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electrical_engineering_and_electronics_1:block11 [2025/11/01 00:31] – mexleadminelectrical_engineering_and_electronics_1:block11 [2025/11/02 16:54] (aktuell) – mexleadmin
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-====== Block 11 — Influence and Displacement Field ======+===== Block 11 — Influence and Displacement Field ======
  
 ===== Learning objectives ===== ===== Learning objectives =====
 <callout> <callout>
 After this 90-minute block, you can 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). 
-  * Explain **electrostatic induction** on conductors: surface charge displacement, $\vec{E}$ perpendicular to the surface, and **field-free interior** in electrostatics (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}$. 
-  * Describe **polarization** of dielectrics (dipole alignment/bending) and its qualitative effect: partial cancellation of the external field inside the material. +  * apply **Gauss’s law** for the displacement field to simple closed surfaces to relate enclosed charge $Q$ and flux $\oint \vec{D}\cdot {\rm d}\vec{A}$. 
-  * Define and use the **electric displacement field** $\vec{D}$ with correct units $[ \vec{D} ] = {\rm C/m^2}$ and apply **Gauss’s law for $\vec{D}$**: +  * determine $E(r)$ for parallel-plate and coaxial geometries starting from $\vec{D}$, then using $\vec{E}=\vec{D}/(\varepsilon_0\varepsilon_{\rm r})$. 
-    \[ +  * reason about **surface charge density** $\varrho_A = \Delta Q/\Delta A$ and the normal field at conductor surfaces. 
-      \displaystyle {\rlap{\rlap{\int_A} \int} \: \LARGE \circ} \vec{D}\cdot{\rm d}\vec{A} \;=\; Q \, . +  * use typical **relative permittivities** $\varepsilon_{\rm r}$ to estimate field reduction in dielectrics. 
-    \] +  * interpret **dielectric strength** $E_0$ (breakdown) and reason about its impact on design limits (safe $E$, spacing, material choice).
-  * Use the **constitutive relation** in linear, isotropic media: $\vec{D}=\varepsilon\,\vec{E}$ with $\varepsilon=\varepsilon_0\varepsilon_{\rm r}$; check dimensions. +
-  * Apply **boundary conditions** at interfaces: $\vec{E}$ normal to ideal conductors; continuity of $\vec{E}_t$; jump of $D_n$ equals surface charge density; equipotential conductor surfaces. +
-  * Interpret **dielectric strength** $E_0$ (breakdown) and reason about its impact on design limits.+
 </callout> </callout>
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== Preparation at Home ===== ===== Preparation at Home =====
  
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   * ...   * ...
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== 90-minute plan ===== ===== 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 ===== ===== Conceptual overview =====
 <callout icon="fa fa-lightbulb-o" color="blue"> <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> </callout>
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== Core content ===== ===== Core content =====
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Electric Field inside of a conductor ==== ==== Electric Field inside of a conductor ====
  
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 </callout> </callout>
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Electrostatic Induction ==== ==== Electrostatic Induction ====
  
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 </WRAP> </WRAP>
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Dielectric Constant (Permittivity) ==== ==== Dielectric Constant (Permittivity) ====
  
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 \begin{align*} \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*} \end{align*}
  
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 </WRAP> </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
 + 600,400 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
 + 600,400 noborder}} \\
 +
 +{{drawio>FieldCylPlates01.svg}}
 +</WRAP></WRAP></WRAP>
 +
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== Dielectric strength of dielectrics ==== ==== Dielectric strength of dielectrics ====
  
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== Common pitfalls ===== ===== 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 ===== ===== Exercises =====