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--- electrical_engineering_and_electronics_1:block18 [2025/12/02 18:40] – mexleadmin
+++ electrical_engineering_and_electronics_1:block18 [2026/01/10 10:40] (aktuell) – mexleadmin
@@ Zeile 1: / Zeile 1: @@
-====== Block 18 — Magnetic Flux and Inductivity ======
+====== Block 18 — Magnetic Flux and Induction ======
-===== Learning objectives =====
+===== 18.0 Intro =====
+==== 18.0.1 Learning objectives ====
 <callout>
 After this 90-minute block, you can
@@ Zeile 7: / Zeile 9: @@
 </callout>
-===== Preparation at Home =====
+==== 18.0.2 Preparation at Home ====
 Well, again
@@ Zeile 14: / Zeile 16: @@
 For checking your understanding please do the following exercises:
-  * ...
+  * Exercise E3 Coil in a magnetic Field
+  * Exercise 4.1.2 Magnetic Field Strength around a horizontal straight Conductor
+  * Exercise 4.1.4 Effects of induction I
-===== 90-minute plan =====
+==== 18.0.3 90-minute plan ====
   - Warm-up (x min):
     - ....
@@ Zeile 24: / Zeile 29: @@
   - Wrap-up (x min): Summary box; common pitfalls checklist.
-===== Conceptual overview =====
+==== 18.0.4 Conceptual overview ====
 <callout icon="fa fa-lightbulb-o" color="blue">
   - ...
 </callout>
-===== Core content =====
+===== 18.1 Core content ====
 We have been considering electric fields created by fixed charge distributions and magnetic fields produced by constant currents, but electromagnetic phenomena are not restricted to these stationary situations. Most of the interesting applications of electromagnetism are, in fact, time-dependent. To investigate some of these applications, we now remove the time-independent assumption we have been making and allow the fields to vary with time. In this and the next several chapters, you will see a wonderful symmetry in the behavior exhibited by time-varying electric and magnetic fields. Mathematically, this symmetry is expressed by an additional term in Ampère’s law and by another key equation of electromagnetism called Faraday’s law. We also discuss how moving a wire through a magnetic field produces a potential difference.
@@ Zeile 37: / Zeile 42: @@
 <WRAP> <imgcaption ImgNr01 | a Credit Card as an magnetic Application> </imgcaption> {{drawio>Creditcard.svg}} </WRAP>
-===== Recap of magnetic Field =====
+==== 18.1.1 Recap of magnetic Field ====
 The first productive experiments concerning the effects of time-varying magnetic fields were performed by Michael Faraday in 1831. One of his early experiments is represented in the simulation in <imgref ImgNr02> - in the tab ''Pickup Coil''. A potential difference is induced when the magnetic field in the coil is changed by pushing a bar magnet into or out of the coil. This potential difference can generate a current when the circuit is closed. Potential differences of opposite signs are produced by motion in opposite directions, and the directions of potential differences are also reversed by reversing poles. The same results are produced if the coil is moved rather than the magnet — it is the relative motion that is important. The faster the motion, the greater the potential difference, and there is no potential difference when the magnet is stationary relative to the coil.
@@ Zeile 87: / Zeile 92: @@
 \end{align*}
-==== Lenz Law ====
+==== 18.1.2 Lenz Law ====
 The direction in which the induced potential difference drives current around a wire loop can be found through the negative sign. However, it is usually easier to determine this direction with Lenz’s law, named in honor of its discoverer, Heinrich Lenz (1804–1865). (Faraday also discovered this law, independently of Lenz.) We state Lenz’s law as follows:
@@ Zeile 124: / Zeile 129: @@
 An animation of this situation can be seen [[https://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/22-Faraday-magnet|here]].
