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electrical_engineering_and_electronics_1:block12 [2025/11/02 17:38] – [Symbols] mexleadminelectrical_engineering_and_electronics_1:block12 [2026/01/10 12:53] (aktuell) – mexleadmin
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 ====== Block 12 - Capacitors and Capacitance ====== ====== Block 12 - Capacitors and Capacitance ======
  
-===== Learning objectives =====+===== 12.0 Intro ===== 
 + 
 +==== 12.0.1 Learning Objectives ====
 <callout> <callout>
 After this 90-minute block, you can  After this 90-minute block, you can 
-  - define a **capacitor** and **capacitance** $C=\dfrac{Q}{U}$ and use $C=\varepsilon_0\,\varepsilon_{\rm r}\,\dfrac{A}{d}$ for an ideal plate capacitor, including unit checks $[C]={\rm F}$.  +  - define a **capacitor** and **capacitance** $C$ and use it for an ideal plate capacitor, including unit checks $[C]={\rm F}$.  
-  - relate fields and material: $\vec{D}=\varepsilon\,\vec{E}$, $Q=\displaystyle\oint \vec{D}\cdot{\rm d}\vec{A}$ (Gauss), and $U=\displaystyle\int \vec{E}\cdot{\rm d}\vec{s}$. +  - relate fields and material. 
   - compute $C$ for key geometries (parallel plates, coaxial, spherical) and explain how $A$, $d$, $\varepsilon_{\rm r}$ scale $C$.    - compute $C$ for key geometries (parallel plates, coaxial, spherical) and explain how $A$, $d$, $\varepsilon_{\rm r}$ scale $C$. 
 </callout> </callout>
  
-===== Preparation at Home =====+==== 12.0.2 Preparation at Home ====
  
 Well, again  Well, again 
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 For checking your understanding please do the following exercises: For checking your understanding please do the following exercises:
-  * ...+  * 5.5.1 
 +  * 5.5.3 
 +  * 5.5.4
  
-===== 90-minute plan =====+==== 12.0.3 90-minute plan ====
   - Warm-up (8 min):   - Warm-up (8 min):
     - Quick recall quiz: $C=\dfrac{Q}{U}$, $\vec{D}=\varepsilon\vec{E}$, units of $E$ (${\rm V/m}$), $D$ (${\rm C/m^2}$).      - Quick recall quiz: $C=\dfrac{Q}{U}$, $\vec{D}=\varepsilon\vec{E}$, units of $E$ (${\rm V/m}$), $D$ (${\rm C/m^2}$). 
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   - Wrap-up (2 min): Summary box + pitfalls checklist; connect to next block (capacitor circuits).   - Wrap-up (2 min): Summary box + pitfalls checklist; connect to next block (capacitor circuits).
  
-===== Conceptual overview =====+==== 12.0.4 Conceptual overview ====
 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
   - A **capacitor** is two conductors separated by a dielectric. It stores **charge** and **energy** in the electric field; no conduction current flows through the ideal dielectric. :contentReference[oaicite:15]{index=15}   - A **capacitor** is two conductors separated by a dielectric. It stores **charge** and **energy** in the electric field; no conduction current flows through the ideal dielectric. :contentReference[oaicite:15]{index=15}
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 </callout> </callout>
  
-===== Core content =====+===== 12.1 Core content =====
  
- +==== 12.1.1 Capacitor ====
-==== Capacitor ====+
  
   * A capacitor can "store" charges. The total charge of a two-plate capacitor is in general 0.    * A capacitor can "store" charges. The total charge of a two-plate capacitor is in general 0. 
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 $\rightarrow$ Thus, for any arrangement of two conductors separated by an insulating material, a capacitance can be specified. $\rightarrow$ Thus, for any arrangement of two conductors separated by an insulating material, a capacitance can be specified.
  
-==== Capacitance $C$ ====+==== 12.1.2 Capacitance $C$ ====
  
 The capacitance $C$ can be derived as follows: The capacitance $C$ can be derived as follows:
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 </WRAP></WRAP> </WRAP></WRAP>
  
-==== Symbols ====+==== 12.1.3 Symbols ====
  
   * The symbol of a general capacitor is given be two parallel lines nearby each other. \\   * The symbol of a general capacitor is given be two parallel lines nearby each other. \\
-  * Since **electrolytic capacitors** can only withstand voltage in one direction, the **polarisation** is often shown by a curved electrode (US) or a unfilled one (EU).+  * Since **electrolytic capacitors** can only withstand voltage in one direction, the **polarisation** is often shown by a curved electrode (US) or a unfilled one (EU). \\ Be aware that electrolytic capacitors can explode, once used in the wrong direction.
  
 {{drawio>electrical_engineering_and_electronics_1:CapSymbols01.svg}} {{drawio>electrical_engineering_and_electronics_1:CapSymbols01.svg}}
  
-==== Designs and types of capacitors ====+==== 12.1.4 Designs and types of capacitors ====
  
 To calculate the capacitance of different designs, the definition equations of $\vec{D}$ and $\vec{E}$ are used. This can be viewed in detail, e.g., in [[https://www.youtube.com/watch?v=kAXg1xMkR_4&ab_channel=PatrickKaploo|this video]]. \\ Based on the geometry, different equations result (see also <imgref ImgNr17>). To calculate the capacitance of different designs, the definition equations of $\vec{D}$ and $\vec{E}$ are used. This can be viewed in detail, e.g., in [[https://www.youtube.com/watch?v=kAXg1xMkR_4&ab_channel=PatrickKaploo|this video]]. \\ Based on the geometry, different equations result (see also <imgref ImgNr17>).
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-===== Common pitfalls =====+===== 12.2 Common pitfalls =====
  
   * **Mixing up geometry symbols.** Use $d$ (or $l$) strictly as **plate spacing** and $A$ as **active area** in $C=\varepsilon_0\varepsilon_{\rm r}\dfrac{A}{d}$. Check units to catch mistakes.    * **Mixing up geometry symbols.** Use $d$ (or $l$) strictly as **plate spacing** and $A$ as **active area** in $C=\varepsilon_0\varepsilon_{\rm r}\dfrac{A}{d}$. Check units to catch mistakes. 
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   * **Real-part issues.** Ignoring polarity of electrolytics and tolerance spreads ($\pm 10~\%$ and more) causes design errors; pick suitable component types.    * **Real-part issues.** Ignoring polarity of electrolytics and tolerance spreads ($\pm 10~\%$ and more) causes design errors; pick suitable component types. 
  
-===== Exercises ===== +===== 12.3 Exercises =====
  
 <panel type="info" title="Task 5.5.1 induced Charges"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Task 5.5.1 induced Charges"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
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 \end{align*} \end{align*}
  
-Since we know that the distance of the air gap is $d_{\rm a} = d_0 - d_{\rm a}$ we can calculate:+Since we know that the distance of the air gap is $d_{\rm a} = d_0 - d_{\rm g}$ we can calculate:
 \begin{align*} \begin{align*}
 E_{\rm g}  &= {{5'000 ~\rm V}\over{0.004 ~{\rm m} + 8 \cdot 0.002 ~{\rm m}}} \\ E_{\rm g}  &= {{5'000 ~\rm V}\over{0.004 ~{\rm m} + 8 \cdot 0.002 ~{\rm m}}} \\