| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung |
| electrical_engineering_and_electronics_1:block13 [2025/11/01 00:12] – [Block xx - xxx] mexleadmin | electrical_engineering_and_electronics_1:block13 [2025/11/22 19:41] (aktuell) – 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 $\rm J$. |
| </callout> | </callout> |
| |
| |
| For checking your understanding please do the following exercises: | For checking your understanding please do the following exercises: |
| * ... | * 5.9.5 |
| |
| ===== 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 $C$. |
| </callout> | </callout> |
| |
| * The capacitors can be discharged again via the lamp. | * The capacitors can be discharged again via the lamp. |
| |
| <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0lwrFWAmZBmMbJgGzYCwAc+yAnIciGmiIvjfAKYC0YYAUAMYjIDslYXjh79wpSpCTZw0ZPEjJ8pZSuW8p2dmEIQiksPnwg9VNMMlYYcazZvCs4SVenOOAZREDDnkKXNQQADMAQwAbAGdGAOR2AHcqYl9hEz8oOONCSVSTAyNIdJNqexxCU3N0uWFUwSrygDdwITLG2oDJfElEJyR4dgAnFua+SiK26F6Ac0HRmqoStvS0RNTkQlLU-Pi0eBGzH1HNn1yjpo544Z5IUovFfS4fW4f4fysIMFl5OVsbGlcK0RuoiwRg43AuwP2yG6zkQyGgSlI8m+cFWGjOR282y8eX+lChpSxl1K+QADlQduBMRT8eAFmTCTSGaIIE52PTqUCKTMFlsuXszAS9ocmXiAaJDjdng9xbixKK8Vc0ucgVD9kL2AAPHhgRCzdACXikYw8IwAYU1PDQAh0JutRhBpQ4WvQRkeild9GN2h4FuQgktsLQrqWxsUIHNzrARrI6nQ9jgobNvv5ChN8bo4AJFoMsdTOuEUIz3vC2YcUfUOcu7RNIAAqhazAJ4Oo0IbwHhE3WG23cCNCEbcEXHd3o3IEqPSg6fVrGzxVa3RzQPV2Zzg8YQW22yNWw-WZ1vxOOeIivVnIyDUeh9BhO25ff6zAWg3MjcuAErsIA 600,600 noborder}} | <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0lwrFWAmZBmMbJgGzYCwAc+yAnIciGmiIvjfAKYC0YYAUAMYjIDslYXjh79wpSviTZw0ZPEjJ8pZSuW8p0yOzCEIRSCFLD9VNMINYYcazZvCs4A1c2xs7AMoiB+SX0pGoQIAzAEMAGwBnRkDkdgB3EBNqe3hKZKh4nnhhAMEc83YAN3AhU2E8w3NA-ANEJyR4dgBzEvszVqocQkCtBJMA5EJugN6qVLKvCdG-cB9Jio4EmeRIbuWa8C5JxQNl7MSpCDBZeTlbGxpnCGnRZdEsSQ5uGYfJtBWD50RkaCVSeXOcEGGmumRmYDmaHGEMkN0oK26UPhqwyAAcxt5JEieCjrj12OjsQiMfMemiSS9xuk8U5MkTbgzKHCssI7kywaIwOJtijme9dvd2loAB48MCICroAS8UgHRQgADC7FFUvAuh4JDVkke3Q4KrQvg2ikNNB1PGVYvU6G+Bp4aG6hsVFuQXJ4pCt7UscskSv19gUGv9dDVVAtEKtAfFrMgwZ0IAiYYcXPU4ZxBkdAFULWYBPB1GgZeA8N6QFnRQXZbg0oRKzhY7rs4XUIh7bLUA6Q7Fyzh4R8Kzx0CWy514YR803SOmNaXG23ua23fXQyqIeBgegDJg9NP3By0u0KuktEA 600,600 noborder}} |
| </WRAP> | </WRAP> |
| |
