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electrical_engineering_and_electronics_1:block10 [2025/10/31 13:08] – mexleadminelectrical_engineering_and_electronics_1:block10 [2025/11/02 17:18] (aktuell) – mexleadmin
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-====== Block 10 - Field patterns of key geometries ======+====== Block 10 - Field Patterns of key Geometries ======
  
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 <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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 ===== 90-minute plan ===== ===== 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 ===== ===== Conceptual overview =====
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   - **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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 <WRAP> <WRAP>
-<imgcaption ImgNr193 | field of a pointy object>+<imgcaption ImgNr193 | field of a pointy object (field line density is not correct)>
 </imgcaption> <WRAP group><WRAP column half> </imgcaption> <WRAP group><WRAP column half>
 {{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+6+310+0.048828125+390%0Ae+1+2+0+256+256+276+276+0%0Ae+1+2+0+256+256+276+276+0%0Ae+0+2+0+256+256+276+276+0%0Ae+0+2+100+126+80+141+120+0%0Ae+0+2+100+147+96+162+100+0%0Ae+0+2+100+125+89+153+108+0%0Aw+0+2+100+112+138+157+99%0Aw+0+2+100+110+66+157+98%0AE+1+2+100+47+57+137+147+65+95+99+129+0%0Ae+0+2+100+119+77+134+83+0%0Ae+0+2+100+131+105+146+111+0%0Ae+0+2+100+128+87+143+93+0%0A {{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+6+310+0.048828125+390%0Ae+1+2+0+256+256+276+276+0%0Ae+1+2+0+256+256+276+276+0%0Ae+0+2+0+256+256+276+276+0%0Ae+0+2+100+126+80+141+120+0%0Ae+0+2+100+147+96+162+100+0%0Ae+0+2+100+125+89+153+108+0%0Aw+0+2+100+112+138+157+99%0Aw+0+2+100+110+66+157+98%0AE+1+2+100+47+57+137+147+65+95+99+129+0%0Ae+0+2+100+119+77+134+83+0%0Ae+0+2+100+131+105+146+111+0%0Ae+0+2+100+128+87+143+93+0%0A
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 </WRAP> </WRAP>
  
-In the <imgref ImgNr194> an example of a "pointy" conductor is given in image (a). The surface of the conductor is always at the same potential.+In the <imgref ImgNr194> an example of a charged and "pointy" conductor is given in image (a). The surface of the conductor is always at the same potential. 
 To cope with this complex shape and the desired charge density, the following path shall be taken: To cope with this complex shape and the desired charge density, the following path shall be taken:
   -  It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm CG} =  \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ with $\vec{E} ={{1} \over {4\pi\cdot\varepsilon}} \cdot {{q} \over {r^2}} \cdot \vec{e}_r $, where $\vec{e}_r$ is the unit vector pointing radially away, ${ \rm C}$ is a point at distance $r_0$ from the charge and ${ \rm G}$ is the ground potential at infinity.   -  It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm CG} =  \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ with $\vec{E} ={{1} \over {4\pi\cdot\varepsilon}} \cdot {{q} \over {r^2}} \cdot \vec{e}_r $, where $\vec{e}_r$ is the unit vector pointing radially away, ${ \rm C}$ is a point at distance $r_0$ from the charge and ${ \rm G}$ is the ground potential at infinity.
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 +===== 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 $|\vec{E}|=\frac{1}{4\pi\varepsilon}\frac{|Q|}{r^2}$.
  
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 ===== Exercises ===== ===== 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%>