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electrical_engineering_and_electronics_1:block09 [2025/10/27 00:27] – mexleadminelectrical_engineering_and_electronics_1:block09 [2025/11/01 00:14] (aktuell) – [Block 09 - Force on charges and electric field strength] mexleadmin
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-====== Block 09 - Force on charges and electric field strength ======+====== Block 09 - Force on Charges and electric Field Strength ======
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
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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. 
   * Distinguish **charge** $Q$ (source) from **electric field** $\vec{E}$ (effect in space) and **force** $\vec{F}$ on a test charge $q$; use formula for Coulomb force with correct vector directions and units ($1~{\rm N/C}=1~{\rm V/m}$).   * Distinguish **charge** $Q$ (source) from **electric field** $\vec{E}$ (effect in space) and **force** $\vec{F}$ on a test charge $q$; use formula for Coulomb force with correct vector directions and units ($1~{\rm N/C}=1~{\rm V/m}$).
   * Explain and apply the **superposition principle** for forces and fields from multiple charges.   * Explain and apply the **superposition principle** for forces and fields from multiple charges.
-  * 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. 
-  * 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). 
   * Compute $|\vec{E}|$ for a **point charge** (Coulomb force), identify $\varepsilon$ and check dimensions.   * Compute $|\vec{E}|$ for a **point charge** (Coulomb force), identify $\varepsilon$ and check dimensions.
   * Determine the force on a charge in an electrostatic field by applying Coulomb's law. Specifically:   * Determine the force on a charge in an electrostatic field by applying Coulomb's law. Specifically:
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 </callout> </callout>
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 +===== Preparation at Home =====
 +
 +And again: 
 +  * Please read through the following chapter.
 +  * Also here, there are some clips for more clarification under 'Embedded resources'. 
 +
 +For checking your understanding please do the following exercise:
 +  * 1.2.3
 +
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 ===== 90-minute plan ===== ===== 90-minute plan =====
   - Warm-up (8–10 min):   - Warm-up (8–10 min):
Zeile 29: Zeile 37:
     - **Field lines**: definition, drawing rules, sources/sinks, no intersections; relate density to magnitude.     - **Field lines**: definition, drawing rules, sources/sinks, no intersections; relate density to magnitude.
     - **Homogeneous vs. inhomogeneous** fields; conductor boundary facts (perpendicular $\vec{E}$, interior field-free).     - **Homogeneous vs. inhomogeneous** fields; conductor boundary facts (perpendicular $\vec{E}$, interior field-free).
-  - Guided simulations (20–25 min) +  - Guided simulation (20–25 min) 
-  - Practice (10–15 min): +    - Place single and multiple charges; measure $\vec{E}$ at points. 
-    - Short worksheet: sketch field lines for two like charges and a dipole; compute $|\vec{E}|$ at a marked point.+  - Practice (10–15 min) 
 +    - net field on-axis of two charges; quick peer check.
   - Wrap-up (5 min):   - Wrap-up (5 min):
-    - Summary map: charges → $\vec{E}$ → $\vec{F}$; key properties and units; preview link to **equipotentials** and energy (next block).+    - Summary map: charges → $\vec{E}$ → $\vec{F}$; key properties and units.
  
 ===== Conceptual overview ===== ===== Conceptual overview =====
 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
   - **Fields separate cause and effect**: charges set up a state in space (the field) that exists whether or not a test charge is present.   - **Fields separate cause and effect**: charges set up a state in space (the field) that exists whether or not a test charge is present.
 +  - **Coulomb field of a point charge:** $\displaystyle \vec{E}(\vec{r})=\frac{1}{4\pi\varepsilon}\frac{Q}{r^2}\,\vec{e}_{\rm r}$ (radial; outward for $Q>0$, inward for $Q<0$). Magnitude $|\vec{E}|$ follows the inverse-square law.
   - The **electric field** is a **vector field** $\vec{E}(\vec{x})$; its **direction** is the direction of the force on a *positive* test charge; its **magnitude** is given by the actinv force and the charge with units $1~{\rm N/C}=1~{\rm V/m}$.   - The **electric field** is a **vector field** $\vec{E}(\vec{x})$; its **direction** is the direction of the force on a *positive* test charge; its **magnitude** is given by the actinv force and the charge with units $1~{\rm N/C}=1~{\rm V/m}$.
   - **Point charge** model: inverse-square law; direction is radial, outward for $Q>0$, inward for $Q<0$.   - **Point charge** model: inverse-square law; direction is radial, outward for $Q>0$, inward for $Q<0$.
   - **Superposition** holds: for multiple sources, $\vec{E}_{\rm total}=\sum_k \vec{E}_k$ (vector sum at the same point).   - **Superposition** holds: for multiple sources, $\vec{E}_{\rm total}=\sum_k \vec{E}_k$ (vector sum at the same point).
-  - **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). 
-  - **Conductors (electrostatics)**: $\vec{E}$ is perpendicular to the surface; interior is field-free; surface charge arranges to enforce these conditions. 
 </callout> </callout>
  
