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electrical_engineering_and_electronics_1:block10 [2025/11/01 00:13] – [Block 10 - Field patterns of key geometries] mexleadminelectrical_engineering_and_electronics_1:block10 [2026/01/10 12:58] (aktuell) – mexleadmin
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 ====== Block 10 - Field Patterns of key Geometries ====== ====== Block 10 - Field Patterns of key Geometries ======
  
-~~PAGEBREAK~~ ~~CLEARFIX~~ +===== 10.0 Intro ===== 
-===== Learning objectives =====+ 
 +==== 10.0.1 Learning objectives ====
 <callout> <callout>
 By the end of this section, you will be able to: By the end of this section, you will be able to:
-  * Sketch the field lines of electric fields. +  * Explain and sketch **electric field lines** for single and multiple charges; state that line **direction** follows the force on a positive test charge and line **density** indicates $|\vec{E}|$. 
-  * Describe and sketch **field lines** for single and multiple charges; relate line **density** to $|\vec{E}|$ and line **direction** to the force on a positive test charge. +  * Distinguish **homogeneous** fields (e.g. ideal parallel plates) from **inhomogeneous** fields (e.g. point charge, edges) and relate $E=\frac{U}{d}$ in plate geometries. 
-  * Classify fields as **homogeneous** (e.g., parallel-plate region) or **inhomogeneous** (e.g., point charge); state typical properties near **conductors** (perpendicular boundary, field-free interior in electrostatics). +  * State conductor boundary facts in electrostatics: $\vec{E}$ is **perpendicular** to conducting surfaces and the **interior is field-free**; surfaces are **equipotentials**. 
-  * Compute $|\vec{E}|$ for a **point charge** (Coulomb force), identify $\varepsilon$ and check dimensions.+  * Use the **superposition principle** to construct field patterns. 
 +  * Compute $|\vec{E}|$ for a **point charge** with $\varepsilon=\varepsilon_0\varepsilon_r$: $\displaystyle |\vec{E}|=\frac{1}{4\pi\varepsilon}\frac{|Q|}{r^2}$.
 </callout> </callout>
  
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-===== Preparation at Home =====+==== 10.0.2 Preparation at Home ====
  
 And again:  And again: 
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-===== 90-minute plan ===== +==== 10.0.3 90-minute plan ==== 
-  - Warm-up (8–10 min): +  * **Warm-up (8–10 min)**   
-  - Concept build & demonstrations (35–40 min): +    Quick sketches: single charge, dipole, parallel plates. Poll for rules of field lines and equipotentials. 
-    - **Field lines**: definition, drawing rules, sources/sinks, no intersections; relate density to magnitude. +  * **Concept build & demonstrations (35–40 min)**   
-    - **Homogeneous vs. inhomogeneous** fields; conductor boundary facts (perpendicular $\vec{E}$, interior field-free). +    - Rules for **field lines**: start at $+$, end at $-$, no intersections; density $\propto |\vec{E}|$; not particle trajectories.   
-  - Guided simulations (20–25 min) +    - **Homogeneous vs. inhomogeneous**: parallel-plate region ($E=\frac{U}{d}$) vs. point/edge fields ($|\vec{E}|\sim 1/r^2$ near a point charge).   
-  - Practice (10–15 min): +    - **Conductors in electrostatics**: interior $E=0$, surface is an **equipotential**, $\vec{E}\perp$ surface; charge crowds near sharp curvature.   
-    - Short worksheet: sketch field lines for two like charges and a dipole; compute $|\vec{E}|$ at a marked point. +    - **Superposition**: build dipole and two-like-charge patterns from single-charge fields. 
-  - Wrap-up (5 min): +  * **Guided simulations (20–25 min)**   
-    - Summary map: link to **equipotentials** and energy (next block).+    Move charges, toggle equipotentials, and compare line density to indicated $|\vec{E}|$; vary plate spacing $d$ and discuss $E=\frac{U}{d}$ (units: $\rm V/m$). 
 +  * **Practice (10–15 min)**   
 +    Mini-worksheet: sketch fields for two like charges and a dipole; mark where $|\vec{E}|$ is largest; short calc: $|\vec{E}|$ at $r$ from a charge. 
 +  * **Wrap-up (5 min)**   
 +    Summary map linking **field lines ↔ equipotentials ↔ potential difference** as bridge to capacitors and energy (next blocks).
  
-===== Conceptual overview =====+==== 10.0.4 Conceptual overview ====
 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
   - **Field lines** visualize $\vec{E}$: start at $+$, end at $-$, never intersect; higher line density ⇔ larger $|\vec{E}|$; lines are **not** particle trajectories.   - **Field lines** visualize $\vec{E}$: start at $+$, end at $-$, never intersect; higher line density ⇔ larger $|\vec{E}|$; lines are **not** particle trajectories.
   - **Homogeneous fields** (ideal between large parallel plates): parallel, equally spaced lines; **inhomogeneous fields** elsewhere (e.g., point charges, edges).   - **Homogeneous fields** (ideal between large parallel plates): parallel, equally spaced lines; **inhomogeneous fields** elsewhere (e.g., point charges, edges).
   - **Conductors (electrostatics)**: $\vec{E}$ is perpendicular to the surface; interior is field-free; surface charge arranges to enforce these conditions.   - **Conductors (electrostatics)**: $\vec{E}$ is perpendicular to the surface; interior is field-free; surface charge arranges to enforce these conditions.
 +
 +  * **What field lines mean:** visual aid for $\vec{E}$. \\ they start on positive charge and end on negative charge; their **density** reflects the **magnitude** $|\vec{E}|$; arrows show the **force direction on a positive test charge**. Lines never intersect.
 +  * **Homogeneous vs. inhomogeneous:** between large, parallel plates the field is approximately uniform with $E=\frac{U}{d}$; \\ around localized or curved conductors and point charges the field varies with position (e.g. $|\vec{E}|\propto 1/r^2$ for a point charge).
 +  * **Conductors (electrostatics):** inside an ideal conductor $E=0$; surfaces are equipotentials; \\ $\vec{E}$ meets the surface **perpendicularly**; surface charge re-arranges to enforce these conditions and concentrates at sharp edges.
 +  * **Superposition:** total field is the vector sum of contributions from all charges; use it to construct patterns for dipoles and multi-charge systems.
 +
 </callout> </callout>
  
