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electrical_engineering_and_electronics_1:block09 [2025/10/20 02:33] – 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:
- +  * 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.
 </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>
  
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 {{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:
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 ==== 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*}
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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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 The direction of the electric field is switchable in <imgref ImgNr02> via the "Electric Field" option on the right. \\ The direction of the electric field is switchable in <imgref ImgNr02> via the "Electric Field" option on the right. \\
 +
 +
 +==== 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).
 +
 +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: $\vec{F_3}= \vec{F_{31}}+\vec{F_{32}}$. \\
 +<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>
 +<imgcaption ImgNr06 | direction of coulomb force>
 +</imgcaption> <WRAP>.
 +{{drawio>DirectionOfCoulombforce.svg}} \\
 +</WRAP>
 +
 +==== Energy required to Displace a Charge in the electric Field ====
 +
 +Now we want to see, whether we can derive the required energy to displace a charge in the electric field. \\
 +
 +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 $\Delta W$:
 +
 +\begin{align*}
 +\Delta W = \int \vec{F} d\vec{r} =  q \int \vec{E} d\vec{r} 
 +\end{align*}
 +
 +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:
 +
 +\begin{align*}
 +\boxed{U = \int \vec{E} d\vec{r} }
 +\end{align*}
 +
 +We will apply this relationship in multiple of the upcoming blocks.
 +
 +~~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 =====
- 
-==== Quick checks ==== 
  
 <panel type="info" title="Task 1.1.1 simple task with charges"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Task 1.1.1 simple task with charges"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
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 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
 +
 +{{page>task_1.2.1_with_calc&nofooter}}
 +{{page>task_1.2.2&nofooter}}
 +{{page>task_1.2.3&nofooter}}
 +
 +<panel type="info" title="Task 1.2.4 Superposition of Charges in 1D"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
 +{{youtube>QWOwK-zyEnE}}
 +</WRAP></WRAP></panel>
  
 ===== Embedded resources ===== ===== Embedded resources =====
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 {{youtube>2GQTfpDE9DQ}} {{youtube>2GQTfpDE9DQ}}
 </WRAP> </WRAP>
 +
 +