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 \end{align*} \end{align*}
  
 +The unit of $E$ is $\rm 1 {{N}\over{As}} =  1 {{V}\over{m}} $
  
 <callout icon="fa fa-exclamation" color="red" title="Note:"> <callout icon="fa fa-exclamation" color="red" title="Note:">
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 </WRAP> </WRAP>
  
-~~PAGEBREAK~~ ~~CLEARFIX~~ +==== Energy required to Displace a Charge in the electric Field ====
-==== Geometric Distribution of Charges ====+
  
-In previous chapters, only single charges (e.g., $Q_1$, $Q_2$) were considered.   +Now we want to see, whether we can derive the required energy to displace a charge in the electric field. \\
-  * The charge $Q$ was previously reduced to a **point charge**. \\ This can be used, for example, for the elementary charge or for extended charged objects from a large distance. The distance is sufficiently large if the ratio between the largest object extent and the distance to the measurement point $P$ is small. +
-  * If the charges are lined up along a line, this is referred to as a **line charge**. \\ Examples of this are a straight trace on a circuit board or a piece of wire. Furthermore, this also applies to an extended charged object, which has exactly an extension that is no longer small in relation to the distance. For this purpose, the charge $Q$ is considered to be distributed over the line. Thus, a (line) charge density $\rho_l$ can be determined: <WRAP centeralign>$\rho_l = {{Q}\over{l}}$</WRAP> or, in the case of different charge densities on subsections: <WRAP centeralign>$\rho_l = {{\Delta Q}\over{\Delta l}} \rightarrow \rho_l(l)={{\rm d}\over{{\rm d}l}} Q(l)$</WRAP> +
-  * It is spoken of as an **area charge** when the charge is distributed over an area. \\ Examples of this are the floor or the plate of a capacitor. Again, an extended charged object can be considered when two dimensions are no longer small in relation to the distance (e.g. surface of the earth). Again, a (surface) charge density $\rho_A$ can be determined: <WRAP centeralign>$\rho_A = {{Q}\over{A}}$</WRAP> or if there are different charge densities on partial surfaces: <WRAP centeralign>$\rho_A = {{\Delta Q}\over{\Delta A}} \rightarrow \rho_A(A) ={{\rm d}\over{{\rm d}A}} Q(A)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}} Q(A)$</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 ====+Since we know the force on a charge in an electrical field $\vec{E}$ (= Coulomb-Force $\vec{F}_C = q \cdot \vec{E} $), we can borrow some relationships from mechanics for the energy $\Delta W$:
  
-Electric field lines result from the (fictitious) path of a sample charge. Thus, also electric field lines of several charges can be determined.  +\begin{align*} 
-However, these also result from a superposition of the individual effects - i.e., electric field - at a measuring point $P$.+\Delta W = \int \vec{F} d\vec{r} =  q \int \vec{E} d\vec{r}  
 +\end{align*}
  
-The superposition is sketched in <imgref ImgNr032>: Two charges $Q_1$ and $Q_2$ act on the test charge $q$ with the forces $F_1$ and $F_2$. Depending on the positions and charges, the forces vary, and so does the resulting force. The simulation also shows a single field line. +Looks familiar? Maybe not on the first sight. But we already had defined the fraction of the energy difference per charge ${{\Delta W}\over{q}}$ as voltage $U$! \\ 
 +Therefore:
  
-<WRAP> +\begin{align*} 
-<imgcaption ImgNr032 | examples of field lines> +\boxed{U = \int \vec{E} d\vec{r} } 
-</imgcaption> <WRAP> +\end{align*}
-{{url>https://www.geogebra.org/material/iframe/id/qIXZJKqj/width/500/height/500/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/true/at/preferhtml5 600,400 noborder}} +
-</WRAP></WRAP>+
  
