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--- electrical_engineering_and_electronics_1:block11 [2025/10/31 21:07] – mexleadmin
+++ electrical_engineering_and_electronics_1:block11 [2025/11/02 16:54] (aktuell) – mexleadmin
@@ Zeile 1: / Zeile 1: @@
-====== Block 11 — Influence and displacement field ======
+===== Block 11 — Influence and Displacement Field ======
 ===== Learning objectives =====
 <callout>
 After this 90-minute block, you can
-  * ...
+  * explain **electrostatic induction** on conductors and argue why the interior of a conductor is field-free (Faraday cage).
+  * distinguish the **electric field strength** $\vec{E}$ from the **electric displacement flux density** $\vec{D}$ and state $ \vec{D} = \varepsilon \vec{E} = \varepsilon_0 \varepsilon_{\rm r}\vec{E}$.
+  * apply **Gauss’s law** for the displacement field to simple closed surfaces to relate enclosed charge $Q$ and flux $\oint \vec{D}\cdot {\rm d}\vec{A}$.
+  * determine $E(r)$ for parallel-plate and coaxial geometries starting from $\vec{D}$, then using $\vec{E}=\vec{D}/(\varepsilon_0\varepsilon_{\rm r})$.
+  * reason about **surface charge density** $\varrho_A = \Delta Q/\Delta A$ and the normal field at conductor surfaces.
+  * use typical **relative permittivities** $\varepsilon_{\rm r}$ to estimate field reduction in dielectrics.
+  * interpret **dielectric strength** $E_0$ (breakdown) and reason about its impact on design limits (safe $E$, spacing, material choice).
 </callout>
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== Preparation at Home =====
@@ Zeile 16: / Zeile 23: @@
   * ...
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== 90-minute plan =====
-  - Warm-up (x min):
+  - Warm-up (10 min):
-    - ....
+    - One-minute recap quiz (Block 10): equipotentials, field lines.
-  - Core concepts & derivations (x min):
+    - Demo: conductor in external field → Faraday cage effect (refer to the embedded sim in this block).
-    - ...
+  - Core concepts & derivations (45 min):
-  - Practice (x min): ...
+    - Induction on conductors: charge displacement, $E_{\rm inside}=0$, field normal to the surface.
-  - Wrap-up (x min): Summary box; common pitfalls checklist.
+    - Polarization of dielectrics; motivation for introducing $\vec{D}$.
+    - Definitions: $\vec{D}$, $\vec{E}$, $\varepsilon_0$, $\varepsilon_{\rm r}$; Gauss’s law with closed surface.
+    - Worked derivations via $\vec{D}$:
+      * Parallel plates: $D=Q/A \;\Rightarrow\; E = D/(\varepsilon_0\varepsilon_{\rm r})$.
+      * Coaxial cylinders: $D(r)=Q/(2\pi l r) \;\Rightarrow\; E(r)=D(r)/(\varepsilon_0\varepsilon_{\rm r})$.
+    - Material data: typical $\varepsilon_{\rm r}$; concept of **dielectric strength** $E_0$ and safe design margins.
+  - Practice (25 min):
+    - Short board tasks using pillbox surfaces to find $\varrho_A$ on a conductor.
+    - Mixed-dielectric capacitor slice: split voltages via constant $D$.
+    - Guided use of the embedded sims to observe field/equipotential behavior and verify normal field at surfaces.
+  - Wrap-up (10 min):
+    - Summary box (key formulas, when to start with $D$ vs. $E$).
+    - Common pitfalls checklist and quick self-test questions.
+    - Preview to Block 12 (capacitors from field viewpoint).
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== Conceptual overview =====
 <callout icon="fa fa-lightbulb-o" color="blue">
-  - ...
+  - **Conductors in electrostatics:** free charges move until $E_{\rm inside}=0$; the surface becomes an equipotential and field lines are perpendicular to it. \\ Induced charges live on the surface (surface density $\varrho_A$).
+  - **Dielectrics (polarization):** bound charges shift slightly → the macroscopic effect is a reduced $E$ compared to vacuum. \\ This motivates $\vec{D}$, which “counts causes” (free/enclosed charge) independent of polarization details.
+  - **Displacement field & Gauss’s law:** for any closed surface, the flux of $\vec{D}$ equals the enclosed charge: $Q=\oint \vec{D}\cdot{\rm d}\vec{A}$. \\ Choose the surface to exploit symmetry, get $\vec{D}$ first, then $\vec{E}$ via material law.
+  - **Permittivity:** $\varepsilon=\varepsilon_0\varepsilon_{\rm r}$ links $\vec{D}$ and $\vec{E}$. Larger $\varepsilon_{\rm r}$ → smaller $E$ for the same $D$ (same free charge).
+  - **Design limit:** when $|E|$ exceeds the **dielectric strength** $E_0$, breakdown occurs → current flows. \\ Safe design keeps $|E|\ll E_0$ by material choice and geometry (spacing, shaping to avoid high curvature).
 </callout>
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== Core content =====
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Electric Field inside of a conductor ====
@@ Zeile 54: / Zeile 82: @@
 </callout>
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Electrostatic Induction ====
@@ Zeile 146: / Zeile 174: @@
 </WRAP>
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Dielectric Constant (Permittivity) ====
@@ Zeile 152: / Zeile 180: @@
 \begin{align*}
-{{D}\over{E}} = \varepsilon = \varepsilon_{ \rm r} \cdot \varepsilon_0 \\
+{{D}\over{E}} = \varepsilon = \varepsilon_0 \cdot \varepsilon_{ \rm r} \\
-\boxed{D = \varepsilon_{ \rm r} \cdot \varepsilon_0 \cdot E}
+\boxed{D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E}
 \end{align*}
@@ Zeile 173: / Zeile 201: @@
 </WRAP>
+~~PAGEBREAK~~ ~~CLEARFIX~~
+==== Typical Geometries ====
+The "new" $D$-field is a nice tool, which helps to derive the $E$-field more easily. \\
+This shall be shown with the two most common geometries (which are the only one necessary for this course).
