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electrical_engineering_and_electronics_1:block10 [2025/10/31 11:20] – 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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 For checking your understanding please do the following exercise: For checking your understanding please do the following exercise:
-  * 1.2.3+  * 1.1.2 
 +  * 1.2.5
  
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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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 In **homogeneous fields**, magnitude and direction are constant throughout the field range.  In **homogeneous fields**, magnitude and direction are constant throughout the field range. 
 This field form is idealized to exist within plate capacitors. e.g., in the plate capacitor (<imgref ImgNr07>), or the vicinity of widely extended bodies. This field form is idealized to exist within plate capacitors. e.g., in the plate capacitor (<imgref ImgNr07>), or the vicinity of widely extended bodies.
 +
 +Here, the electric field $E$ is given as: $E = {{U}\over{d}}$
  
 <WRAP> <WRAP>
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 For **inhomogeneous fields**, the magnitude and/or direction of the electric field changes from place to place.  For **inhomogeneous fields**, the magnitude and/or direction of the electric field changes from place to place. 
 This is the rule in real systems, even the field of a point charge is inhomogeneous (<imgref ImgNr08>).   This is the rule in real systems, even the field of a point charge is inhomogeneous (<imgref ImgNr08>).  
 +
 +For the given  example of a cylindrical configuration woth tha radius $r$, the electric field $E$ is given as: $E \\sim {{1}\over{r}}$
 +
  
 <WRAP> <WRAP>
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 </WRAP> </WRAP>  </WRAP> </WRAP> 
 +
 +==== 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>. 
 +The additional charges create an electric field. Thus, a resultant force acts on each charge. 
 +The causes of this force are the electric fields of the surrounding electric charges. So the charges repel and move apart. \\
 +
 +<WRAP>
 +<imgcaption ImgNr10 | Viewing a charged metal ball>
 +</imgcaption>
 +{{drawio>LoadedMetalBall.svg}}
 +<WRAP>
 +
 +The movement of the charge continues until a force equilibrium is reached. 
 +In this steady state, there is no longer a resultant force acting on the single charge. 
 +In <imgref ImgNr10> this can be seen on the right: the repulsive forces of the charges are counteracted by the attractive forces of the atomic shells. \\
 +Results:
 +  * The charge carriers are distributed on the surface.
 +  * Due to the dispersion of the charges, the interior of the conductor is free of fields.
 +  * All field lines are perpendicular to the surface. Because: if they were not, there would be a parallel component of the field, i.e., along the surface. Thus, a force would act on charge carriers, and they would move accordingly.
 +
 +<wrap #edu_task_1 />
 +<panel type="info" title="Educational Task - Why is there a discharge at pointy ends of conductors?"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
 +
 +Point discharge is a well-known phenomenon, which can be seen as {{wp>corona discharge}} on power lines (where it also creates the summing sound) or is used in {{wp>spark plug}}s. The phenomenon addresses the effect that there are many more charges at the corners and edges of a conductor. But why is that so? For this, it is feasible to try to calculate the charge density at different spots of a conductor.
 +
 +<WRAP>
 +<imgcaption ImgNr193 | field of a pointy object (field line density is not correct)>
 +</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
 + 600,400 noborder}}
 +</WRAP><WRAP column half>
 +{{drawio>electrical_engineering_and_electronics_1:Fieldofapointyobject3d.svg}}
 +</WRAP></WRAP></WRAP>
 +
 +<WRAP>
 +<imgcaption ImgNr194| examples for an arbitrarily formed conductor>
 +</imgcaption>
 +{{drawio>StrangelyFormedMetalObject.svg}}
 +</WRAP>
 +
 +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:
 +  -  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.
 +  - Compare the field and the potentials of the different spherical conductors in <imgref ImgNr194>, image (b). 
 +    - Are there differences for the electric field $\vec{E}$ outside the spherical conductors? Are the potentials on the surface the same? 
 +    - What can be conducted for the field of the three situations in (b) and (d), when the total charge on the surface is considered to be always the same?
 +  - For spherical conductors, the surface charge density is constant. Given that this charge density leads to the overall charge $q$, how does $\varrho_A$ depend on the radius $r$ of a sphere?
 +  - Now, the situation in (c) shall be considered. Here, all components are conducting, i.e., the potentials on the surface are similar. Both spheres shall be considered to be as far away from each other, so that they show an undisturbed field near their surfaces. In this case, charges on the surface of the curvature to the left and the right represent the same situation as in (a). For the next step, it is important that by this, the potentials of the left sphere with $q_1$ and $r_1$ and the right sphere with $q_2$ and $r_2$ are the same.  
 +    - Set up this equality formula based on the formula for the potential from question 1. 
 +    - Insert the relationship for the overall charges $q_1$ and $q_2$ based on the surface charge densities $\varrho_{A1}$ and $\varrho_{A2}$ of a sphere and their radii $r_1$ and $r_2$.
 +    - What is the relationship between the bending of the surface and the charge density?
 +
 +</WRAP></WRAP></panel>
  
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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%>