-For a full picture of the field lines between charges, one has to start with a single charge. The in- and outgoing lines on this charge are drawn equidistant from the charge. This is also true for the situation with multiple charges. However, there, the lines are not necessarily run radially anymore. The test charge is influenced by all the single charges, and therefore, the field lines can get bent.  +We will apply this relationship in multiple of the upcoming blocks.
- +
-<WRAP> +
-<imgcaption ImgNr03 | examples of field lines> +
-</imgcaption> <WRAP> +
-{{drawio>ExamplesForFieldLines.svg}}  +
-</WRAP></WRAP> +
- +
-In <imgref ImgNr031> the field lines are shown. The additional "equipotential lines" will be discussed later and can be deactivated by clearing the checkmark ''Show Equipotentials''. +
-Try the following in the simulation: +
-  * Get accustomed to the simulation. You can... +
-    * ... move the charges by drag and drop. +
-    * ... add another Charge with ''Add'' >> ''Add Point Charge''. +
-    * ... delete components with a right click on them and ''delete'' +
-  * Where is the density of the field lines higher?  +
-  * How does the field between two positive charges look? How does it look between two different charges? +
- +
-<WRAP> +
-<imgcaption ImgNr031 | examples of field lines> +
-</imgcaption> <WRAP> +
-{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+1+435+0.048828125+403%0Ac+0+0+128+158+-1e-9%0Ac+0+0+128+98+1e-9%0A 600,600 noborder}} +
-</WRAP></WRAP> +
- +
-<callout icon="fa fa-exclamation" color="red" title="Note:"> +
-  - The electrostatic field is a source field. This means there are sources and sinks.  +
-  - From the field line diagrams, the following can be obtained: +
-    - Direction of the field ($\hat{=}$ parallel to the field line). +
-    - Magnitude of the field ($\hat{=}$ number of field lines per unit area). +
-  - The magnitude of the field along a field line is usually __not__ constant. +
-</callout> +
- +
-<callout icon="fa fa-exclamation" color="red" title="Note:"> +
-Field lines have the following properties: +
-  * The electric field lines have a beginning (at a positive charge) and an end (at a negative charge). +
-  * The direction of the field lines represents the direction of a force onto a positive test charge. +
-  * There are **no closed field lines** in electrostatic fields. The reason for this can be explained by considering the energy of the moved particle (see later subchapters). +
-  * Electric **field lines cannot cut** each other: This is based on the fact that the direction of the force at a cutting point would not be unique. +
-  * The field lines are **always perpendicular to conducting surfaces**. This is also based on energy considerations; more details later. +
-  * The **inside of a conducting component is always field-free**. Also, this will be discussed in the following. +
- +
-</callout>+
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== Types of Fields depending on the Charge Distribution ==== 
  
-There are two different types of fields: 
- 
-<WRAP group><WRAP column half> 
-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. 
- 
-<WRAP> 
-<imgcaption ImgNr07 | Field lines of a homogeneous field> 
-</imgcaption> \\ 
-{{drawio>FieldLinesOfAHomogeneousField.svg}} \\ 
-</WRAP> 
- 
-</WRAP><WRAP column half> 
- 
-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>).   
- 
-<WRAP> 
-<imgcaption ImgNr08 | Field lines of an inhomogeneous field> 
-</imgcaption> \\ 
-{{drawio>FieldLinesOfAnInhomogeneousField.svg}} \\ 
-</WRAP> 
- 
-</WRAP> </WRAP> </WRAP> 
- 
-~~PAGEBREAK~~ ~~CLEARFIX~~ 
 ===== Common pitfalls ===== ===== Common pitfalls =====
   * Treating **force** and **field** as the same thing; forgetting $\vec{F}=q\,\vec{E}$ and the positive-test-charge convention.   * Treating **force** and **field** as the same thing; forgetting $\vec{F}=q\,\vec{E}$ and the positive-test-charge convention.
Zeile 275: Zeile 211:
  
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
- 
-<panel type="info" title="Task 1.1.2 Field lines"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> 
- 
-Sketch the field line plot for the charge configurations given in <imgref ImgNr04>. \\ 
-Note: 
-  * The __overlaid__ picture is requested. 
-  * Make sure that it is a source field. 
- 
-You can prove your result with the simulation <imgref ImgNr032>. 
- 
-<WRAP> 
-<imgcaption ImgNr04| Task on field lines> 
-</imgcaption> <WRAP> 
-{{drawio>TaskOnFieldLines.svg}} \\ 
-</WRAP> 
- 
-</WRAP></WRAP></WRAP></panel> 
  
  
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 {{youtube>QWOwK-zyEnE}} {{youtube>QWOwK-zyEnE}}
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
- 
-{{page>task_1.2.5_with_calc&nofooter}} 
-{{page>task_1.2.6&nofooter}} 
-{{page>task_1.2.7&nofooter}} 
- 
  
 ===== Embedded resources ===== ===== Embedded resources =====
Zeile 320: Zeile 234:
 Intro into  electric field Intro into  electric field
 {{youtube>2GQTfpDE9DQ}} {{youtube>2GQTfpDE9DQ}}
-</WRAP> 
- 
-<WRAP column half> 
-Field lines of various extended charged objects 
-{{youtube>LB8Rhcb4eQM}} 
 </WRAP> </WRAP>