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electrical_engineering_and_electronics_1:the_magnetostatic_field [2025/09/19 12:33] – ↷ Seite von electrical_engineering_and_electronics_2:the_magnetostatic_field nach electrical_engineering_and_electronics_1:the_magnetostatic_field verschoben mexleadminelectrical_engineering_and_electronics_1:the_magnetostatic_field [2025/09/19 17:23] (aktuell) – mexleadmin
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-====== 3 The magnetostatic Field ======+====== 7 The magnetostatic Field ======
  
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 </callout> </callout>
-===== 3.1 Magnetic Phenomena =====+===== 7.1 Magnetic Phenomena =====
  
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 </WRAP> </WRAP>
  
-First, permanent magnets made of magnetic magnetite ($\rm Fe_{3} O_{4}$) were found in Greece in the region around Magnesia. Besides the iron materials, other elements also show a similar "strong and permanent magnetic force effect", which is also called ferromagnetism after iron: Cobalt and nickel, as well as many of their alloys, also exhibit such an effect. Chapter [[#3.5 Matter in the magnetic field]] describes the subdivision of magnetic materials in detail.+First, permanent magnets made of magnetic magnetite ($\rm Fe_{3} O_{4}$) were found in Greece in the region around Magnesia. Besides the iron materials, other elements also show a similar "strong and permanent magnetic force effect", which is also called ferromagnetism after iron: Cobalt and nickel, as well as many of their alloys, also exhibit such an effect. Chapter [[#7.5 Matter in the magnetic field]] describes the subdivision of magnetic materials in detail.
  
 Here now the "magnetic force effect" is to be looked at more near. For this purpose, a few thought experiments are carried out with a magnetic iron stone <imgref BildNr01> ([[https://www.youtube.com/watch?v=IgtIdttfGVw|This video]] gives a similar introduction). Here now the "magnetic force effect" is to be looked at more near. For this purpose, a few thought experiments are carried out with a magnetic iron stone <imgref BildNr01> ([[https://www.youtube.com/watch?v=IgtIdttfGVw|This video]] gives a similar introduction).
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 ~~PAGEBREAK~~~~CLEARFIX~~ ~~PAGEBREAK~~~~CLEARFIX~~
  
-===== 3.2 Magnetic Field Strength =====+===== 7.2 Magnetic Field Strength =====
  
 <callout> <callout>
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 \boxed{\theta = \sum I = N \cdot I} \boxed{\theta = \sum I = N \cdot I}
 \end{align*} \end{align*}
-The unit of $\theta$ is: $[\theta]= 1~ \rm A$ (obsoletely called ampere-turn). For the magnetic voltage the currents which flow through the surface enclosed by the closed path have to be considered. A detailed definition will be given below after more analysis. +The unit of $\theta$ is: $[\theta]= 1~ \rm A$ (obsoletely called ampere-turn). For the magnetic voltage the currents which flow through the surface enclosed by the closed path have to be considered.  
 + 
 +Thus, the magnetic field strength $H$ of the toroidal coil is then given by:  
 + 
 +\begin{align*} 
 +H= {{\theta}\over{l}} 
 +\end{align*}
  
 <WRAP> <WRAP>
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 </WRAP> </WRAP>
  
-Thus, the magnetic field strength $H$ of the toroidal coil is then given by: $H= {{\theta}\over{l}}$+But why is that thing called a voltage?
  
-<callout icon="fa fa-exclamation" color="red" title="Notice:"> 
-  * For the sign of the magnetic voltage, one has to consider the orientation of the current and way on the enclosing path. The <imgref BildNr05> shows the positive orientation: The positive orientation is given when the currents show out of the drawing plane and the path shows a counterclockwise orientation.  
-  * This is again given as the right-hand rule (see <imgref BildNr76>): For the positive orientation the current shows along the thumb of the right hand, while the path is counted along the direction of the fingers of the right hand. 
- 
-<WRAP> 
-<imgcaption BildNr76 | Right hand rule> 
-</imgcaption> \\ 
-{{drawio>Righthandrule2.svg}}  
-</WRAP> 
- 
-</callout> 
- 
-In the English literature often the name **{{wp>Magnetomotive Force}}** $\mathcal{F}$ is used instead of magnetic voltage $\theta$. The naming refers to the {{wp>Electromotive Force}}. The electromotive force describes the root cause of a (voltage) source to be able to drive a current and therefore generate a defined voltage. Both "forces" shall not be confused with the mechanical force $\vec{F}= m \cdot \vec{a}$. They only describe the driving cause behind the electric or magnetic fields. The German courses in higher semesters use the term //Magnetische Spannung// - therefore, the English equivalent is introduced here.  
- 
-==== Magnetic Field Strength part 3: Generalization ==== 
- 
-So far, only rotational symmetric problems could be solved. Now, this shall be generalized.  
 For this purpose, we will have a look back at the electric field. For the electric field strength $E$ of a capacitor with two plates at a distance of $s$ and the potential difference $U$ holds: For this purpose, we will have a look back at the electric field. For the electric field strength $E$ of a capacitor with two plates at a distance of $s$ and the potential difference $U$ holds:
  
