true/at/preferhtml5 600,400 noborder}}
</WRAP></WRAP>

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. 

<WRAP>
<imgcaption ImgNr03 | examples of field lines>
</imgcaption> <WRAP>
{{drawio>ExamplesForFieldLines.svg}} 
</WRAP></WRAP>

In <imgref ImgNr031> 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?

<WRAP group>
<WRAP half column>
<imgcaption ImgNr031 | examples of field lines>
</imgcaption> <WRAP>
{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+1+435+0.048828125+403%0Ac+0+0+128+158+-1e-9%0Ac+0+0+128+98+1e-9%0A 600,600 noborder}}
</WRAP></WRAP>

<WRAP half column>
<imgcaption ImgNr032 | examples of field lines in 3D>
</imgcaption> <WRAP> \\
{{url>https://www.falstad.com/vector3de/vector3de.html?f=InverseSquaredRadialDipole&d=streamlines&sl=none&st=20&ld=7&a1=30&rx=-15&ry=76&rz=-97&zm=1.473 700,400 noborder}}
</WRAP></WRAP>
</WRAP>

<callout icon="fa fa-exclamation" color="red" title="Note:">
  - The electrostatic field is a source field. This means there are sources and sinks. 
  - From the field line diagrams, the following can be obtained:
    - Direction of the field ($\hat{=}$ parallel to the field line).
    - Magnitude of the field ($\hat{=}$ number of field lines per unit area).
  - The magnitude of the field along a field line is usually __not__ constant.
</callout>

<callout icon="fa fa-exclamation" color="red" title="Note:">
Field lines have the following properties:
  * The electric field lines have a beginning (at a positive charge) and an end (at a negative charge).
  * The direction of the field lines represents the direction of a **force onto a positive test charge**.
  * There are **no closed field lines** in electrostatic fields. The reason for this can be explained by considering the energy of the moved particle (see later subchapters).
  * Electric **field lines cannot cut** each other: This is based on the fact that the direction of the force at a cutting point would not be unique.
  * The field lines are **always perpendicular to conducting surfaces**
  * The **inside of a conducting component is always field-free**. 
  * The density of the field lines is a measure for the electric field density.

</callout>

~~PAGEBREAK~~ ~~CLEARFIX~~
==== Types of Fields depending on the Charge Distribution ====

There are two different types of fields:

<WRAP group><WRAP column half>
In **homogeneous fields**, magnitude and direction are constant throughout the field range. 
This field form is idealized to exist within plate capacitors. e.g., in the plate capacitor (<imgref ImgNr07>), or the vicinity of widely extended bodies.

Here, the electric field $E$ is given as: $E = {{U}\over{d}}$

<WRAP>
<imgcaption ImgNr07 | Field lines of a homogeneous field>
</imgcaption> \\
{{drawio>FieldLinesOfAHomogeneousField.svg}} \\
</WRAP>

</WRAP><WRAP column half>

For **inhomogeneous fields**, the magnitude and/or direction of the electric field changes from place to place. 
This is the rule in real systems, even the field of a point charge is inhomogeneous (<imgref ImgNr08>).  

For the given  example of a cylindrical configuration woth tha radius $r$, the electric field $E$ is given as: $E \\sim {{1}\over{r}}$


<WRAP>
<imgcaption ImgNr08 | Field lines of an inhomogeneous field>
</imgcaption> \\
{{drawio>FieldLinesOfAnInhomogeneousField.svg}} \\
</WRAP>

</WRAP> </WRAP> 

==== 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 <imgref ImgNr10>. 
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. \\

<WRAP>
<imgcaption ImgNr10 | Viewing a charged metal ball>
</imgcaption>
{{drawio>LoadedMetalBall.svg}}
<WRAP>

The movement of the charge continues until a force equilibrium is reached. 
In this steady state, there is no longer a resultant force acting on the single charge. 
In <imgref ImgNr10> this can be seen on the right: the repulsive forces of the charges are counteracted by the attractive forces of the atomic shells. \\
Results:
  * The charge carriers are distributed on the surface.
  * Due to the dispersion of the charges, the interior of the conductor is free of fields.
  * All field lines are perpendicular to the surface. Because: if they were not, there would be a parallel component of the field, i.e., along the surface. Thus, a force would act on charge carriers, and they would move accordingly.

<wrap #edu_task_1 />
<panel type="info" title="Educational Task - Why is there a discharge at pointy ends of conductors?"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>

Point discharge is a well-known phenomenon, which can be seen as {{wp>corona discharge}} on power lines (where it also creates the summing sound) or is used in {{wp>spark plug}}s. The phenomenon addresses the effect that there are many more charges at the corners and edges of a conductor. But why is that so? For this, it is feasible to try to calculate the charge density at different spots of a conductor.

<WRAP>
<imgcaption ImgNr193 | field of a pointy object (field line density is not correct)>
</imgcaption> <WRAP group><WRAP column half>
{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+6+310+0.048828125+390%0Ae+1+2+0+256+256+276+276+0%0Ae+1+2+0+256+256+276+276+0%0Ae+0+2+0+256+256+276+276+0%0Ae+0+2+100+126+80+141+120+0%0Ae+0+2+100+147+96+162+100+0%0Ae+0+2+100+125+89+153+108+0%0Aw+0+2+100+112+138+157+99%0Aw+0+2+100+110+66+157+98%0AE+1+2+100+47+57+137+147+65+95+99+129+0%0Ae+0+2+100+119+77+134+83+0%0Ae+0+2+100+131+105+146+111+0%0Ae+0+2+100+128+87+143+93+0%0A
 600,400 noborder}}
</WRAP><WRAP column half>
{{drawio>electrical_engineering_and_electronics_1:Fieldofapointyobject3d.svg}}
</WRAP></WRAP></WRAP>