+<WRAP group>
+<WRAP half column>
+=== Field of a parallel Plates ===
+\\
+  * Nearly all of the field is between the plate (see <imgref ImgNr294> top), when the distance between the plates is much smaller than the width of the plates. \\  → Idealization: all of the field is inside. There is neither a stray field on the side, nor a field on top / below the structure. \\ \\
+  * All of the $D$-field of the charges is between the plates, and therefore through the area $A$ of the plates (see <imgref ImgNr294> bottom).  \\  → The $D$-Field is given as: $$ D = {{Q}\over{A}}$$
+  * Given $D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E$, the eletric field $E$ is: $$  E = {{Q}\over{ \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot A}} $$\\
+<imgcaption ImgNr294 | field of parallel plates (field line breaks are not correct)>
+</imgcaption> <WRAP>
+{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+10+348+0.048828125+461%0Ab+0+2+10+30+130+220+134+0%0Ab+0+2+-10+30+110+220+114+0%0A
+,400 noborder}} \\
+{{drawio>FieldParallelPlates01.svg}}
+</WRAP></WRAP>
+<WRAP half column>
+=== Field of a coaxial cylindrical Plates ===
+\\
+  * All of the field is between the plate (see <imgref ImgNr295> top). \\ \\ \\ \\ \\
+  * All of the $D$-field of the charges on the inner plate penetrates through any cylintrical area $A(l,r) = 2 \pi \cdot l \cdot r$.  \\  → The $D$-Field is given as: $$ D = {{Q}\over{A(l,r)}} = {{Q}\over{2 \pi \cdot l \cdot r}}$$
+  * Again given $D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E$, the eletric field $E$ is: $$  E = {{Q}\over{ \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot 2 \pi \cdot l \cdot r}} $$\\
+<imgcaption ImgNr295 | field of coaxial cylindrical plates >
+</imgcaption> <WRAP >
+{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+10+355+0.048828125+355%0AE+1+2+0+30+30+230+230+40+40+220+220+0%0Ae+1+2+-80+111+111+149+149+0%0A
+,400 noborder}} \\
+{{drawio>FieldCylPlates01.svg}}
+</WRAP></WRAP></WRAP>
+~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Dielectric strength of dielectrics ====
@@ Zeile 199: / Zeile 264: @@
 ~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== Common pitfalls =====
-  * ...
+  * Mixing up **cause** and **effect**: using $\oint \vec{E}\cdot{\rm d}\vec{A}$ to count charge. Use **$\vec{D}$** for Gauss’s law with charge; convert to $\vec{E}$ only via $\vec{E}=\vec{D}/(\varepsilon_0\varepsilon_{\rm r})$.
+  * Forgetting that the **interior of a conductor is field-free** in electrostatics and that $E$ is **normal** to an ideal conducting surface (no tangential $E$ on the surface).
+  * Assuming induced charges fill the **volume** of a conductor. They reside on the **surface**; use $\varrho_A$, not a volume density.
+  * Ignoring that **$D$ is continuous** in the normal direction across simple dielectric interfaces when no free surface charge is present; consequently, the **electric field changes** with $\varepsilon_{\rm r}$.
+  * Treating $\varepsilon_{\rm r}$ as a constant in all contexts. Real materials can be frequency/temperature dependent; here we use low-frequency values as stated.
+  * Checking breakdown with voltage only. The limit is on **field** $E$; always relate geometry (e.g., plate spacing, curvature) to $E$ and compare to **$E_0$** with units (e.g., $\,{\rm kV/mm}$).
 ===== Exercises =====
@@ Zeile 387: / Zeile 457: @@
 $d_{ \rm g} = 2.10~{ \rm mm}$
 #@HiddenEnd_HTML~1532R,Results~@#
-</WRAP></WRAP></panel>
-<panel type="info" title="Task 5.5.4 Spherical capacitor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
-Two concentric spherical conducting plates set up a spherical capacitor.
-The radius of the inner sphere is $r_{ \rm i} = 3~{ \rm mm}$, and the inner radius from the outer sphere is $r_{ \rm o} = 9~{ \rm mm}$.
-  - What is the capacity of this capacitor, given that air is used as a dielectric?
-  - What would be the limit value of the capacity when the inner radius of the outer sphere goes to infinity ($r_{ \rm o} \rightarrow \infty$)?
-<button size="xs" type="link" collapse="Loesung_1_5_4_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_1_5_4_Tipps" collapsed="true">
-  * What is the displacement flux density of the inner sphere?
-  * Out of this derive the strength of the electric field $E$
-  * What ist the general relationship between $U$ and $\vec{E}$? Derive from this the voltage between the spheres.
-</collapse>
-<button size="xs" type="link" collapse="Loesung_1_5_4_Endergebnis">{{icon>eye}} Result</button><collapse id="Loesung_1_5_4_Endergebnis" collapsed="true">
-  - $C = 0.5~{ \rm pF}$
-  - $C_{\infty} = 0.33~{ \rm pF}$
-</collapse>
 </WRAP></WRAP></panel>