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 In words: The potential difference is given by adding up the field strength along the path of a probe from one plate to the other. This was extended to $U = \int_s E {\rm d}s$. In words: The potential difference is given by adding up the field strength along the path of a probe from one plate to the other. This was extended to $U = \int_s E {\rm d}s$.
- If we compare this idea to the magnetic field strength $H$ of a toroidal coil with the mean magnetic path length $l$, we had+If we compare this idea to the magnetic field strength $H$ of a toroidal coil with the mean magnetic path length $l$, we had
  
 \begin{align*} \begin{align*}
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 Can you see the similarities? Again, the magnitude of the field strength is summed up along a path to arrive at another field-describing quantity (here, the magnetic voltage $\theta$).  Can you see the similarities? Again, the magnitude of the field strength is summed up along a path to arrive at another field-describing quantity (here, the magnetic voltage $\theta$). 
-Because of the similarity - which continues below - the so-called **magnetic potential difference $V_m$** is introduced:+Because of the similarity we call it magnetic voltage.
  
-\begin{align*} +In the English literature often the name **{{wp>Magnetomotive Force}}** $\mathcal{F}$ is used instead of magnetic voltage $\theta$. The naming refers to the {{wp>Electromotive Force}}. The electromotive force describes the root cause of a (voltage) source to be able to drive a current and therefore generate a defined voltage. Both "forces" shall not be confused with the mechanical force $\vec{F}= m \cdot \vec{a}$. They only describe the driving cause behind the electric or magnetic fields. The German courses in higher semesters use the term //Magnetische Spannung// - therefore, the English equivalent is introduced here. 
-V_m = H \cdot s \quad \quad | \quad \text{applies to toroidal coil only} +
-\end{align*} +
- +
-Now, what is the difference between the magnetic potential difference $V_m$ and the magnetic voltage $\theta$? +
-  - The first equation of the toroidal coil ($\theta = H \cdot l$) is valid for exactly __one turn__ along a field line with the length $l$. In addition, the magnetic voltage was equal to the current times the number of windings: $\theta = N \cdot I$. +
-  - The second equation ($V_{\rm m} = H \cdot s$) is independent of the length of the field line $l$. Only if $s = l$ is chosen, the magnetic voltage equals the magnetic potential difference. The path length $s$ can be a fraction or multiple of a single revolution $l$ for the magnetic potential difference. +
- +
-Thus, for each infinitesimally small path ${\rm d}s$ along a field line, the resulting infinitesimally small magnetic potential difference ${\rm d}V_{\rm m} = H \cdot {\rm d}s$ can be determined. If now along the field line the magnetic field strength $H = H(\vec{s})$ changes, then the magnetic potential difference from point $\vec{s_1}$ to point $\vec{s_2}$ results to: +
- +
-\begin{align*} +
-V_{\rm m12} = V_{\rm m}(\vec{s_1}, \vec{s_2})  +
-            = \int_\vec{s_1}^\vec{s_2} H(\vec{s}) {\rm d}s +
-\end{align*} +
- +
-Up to now, only the situation was considered that one always walks along one single field line. $\vec{s}$ therefore always arrived at the same spot of the field line.  +
-If one wants to extend this to arbitrary directions (also perpendicular to field lines), then only that part of the magnetic field strength $\vec{H}$ may be used in the formula, which is parallel to the path ${\rm d} \vec{s}$. This is made possible by scalar multiplication. Thus, it is generally valid: +
- +
-\begin{align*} +
-\boxed{V_{\rm m12} = \int_\vec{s_1}^\vec{s_2} \vec{H} \cdot {\rm d} \vec{s}} +
-\end{align*} +
- +
-The magnetic voltage $\theta$ (and therefore the current) is the cause of the magnetic field strength.  +
-From the chapter [[electrical_engineering_2:The stationary Electric Flow]] the general representation of the current through a surface is known.  +
-This leads to the **{{wp>Ampere's Circuital Law}}**+
  