<WRAP>
<imgcaption ImgNr194| examples for an arbitrarily formed conductor>
</imgcaption>
{{drawio>StrangelyFormedMetalObject.svg}}
</WRAP>

In the <imgref ImgNr194> an example of a charged and "pointy" conductor is given in image (a). The surface of the conductor is always at the same potential. 
To cope with this complex shape and the desired charge density, the following path shall be taken:
  -  It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm CG} =  \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ with $\vec{E} ={{1} \over {4\pi\cdot\varepsilon}} \cdot {{q} \over {r^2}} \cdot \vec{e}_r $, where $\vec{e}_r$ is the unit vector pointing radially away, ${ \rm C}$ is a point at distance $r_0$ from the charge and ${ \rm G}$ is the ground potential at infinity.
  - Compare the field and the potentials of the different spherical conductors in <imgref ImgNr194>, image (b). 
    - Are there differences for the electric field $\vec{E}$ outside the spherical conductors? Are the potentials on the surface the same? 
    - What can be conducted for the field of the three situations in (b) and (d), when the total charge on the surface is considered to be always the same?
  - For spherical conductors, the surface charge density is constant. Given that this charge density leads to the overall charge $q$, how does $\varrho_A$ depend on the radius $r$ of a sphere?
  - Now, the situation in (c) shall be considered. Here, all components are conducting, i.e., the potentials on the surface are similar. Both spheres shall be considered to be as far away from each other, so that they show an undisturbed field near their surfaces. In this case, charges on the surface of the curvature to the left and the right represent the same situation as in (a). For the next step, it is important that by this, the potentials of the left sphere with $q_1$ and $r_1$ and the right sphere with $q_2$ and $r_2$ are the same.  
    - Set up this equality formula based on the formula for the potential from question 1. 
    - Insert the relationship for the overall charges $q_1$ and $q_2$ based on the surface charge densities $\varrho_{A1}$ and $\varrho_{A2}$ of a sphere and their radii $r_1$ and $r_2$.
    - What is the relationship between the bending of the surface and the charge density?

</WRAP></WRAP></panel>

~~PAGEBREAK~~ ~~CLEARFIX~~
==== Dielectric strength ====

In [[block03#conductivity_of_matter|Block03]] we had a short look on conductivity of matter. \\
Here, we want to have again a look onto isolators. 

  * The ability to insulate is dependent on the material.
  * If a maximum electric field $E_0$ is exceeded, the insulating ability is eliminated.
    * One says: The insulator breaks down. This means that above this electric field, a current can flow through the insulator.
    * Examples are: Lightning in a thunderstorm, ignition spark, glow lamp in a {{wp>Test_light#One-contact_neon_test_lights|phase tester}}
    * The maximum electric field $E_0$ is referred to as ** dielectric strength** (in German: //Durchschlagfestigkeit// or //Durchbruchfeldstärke//).
    * $E_0$ depends on the material (see <tabref tab02>), but also on other factors (temperature, humidity, ...).

<WRAP 30em>
<tabcaption tab02| Dielectric strength>
^ Material        ^ Dielectric strength $E_0$ in ${ \rm kV/mm}$ ^
| air             | $\rm 0.1...0.3$  |
| SF6 gas         | $\rm 8$          |
| insulating oils | $\rm 5...30$     |
| vacuum          | $\rm 20...30$    |
| quartz          | $\rm 30...40$    |
| PP, PE          | $\rm 50$         |
| PS              | $\rm 100$        |
| distilled water | $\rm 70$         |
</tabcaption>
</WRAP>

~~PAGEBREAK~~ ~~CLEARFIX~~
===== Common pitfalls =====
  * Treating field lines as **charge paths**: they are drawings of direction/magnitude of $\vec{E}$, **not** particle trajectories.
  * Forgetting the **reference charge sign**: line arrows indicate the force on a **positive** test charge; forces on electrons point opposite to the arrows.
  * Mixing up **equipotentials** and field lines: equipotentials are everywhere **perpendicular** to field lines; they do **not** indicate current.
  * Assuming the plate field is always perfectly uniform: edge effects make real plate fields **inhomogeneous** away from the central region.
  * Ignoring conductor boundary conditions: in electrostatics the interior of a conductor is **field-free** and $\vec{E}$ is **normal** to the surface; any tangential $\vec{E}$ would drive charges until it vanishes.
  * Confusing $\vec{E}$ with $\vec{D}$: here we use $\vec{E}$ and **permittivity** $\varepsilon=\varepsilon_0\varepsilon_r$ for $|\vec{E}|=\frac{1}{4\pi\varepsilon}\frac{|Q|}{r^2}$.

~~PAGEBREAK~~ ~~CLEARFIX~~
===== Exercises =====
<panel type="info" title="Task 1.1.2 Field lines"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>

Sketch the field line plot for the charge configurations given in <imgref ImgNr04>. \\
Note:
  * The __overlaid__ picture is requested.
  * Make sure that it is a source field.

You can prove your result with the simulation <imgref ImgNr032>.

<WRAP>
<imgcaption ImgNr04| Task on field lines>
</imgcaption> <WRAP>
{{drawio>TaskOnFieldLines.svg}} \\
</WRAP>

</WRAP></WRAP></WRAP></panel>

{{page>task_1.2.5_with_calc&nofooter}}
{{page>task_1.2.6&nofooter}}
{{page>task_1.2.7&nofooter}}


===== Embedded resources =====


<WRAP column half>
Field lines of various extended charged objects
{{youtube>LB8Rhcb4eQM}}
</WRAP>