Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

Link zu dieser Vergleichsansicht

Beide Seiten der vorigen Revision Vorhergehende Überarbeitung
Nächste Überarbeitung		 Vorhergehende Überarbeitung
electrical_engineering_and_electronics_1:block09 [2025/10/20 02:46] – mexleadminelectrical_engineering_and_electronics_1:block09 [2025/11/01 00:14] (aktuell) – [Block 09 - Force on charges and electric field strength] mexleadmin
Zeile 1: Zeile 1:
-====== Block 09 — Force on charges and electric field strength ======+====== Block 09 - Force on Charges and electric Field Strength ======
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
Zeile 5: Zeile 5:
 <callout> <callout>
 By the end of this section, you will be able to: By the end of this section, you will be able to:
- +  * 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 ($1~{\rm N/C}=1~{\rm V/m}$). 
-  - Know that an electric field is formed around a charge. +  * Explain and apply the **superposition principle** for forces and fields from multiple charges. 
-  - Sketch the field lines of electric fields. +  * Compute $|\vec{E}|$ for a **point charge** (Coulomb force), identify $\varepsilon$ and check dimensions. 
-  - Represent the field vectors in a sketch when given several charges. +  * Determine the force on a charge in an electrostatic field by applying Coulomb's law. Specifically: 
-  - Determine the resulting field vector by superimposing several field vectors using vector calculus. +    * The force vector in coordinate representation 
-  - Determine the force on a charge in an electrostatic field by applying Coulomb's law. Specifically: +    * The magnitude of the force vector 
-    - The force vector in coordinate representation +    * The angle of the force vector 
-    - The magnitude of the force vector +    * The direction of the force 
-    - The angle of the force vector +  * Determine a force vector by superimposing several force vectors using vector calculus.
-  - Determine the direction of the forces using the given charges. +
-  - Represent the acting force vectors in a sketch. +
-  - Determine a force vector by superimposing several force vectors using vector calculus. +
-  - State the following quantities for a force vector: +
-    - The force vector in coordinate representation +
-    - The magnitude of the force vector +
-    - The angle of the force vector+
 </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 (5–10 min):  +  - Warm-up (8–10 min): 
-    - Recall / Quick quiz ... +    - Quick recall quiz: units of $Q$, $\vec{E}$, $\vec{F}$; passive sign convention for forces on a **positive** test charge. 
-  - Core concepts & derivations (60–70 min):   +    - Dimensions check: show $1~{\rm N/C}=1~{\rm V/m}$. 
-    - ... +  - Concept build & demonstrations (35–40 min): 
-  - Practice (10–20 min): ... +    - Cause–field–effect chain: charges $\Rightarrow \vec{E}(\vec{x}) \Rightarrow \vec{F}=q\,\vec{E}$. 
-  - Wrap-up (5 min): ...+    - Coulomb law $\Rightarrow$ point-charge field magnitude and direction. 
 +    - **Superposition** for two/three charges; vector addition. 
 +    - **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). 
 +  - Guided simulation (20–25 min) 
 +    - Place single and multiple charges; measure $\vec{E}$ at points. 
 +  - Practice (10–15 min) 
 +    - net field on-axis of two charges; quick peer check. 
 +  - Wrap-up (5 min): 
 +    - 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. 
 +  - **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 $1~{\rm N/C}=1~{\rm V/m}$. 
 +  - **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).
 </callout> </callout>
  
Zeile 48: Zeile 65:
 {{url>https://phet.colorado.edu/sims/html/john-travoltage/latest/john-travoltage_de.html 500,400 noborder}} {{url>https://phet.colorado.edu/sims/html/john-travoltage/latest/john-travoltage_de.html 500,400 noborder}}
 </WRAP> </WRAP>
- 
-We had already considered the charge as the central quantity of electricity in the first chapter of the previous semester and recognized it as a multiple of the elementary charge. There was already a mutual force action ([[electrical_engineering_1:preparation_properties_proportions#coulomb-force|the Coulomb-force]]) derived. This will be more fully explained. 
  