| In the simulation below, again, besides the parallel connected capacitors $C_1$, $C_2$,$C_3$, an ideal voltage source $U_q$, a resistor $R$, a switch $S$, and a lamp are installed. | In the simulation below, again, besides the parallel connected capacitors $C_1$, $C_2$,$C_3$, an ideal voltage source $U_q$, a resistor $R$, a switch $S$, and a lamp are installed. |
| |
| <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l3EaQMxjAFhQJi8gnBgKx4DsJ45hI6yIhApgLRoBQAxiFpABwUBsnHuDxYoUWJFHpoeWXPkLxCFmG4RkfXhnQgNvLIQGRwuacnQHuyQujBZu3Hcjjgl8BR8VJ3n3ywDKglroOly8eEZiAGYAhgA2AM70YlgsAO66XCARuprZRum54QJ64CFQhaUGAvb6hhUZ6MUgjE35FQBuLW3V3c3Gxph0YgPQhCwA5n2c9a36DiOVWTk4OjmQhWEzAnZ1BRlb2kH84IW7ZTrnYCQ77CZaN-fCoqMI1DK+Hm5ghW1Hf48fhwAQI-iIRhJIFRpJ8vt4fgchEdahcGsdeijehsAA6cBbIha9RBonr1c5Ys72Z5PLYbRFacFbWm-BlSfRCOnUZrndaU7ng3n01FXcqckU6P6ilgAJ3RQiZ9VekB+svJSKpWLcqVlPSEpMi3nQhTm2y5po2AA8WtxyGBCORGCRjGA+INODoAKosK2MPh4cB8CC+518CXgXgJb14ii0ZAPUHh05WsAh-1YW0kKTukAAYSjeFoLv9aGdmeo2bzybQMZMEGuCdUnCj10LfuEpazWB0leEdZItC4fYbvGQLCAA 700,500 noborder}} | <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l3EaQMxjAFhQJi8gnBgKx4DsJ45hI6yIhApgLRoBQAxiFpABwUBsnHuDxZqUWJFHpoeWXPkLxCKCzDcIyPrzwDNvLIQGRwOQtDVbCk5N0LqQyOOCXwFbxUlfvvLAMqDeDHQAkB0ocIAzAEMAGwBnenCsFgB3By1OQ05ufSzIVOFRMMZ0bSMWADcQEtyBGtCjcMw6cOMYQhYAc2rSzLrerBzWgr0GzixgsPy0rlqTOemQoKWSATACsEHwdGDNwNXwdnn+Y7ARMU8IaW93F0QC3uXHg-WOZ4FH8+DPSCprm48EnuMyEy0GgR2KhBc3BfRUAAdshDgrCDGtWlDqHM9nDFjizqIcWjMbNCiFiYtPlIFgVSct8ZC8VsnsjMfViQM8iwAB7VbjkMCEciMEjGMB8Ywo4IAVR51T4eHAfAgjAlSuCu14cTlWxwGn0EzEakOvLAxiwYHIWCtXDEhoAwnK8LQLQKzeNJZxgo7TWhxhBNhAcBrwPo5ZaXZbhObbVKQD7hEGLYIkyHjcgWEA 700,500 noborder}} |
| </WRAP> | </WRAP> |
| |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ |
| ===== 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 ===== |
| </WRAP></WRAP></panel> | </WRAP></WRAP></panel> |
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| {{page>electrical_engineering_and_electronics_2:task_5.9.3_with_calculation&nofooter}} | {{page>electrical_engineering_and_electronics:task_5.9.3_with_calculation&nofooter}} |
| | {{page>electrical_engineering_and_electronics:task_k4wrrhf8v46gct49_with_calculation&nofooter}} |
| | {{page>electrical_engineering_and_electronics:task_y7dozgdsljqvnqge_with_calculation&nofooter}} |
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| ===== Embedded resources ===== | ===== Embedded resources ===== |