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electrical_engineering_and_electronics_1:block13 [2025/11/01 00:12] – [Block xx - xxx] mexleadminelectrical_engineering_and_electronics_1:block13 [2025/11/02 17:48] (aktuell) – [Learning objectives] mexleadmin
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 <callout> <callout>
 After this 90-minute block, you can After this 90-minute block, you can
-  * Recognize a series connection of capacitors and distinguish it from a parallel connection, +  * identify series vs. parallel connections of capacitors from a circuit diagram, 
-  * Calculate the resulting total capacitance of a series or parallel circuit, +  * compute equivalent capacitance $C_{\rm eq}$ for series and parallelnetworks, 
-  * Know how the total charge is distributed among the individual capacitors in a parallel circuit, +  * use the key sharing rules: in **series** $Q_k=\text{const.}$ and voltages divide; in **parallel** $U_k=\text{const.}$ and charges divide, 
-  * Determine the voltage across a single capacitor in a series circuit.+  * apply the capacitor divider relation (two series capacitors), 
 +  * determine stored energy, including a dimensional check to $\rm J$.
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 ===== 90-minute plan ===== ===== 90-minute plan =====
-  - Warm-up (x min):  +  - Warm-up (10 min): 
-    - ....  +    - Quick quiz (2–3 items): series or parallel? which rule applies (constant $U$ or constant $Q$)? 
-  - Core concepts & derivations (x min): +    - Recall $Q=C\,U$ and energy $W=\tfrac12 C U^2$ (units). 
-    - ... +  - Core concepts & derivations (35 min): 
-  - Practice (x min): ... +    - Derive $C_{\rm eq}$ for **series** from Kirchhoff’s voltage law and $Q=\text{const.}$; derive voltage division $U_k=\dfrac{Q}{C_k}$. 
-  - Wrap-up (x min): Summary box; common pitfalls checklist.+    - Derive $C_{\rm eq}$ for **parallel** from Kirchhoff’s current/charge balance and $U=\text{const.}$; obtain $Q_k=C_k U$. 
 +    - Energy in the electric field: integrate $dW=U\,dq$ → $W=\tfrac12 C U^2$; short dimensional check. 
 +  - Practice (35 min): 
 +    - Two short worked examples: mixed series/parallel network; two-capacitor divider with given $U$ (find $U_1$, $U_2$, $W$ on each). 
 +    - Short simulation tasks (use the two embedded Falstad circuits in this page): observe $U_k$, $Q_k$ when toggling the switch or changing values. 
 +    - Mini-problems: “double a plate area / halve distance” reasoning on $C$ and $W$. 
 +  - Wrap-up (10 min): 
 +    - Common-pitfalls checklist and one exit-ticket calculation.
  
 ===== Conceptual overview ===== ===== Conceptual overview =====
 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
-  - ...+  - **What stays the same?** In **series** all capacitors carry the **same charge** $Q$; in **parallel** all capacitors see the **same voltage** $U$. 
 +  - **How do totals form?** Capacitances **add inversely** in series and **add directly** in parallel. This mirrors resistors but with the roles swapped. 
 +  - **Voltage/charge sharing:** In series, the **smaller** $C_k$ takes the **larger** $U_k$ ($U_k=Q/C_k$). In parallel, the **larger** $C_k$ takes the **larger** $Q_k$ ($Q_k=C_k U$). 
 +  - **Energy viewpoint:** Charging needs work against the field; $W=\tfrac12 C U^2=\tfrac12 Q U=\dfrac{Q^2}{2C}$. Dimensional check: $[C]=\rm F=\dfrac{A\,s}{V}$, so $[C U^2]=\dfrac{A\,s}{V}\,V^2=A\,s\,V=J$. 
 +  - **Design intuition:** Increasing plate area $A$ or dielectric $\varepsilon_r$ raises $C$ and thus stored $W$ at the same $U$; increasing gap $d$ lowers $C$.
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 ===== Common pitfalls ===== ===== Common pitfalls =====
-  * ...+  * Mixing up the rules: writing $C_{\rm eq}=C_1+C_2$ for **series** (wrong) or $\dfrac{1}{C_{\rm eq}}=\dfrac{1}{C_1}+\dfrac{1}{C_2}$ for **parallel** (wrong). 
 +  * Forgetting which quantity is equal: **series $\Rightarrow Q_k=\text{const.}$**, **parallel $\Rightarrow U_k=\text{const.}$**. 
 +  * Applying the **resistive** voltage divider $U_1=\dfrac{R_1}{R_1+R_2}U$ to capacitors. For capacitors in series it inverts: $U_1=\dfrac{C_2}{C_1+C_2}U$. 
 +  * Ignoring **initial charge states**: pre-charged capacitors reconnected will redistribute charge; use charge conservation on isolated nodes before using $Q=C\,U$. 
 +  * Dropping units or mixing forms of energy: always keep $W=\tfrac12 C U^2=\tfrac12 Q U=\dfrac{Q^2}{2C}$ and check $\rm J$.
  
 ===== Exercises ===== ===== Exercises =====