 First, we shall define certain terms: First, we shall define certain terms:
Zeile 101: Zeile 116:
 ==== The electric Field ==== ==== The electric Field ====
  
-To determine the electric field, a measurement of its magnitude and direction is now required. The Coulomb force between two charges $Q_1$ and $Q_2$ is known from the first chapter of the previous semester:+We had already considered the charge as the central quantity of electricity in [[block02]] and recognized it as a multiple of the elementary charge.  
 +Now, we want to determine the electric field of charges. For this, a measurement of its magnitude and direction is now required. The **Coulomb force** between two charges $Q_1$ and $Q_2$ is:
  
 \begin{align*} \begin{align*}
Zeile 125: Zeile 141:
 \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:">
Zeile 143: Zeile 160:
  
  
-==== Electric Field Lines ==== +==== Direction of the Coulomb force and Superposition ====
- +
-Electric field lines result from the (fictitious) path of a sample charge. Thus, also electric field lines of several charges can be determined.  +
-However, these also result from a superposition of the individual effects - i.e., electric field - at a measuring point $P$. +
- +
-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.  +
- +
-<WRAP> +
-<imgcaption ImgNr032 | examples of field lines> +
-</imgcaption> <WRAP> +
-{{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.  +
- +
-<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~~ +
- +
-==== Electric Charge and Coulomb Force (reloaded) ==== +
- +
-The electric charge and Coulomb force have already been described last semester. However, some points are to be caught up here. +
- +
-=== Direction of the Coulomb force and Superposition ===+
  
 In the case of the force, only the direction has been considered so far, e.g., direction towards the sample charge. For future explanations, it is important to include the cause and effect in the naming. This is done by giving the correct labeling of the subscript of the force. In <imgref ImgNr06> (a) and (b), the convention is shown: A force $\vec{F}_{21}$ acts on charge $Q_2$ and is caused by charge $Q_1$. As a mnemonic, you can remember "tip-to-tail" (first the effect, then the cause). In the case of the force, only the direction has been considered so far, e.g., direction towards the sample charge. For future explanations, it is important to include the cause and effect in the naming. This is done by giving the correct labeling of the subscript of the force. In <imgref ImgNr06> (a) and (b), the convention is shown: A force $\vec{F}_{21}$ acts on charge $Q_2$ and is caused by charge $Q_1$. As a mnemonic, you can remember "tip-to-tail" (first the effect, then the cause).
  
 Furthermore, several forces on a charge can be superimposed, resulting in a single, equivalent force. \\ Furthermore, several forces on a charge can be superimposed, resulting in a single, equivalent force. \\
-Strictly speaking, it must hold that $\varepsilon$ is constant in the structure. For example, the resultant force in <imgref ImgNr06> Fig. (c) on $Q_3$ becomes equal to: $\vec{F_3}= \vec{F_{31}}+\vec{F_{32}}$.+Strictly speaking, it must hold that $\varepsilon$ is constant in the structure. For example, the resultant force in <imgref ImgNr06> Fig. (c) on $Q_3$ becomes equal to: $\vec{F_3}= \vec{F_{31}}+\vec{F_{32}}$. \\ 
 +<imgref ImgNr06> Fig. (d) shows that for a charged surface, the force on a charge on top of this surface is always perpendicular to the surface itself. 
  
 <WRAP> <WRAP>
Zeile 217: Zeile 174:
 </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>+
  
-In the following, area charges and their interactions will be considered.+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 $\Delta W$:
  
-=== Types of Fields depending on the Charge Distribution ===+\begin{align*} 
 +\Delta W = \int \vec{F} d\vec{r} =  q \int \vec{E} d\vec{r}  
 +\end{align*}
  
-There are two different types of fields:+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 group><WRAP column half> +\begin{align*} 
-In **homogeneous fields**, magnitude and direction are constant throughout the field range.  +\boxed{U = \int \vec{E} d\vec{r} } 
-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.+\end{align*}
  
-<WRAP> +We will apply this relationship in multiple of the upcoming blocks.
-<imgcaption ImgNr07 | Field lines of a homogeneous field> +
-</imgcaption> \\ +
-{{drawio>FieldLinesOfAHomogeneousField.svg}} \\ +
-</WRAP>+
  
-</WRAP><WRAP column half>+~~PAGEBREAK~~ ~~CLEARFIX~~
  
-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. 
-  ...  +  * Mixing units (${\rm N}$, ${\rm C}$, ${\rm V}$, ${\rm m}$): not recognizing $1~{\rm N/C}=1~{\rm V/m}$. 
 +  * Drawing **field lines** as closed loops or allowing them to **intersect** (source field: start at $+$, end at $-$; no crossings). 
 +  * Ignoring **vector addition** in superposition (adding magnitudes instead of vectors). 
 +  * Assuming field exists **only** when a test charge is present; the field is a property of space due to sources. 
 +  * Using point-charge formulas too near extended objects; not identifying **homogeneous vs. inhomogeneous** regions. 
 +  * Forgetting conductor boundary facts: lines must be **perpendicular** to ideal conducting surfaces; interior **$|\vec{E}|=0$** in electrostatics.
      
 ===== Exercises ===== ===== Exercises =====
Zeile 267: 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> 
  
  
Zeile 293: Zeile 220:
 {{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 312: 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>