 +In mathematical terms this leads to a rather ugly monster: 
 \begin{align*} \begin{align*}
-\boxed{\oint_{s} \vec{H} \cdot {\rm d} \vec{s} = \iint_A \; \vec{S} {\rm d}\vec{A} = \theta}+\oint_s \vec{H} \cdot {\rm d} \vec{s} &= \sum_n \cdot I_n = \theta 
 \end{align*} \end{align*}
  
-  * The path integral of the magnetic field strength along an arbitrary closed path is equal to the free currents (= current density) through the surface enclosed by the path.+  * The path integral of the magnetic field strength along an arbitrary closed path is equal to the currents through the surface enclosed by the path.
   * The magnetic voltage $\theta$ can be given as   * The magnetic voltage $\theta$ can be given as
     * for a single conductor: $\theta = I$     * for a single conductor: $\theta = I$
     * for a coil: $\theta = N \cdot I$     * for a coil: $\theta = N \cdot I$
     * for multiple conductors: $\theta = \sum_n \cdot I_n$     * for multiple conductors: $\theta = \sum_n \cdot I_n$
-    * for spatial distribution: $\theta = \iint_A \; \vec{S} {\rm d}\vec{A}$   +    * (for spatial distribution: $\theta = \iint_A \; \vec{S} {\rm d}\vec{A}$, see chapter 6) 
-  * ${\rm d}\vec{s}$ and ${\rm d}\vec{A}$ build a right-hand system: once the thumb of the right hand is pointing along ${\rm d}\vec{A}$, the fingers of the right hand show the correct direction for ${\rm d}\vec{s}$ for positive $\vec{H}$ and $\vec{S}$+ 
 +<callout icon="fa fa-exclamation" color="red" title="Notice:"> 
 +  * For the sign of the magnetic voltage, one has to consider the orientation of the current and way on the enclosing path. The <imgref BildNr05> shows the positive orientation: The positive orientation is given when the currents show out of the drawing plane and the path shows a counterclockwise orientation.  
 +  * This is again given as the right-hand rule (see <imgref BildNr76>): For the positive orientation the current shows along the thumb of the right hand, while the path is counted along the direction of the fingers of the right hand. 
 +  * An alternative view is shown in <imgref BildNr1065>: Each current $I$ penetrating an area $A$ will create an $H$-field along the boundary of this area. Both direction of the current and the $H$-field are interlinked and their directions can be transformed with the right hand rule.
  
 <WRAP> <WRAP>
-<imgcaption BildNr1065 | Right hand rule>+<imgcaption BildNr76 | Right hand rule> 
 +</imgcaption> \\ 
 +{{drawio>Righthandrule2.svg}}  
 +</WRAP> 
 + 
 +<WRAP> 
 +<imgcaption BildNr1065 | "alternative view" on the Right hand rule >
 </imgcaption> \\ </imgcaption> \\
 {{drawio>Righthandrule.svg}}  {{drawio>Righthandrule.svg}} 
 </WRAP> </WRAP>
 +</callout>
 +
  
 ==== Recap: Application of magnetic Field Strength ==== ==== Recap: Application of magnetic Field Strength ====
  
-Ampere's Circuital Law shall be applied to find the magnetic field strength $H$ inside the toroidal coil (<imgref BildNr25>).+The magnetic voltage shall now be applied to find the magnetic field strength $H$ inside the toroidal coil (<imgref BildNr25>) - just to check how we can work with it.
  