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-===== Core content =====+===== 10.1 Core content =====
  
-==== Geometric Distribution of Charges ====+==== 10.1.1 Geometric Distribution of Charges ====
  
 In previous chapters, only single charges (e.g., $Q_1$, $Q_2$) were considered.   In previous chapters, only single charges (e.g., $Q_1$, $Q_2$) were considered.  
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   * Finally, a **space charge** is the term for charges that span a volume. \\ Here, examples are plasmas or charges in extended objects (e.g., the doped volumes in a semiconductor). As with the other charge distributions, a (space) charge density $\rho_V$ can be calculated here: <WRAP centeralign>$\rho_V = {{Q}\over{V}}$</WRAP> or for different charge density in partial volumes: <WRAP centeralign>$\rho_V = {{\Delta Q}\over{\Delta V}} \rightarrow \rho_V(V) ={{\rm d}\over{{\rm d}V}} Q(V)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}}{{\rm d}\over{{\rm d}z}} Q(V)$</WRAP>   * Finally, a **space charge** is the term for charges that span a volume. \\ Here, examples are plasmas or charges in extended objects (e.g., the doped volumes in a semiconductor). As with the other charge distributions, a (space) charge density $\rho_V$ can be calculated here: <WRAP centeralign>$\rho_V = {{Q}\over{V}}$</WRAP> or for different charge density in partial volumes: <WRAP centeralign>$\rho_V = {{\Delta Q}\over{\Delta V}} \rightarrow \rho_V(V) ={{\rm d}\over{{\rm d}V}} Q(V)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}}{{\rm d}\over{{\rm d}z}} Q(V)$</WRAP>
  
-==== Electric Field Lines ====+==== 10.1.2 Electric Field Lines ====
  
 Electric field lines result from the (fictitious) path of a sample charge. Thus, also electric field lines of several charges can be determined.  Electric field lines result from the (fictitious) path of a sample charge. Thus, also electric field lines of several charges can be determined. 
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-==== Types of Fields depending on the Charge Distribution ====+==== 10.1.3 Types of Fields depending on the Charge Distribution ====
  
 There are two different types of fields: There are two different types of fields:
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 </WRAP> </WRAP>  </WRAP> </WRAP> 
  
-==== Stationary Situation of a charged conducting Object (without an external Field) ====+==== 10.1.4 Stationary Situation of a charged conducting Object (without an external Field) ====
  
 In the first thought experiment, a conductor (e.g., a metal plate) is charged, see <imgref ImgNr10>.  In the first thought experiment, a conductor (e.g., a metal plate) is charged, see <imgref ImgNr10>. 
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-==== Dielectric strength ====+==== 10.1.5 Dielectric strength ====
  
 In [[block03#conductivity_of_matter|Block03]] we had a short look on conductivity of matter. \\ In [[block03#conductivity_of_matter|Block03]] we had a short look on conductivity of matter. \\
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
 +===== 10.2 Common pitfalls =====
 +  * Treating field lines as **charge paths**: they are drawings of direction/magnitude of $\vec{E}$, **not** particle trajectories.
 +  * Forgetting the **reference charge sign**: line arrows indicate the force on a **positive** test charge; forces on electrons point opposite to the arrows.
 +  * Mixing up **equipotentials** and field lines: equipotentials are everywhere **perpendicular** to field lines; they do **not** indicate current.
 +  * Assuming the plate field is always perfectly uniform: edge effects make real plate fields **inhomogeneous** away from the central region.
 +  * Ignoring conductor boundary conditions: in electrostatics the interior of a conductor is **field-free** and $\vec{E}$ is **normal** to the surface; any tangential $\vec{E}$ would drive charges until it vanishes.
 +  * Confusing $\vec{E}$ with $\vec{D}$: here we use $\vec{E}$ and **permittivity** $\varepsilon=\varepsilon_0\varepsilon_r$ for $|\vec{E}|=\frac{1}{4\pi\varepsilon}\frac{|Q|}{r^2}$.
  
-===== Exercises =====+~~PAGEBREAK~~ ~~CLEARFIX~~ 
 +===== 10.3 Exercises =====
 <panel type="info" title="Task 1.1.2 Field lines"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Task 1.1.2 Field lines"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
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 </WRAP></WRAP></WRAP></panel> </WRAP></WRAP></WRAP></panel>
  
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-{{page>task_1.2.6&nofooter}} +{{page>electrical_engineering_and_electronics:task_1.2.6&nofooter}} 
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 ===== Embedded resources ===== ===== Embedded resources =====