-==== Moving single Charge in a magnetic Field ====
+==== 18.1.3 Moving single Charge in a magnetic Field ====
 Instead of a current in the magnetic field, we will now have a look on a charge moving in the magnetic field. \\
@@ Zeile 168: / Zeile 173: @@
 </callout>
-==== Moving single Rod in a magnetic Field ====
+==== 18.1.4 Moving single Rod in a magnetic Field ====
 Coming from a single free charge, let us have a look onto free charges in a conductor, when the conductor is moving. \\
@@ Zeile 201: / Zeile 206: @@
 \end{align*}
-==== Rod in Circuit ====
+==== 18.1.5 Rod in Circuit ====
 Now let’s look at the conducting rod pulled in a circuit, changing magnetic flux. The area enclosed by the circuit '0123' of <imgref ImgNr08> is $l\cdot x$ and is perpendicular to the magnetic field.
@@ Zeile 238: / Zeile 243: @@
 which is identical to the potential difference between the ends of the rod that we determined earlier.
-==== Linked Flux ====
+<WRAP> <imgcaption ImgNr10 | motional Induction on a single Rod revisited> </imgcaption> {{drawio>MotionalInductionExampleRod2.svg}} </WRAP>
+==== 18.1.6 Linked Flux ====
 When looking at the magnetic field in a coil multiple windings capture the passing flux, see <imgref ImgNr14> (a).
@@ Zeile 265: / Zeile 272: @@
 </callout>
-===== Common pitfalls =====
+===== 18.2 Common pitfalls =====
   * ...
-===== Exercises =====
+===== 18.3 Exercises =====
+{{page>electrical_engineering_and_electronics:task_rdz03rspbwusy7wk_with_calculation&nofooter}}
+{{page>electrical_engineering_and_electronics:task_ludzwiuhjxitz85b_with_calculation&nofooter}}
+<panel type="info" title="Exercise 4.1.4 Effects of induction I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A change of magnetic flux is passing a coil with a single winding. The following pictures <imgref ImgNrEx01> show different flux-time-diagrams as examples.
+  * Create for each $\Phi(t)$-diagram the corresponding $u_{\rm ind}(t)$-diagram!
+  * Write down each maximum value of $u_{\rm ind}(t)$
+<WRAP> <imgcaption ImgNrEx01| Flux-Time-Diagrams> </imgcaption> <WRAP> {{drawio>FluxTimeDia1.svg}} \\ </WRAP></WRAP>
+<button size="xs" type="link" collapse="Solution_4_1_4_1_Solution">{{icon>eye}} Solution for (a)</button><collapse id="Solution_4_1_4_1_Solution" collapsed="true">
+For partwise linear $u_{\rm ind}$ one can derive:
+\begin{align*}
+u_{\rm ind} &= -{{{\rm d}\Phi}\over{{\rm d}t}} \\
+            &= -{{\Delta \Phi}\over{\Delta t}}
+\end{align*}
+For diagram (a):
+  * $t= 0.0 ... 0.6 ~\rm s$: $u_{\rm ind} = -{{0 ~\rm Vs}\over{0.6 ~\rm s}}= 0$
+  * $t= 0.6 ... 1.5 ~\rm s$: $u_{\rm ind} = -{{-3.75\cdot 10^{-3} ~\rm Vs}\over{0.9 ~\rm s}}= +4.17 ~\rm mV$
+  * $t= 1.5 ... 2.1 ~\rm s$: $u_{\rm ind} = -{{0 ~\rm Vs}\over{0.6 ~\rm s}}= 0$
+</collapse>
+<button size="xs" type="link" collapse="Solution_4_1_4_1_Finalresult">
+{{icon>eye}} Final result for (a)</button><collapse id="Solution_4_1_4_1_Finalresult" collapsed="true">
+<WRAP> <imgcaption ImgNrEx01| Flux-Time-Diagrams Solution> </imgcaption> <WRAP> {{drawio>FluxTimeDia1Solution.svg}} \\
+</WRAP></WRAP> \\
+</collapse>
+</WRAP></WRAP></panel>
+<panel type="info" title="Exercise 4.1.5 Effects of induction II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A changing of magnetic flux is passing a coil with a single winding and induces the voltage $u_{\rm ind}(t)$.
+The following pictures <imgref ImgNrEx02> show different voltage-time diagrams as examples.
+  * Create for each $u_{\rm ind}(t)$-diagram the corresponding $\Phi(t)$-diagram!