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 </WRAP> </WRAP>
  
-  * The closed path ${\rm s}$ is on a revolution of a field line in the center of the coil +We see, that the current $I$ is going through the area $A$ $N$-times. \\ 
-  * The surface $A$ is the enclosed surface  +The magnetic voltage is the current through the surface and therefore is given as $N\cdot I$: 
 This leads to:  This leads to: 
- 
-\begin{align*} 
-\oint_s \vec{H} \cdot {\rm d} \vec{s} &= \iint_A \vec{S} {\rm d}\vec{A} = \theta  
-\end{align*} 
- 
-Since $\vec{H} \uparrow \uparrow {\rm d} \vec{s}$ the term $\vec{H} \cdot {\rm d} \vec{s}$ can be substituted by $H {\rm d}s$: 
- 
-\begin{align*} 
-\oint_s H \cdot {\rm d}s              &= \iint_A \vec{S} {\rm d}\vec{A}  
-\end{align*} 
- 
-The magnetic voltage is the current through the surface and is given as $N\cdot I$:  
  
 \begin{align*} \begin{align*}
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 ~~PAGEBREAK~~~~CLEARFIX~~ ~~PAGEBREAK~~~~CLEARFIX~~
  
-===== 3.3 Magnetic Flux Density and Lorentz Law =====+===== 7.3 Magnetic Flux Density and Lorentz Law =====
    
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-Please have a look at the German contents (text, videos, exercises) on the page of the [[https://obkp.mint-kolleg.kit.edu/#OBKP_EDYNAMIK_LADUNGSBEWEGUNG|KIT-Brückenkurs >> Lorentz-Kraft]]. The last part "Magnetic field within matter" can be skipped.+For further reading you might have a look at the German contents (text, videos, exercises) on the page of the [[https://obkp.mint-kolleg.kit.edu/#OBKP_EDYNAMIK_LADUNGSBEWEGUNG|KIT-Brückenkurs >> Lorentz-Kraft]]. The last part "Magnetic field within matter" can be skipped.
  
-===== 3.4 Matter in the Magnetic Field =====+===== 7.4 Matter in the Magnetic Field (*) ===== 
 + 
 +<button size="xs" type="link" collapse="NotNeededChapter74">{{icon>eye}} not necessary for the course, but you can still find it , when you click here... </button> 
 +<collapse id="NotNeededChapter74" collapsed="true">
  
 <callout> <callout>
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 {{https://upload.wikimedia.org/wikipedia/commons/0/06/Moving_magnetic_domains_by_Zureks.gif|}} {{https://upload.wikimedia.org/wikipedia/commons/0/06/Moving_magnetic_domains_by_Zureks.gif|}}
  
 +</collapse>
  
-===== 3.5 Poynting Vector (not part of the curriculum) ===== 
- 
-  * Clear picture of the Poynting vector along an electric circuit: https://de.cleanpng.com/png-jyy1vj/ 
-  * Good explanation of the Energy flow via a current model: http://amasci.com/elect/poynt/poynt.html 
-  * Very detailed view of the energy flow in an electric circuit: http://sharif.edu/~aborji/25733/files/Energy%20transfer%20in%20electrical%20circuits.pdf 
  
 ===== Tasks ===== ===== Tasks =====
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-<panel type="info" title="Task 3.2.1 Magnetic Field Strength around a horizontal straight Conductor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Task 7.2.1 Magnetic Field Strength around a horizontal straight Conductor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
 The current $I_0 = 100~\rm A$ flows in a long straight conductor with a round cross-section. The current $I_0 = 100~\rm A$ flows in a long straight conductor with a round cross-section.
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 \end{align*} \end{align*}
  
-#@HiddenEnd_HTML~202,Result~@#+
  
  
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
-<panel type="info" title="Task 3.2.2 Superposition"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Task 7.2.2 Superposition"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
 <WRAP> <WRAP>
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 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
-<panel type="info" title="Task 3.2.3 Magnetic Potential Difference"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Task 7.2.3 Magnetic Potential Difference"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
 <WRAP> <WRAP>
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-<panel type="info" title="Task 3.3.1 magnetic Flux Density"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Task 7.3.1 magnetic Flux Density"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
 A $\rm NdFeB$ magnet can show a magnetic flux density up to $1.2 ~\rm T$ near the surface.  A $\rm NdFeB$ magnet can show a magnetic flux density up to $1.2 ~\rm T$ near the surface. 
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 <wrap #task3_3_2 /> <wrap #task3_3_2 />
  
-<panel type="info" title="Task 3.3.2 Electron in Plate Capacitor with magnetic Field"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Task 7.3.2 Electron in Plate Capacitor with magnetic Field"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
 An electron enters a plate capacitor on a trajectory parallel to the plates.  An electron enters a plate capacitor on a trajectory parallel to the plates.