+  * Write down each maximum value of $\Phi(t)$
+Note the given start value $\Phi_0$ for each flux.
+<WRAP> <imgcaption ImgNrEx02| Voltage-Time-Diagrams> </imgcaption> <WRAP> {{drawio>FluxTimeDia2.svg}} \\ </WRAP></WRAP>
+#@HiddenBegin_HTML~415_1S,Solution for (a)~@#
+For partwise linear $u_{\rm ind}$ one can derive:
+\begin{align*}
+u_{\rm ind}        &= -{{{\rm d}\Phi}\over{{\rm d}t}} \\
+\rightarrow  \Phi  &= -\int_0^t{ u_{\rm ind} \;{\rm d}t} \\
+\Phi               &= \Phi_0 -\sum_k {u_{{\rm ind},~k} \; \Delta t} \\
+\end{align*}
+For diagram (a):
+  * $t= 0.00 ... 0.04 ~\rm s\quad$: $\quad \Phi = \Phi_0 - {0 \cdot \; \Delta t} \quad\quad\quad\quad\quad\quad\quad= 0 ~\rm Wb$
+  * $t= 0.04 ... 0.10 ~\rm s\quad$: $\quad \Phi =   0 {~\rm Wb} - {{30 ~\rm mV} \cdot \; (t - 0.04 ~\rm s)} = \quad {1.2 ~\rm mWb} - t \cdot 30 ~\rm mV$
+  * $t= 0.10 ... 0.14 ~\rm s\quad$: $\quad \Phi =   {1.2 ~\rm mWb} - {0.10 ~\rm s} \cdot 30 ~\rm mV \quad = - {1.8 ~\rm mWb}$
+#@HiddenEnd_HTML~415_1S,Solution ~@#
+#@HiddenBegin_HTML~415_1R,Result for (a)~@#
+{{drawio>FluxTimeDia2Res.svg}}
+#@HiddenEnd_HTML~415_1R,Result~@#
+</WRAP></WRAP></panel>
 <panel type="info" title="Exercise 4.1.1 Magnetic Field Strength around a horizontal straight Conductor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
@@ Zeile 467: / Zeile 549: @@
 </collapse> </WRAP></WRAP></panel>
+<panel type="info" title="Exercise 4.1.6 Coil in magnetic Field I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A single winding is located in a homogenous magnetic field ($B = 0.5 ~\rm T$) between the pole pieces.
+The winding has a length of $150 ~\rm mm$ and a distance between the conductors of $50 ~\rm mm$ (see <imgref ImgNrEx03>).
+  * Determine the function $u_{\rm ind}(t)$, when the coil is rotating with $3000 ~\rm min^{-1}$.
+  * Given a current of $20 ~\rm A$ through the winding: What is the torque $M(\varphi)$ depending on the angle between the surface vector of the winding and the magnetic field?
+<WRAP> <imgcaption ImgNrEx03| Winding between Pole Pieces> </imgcaption> <WRAP> {{drawio>WindingPolePieces.svg}} \\ </WRAP></WRAP>
+<button size="xs" type="link" collapse="Solution_4_1_6_1_Solution">{{icon>eye}} Solution</button><collapse id="Solution_4_1_6_1_Solution" collapsed="true">
+\begin{align*}
+u_{\rm ind} &= -   {{{\rm d}\Phi}\over{{\rm d}t}} \\
+            &= -         {{\rm d}\over{{\rm d}t}} B \cdot A \\
+            &= - B \cdot {{\rm d}\over{{\rm d}t}} A\\
+            &= - B \cdot {{\rm d}\over{{\rm d}t}} \left(l \cdot d \cdot \cos(\omega t) \right)\\
+            &= + B \cdot l \cdot d \cdot \omega \cdot \sin(\omega t)\\
+\end{align*}
+</collapse> </WRAP></WRAP></panel>
+<panel type="info" title="Exercise 4.1.7 Coil in magnetic Field II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A rectangular coil is given by the sizes $a=10 ~\rm cm$, $b=4 ~\rm cm$, and the number of windings $N=200$.
+This coil moves with a constant speed of $v=2 ~\rm m/s$ perpendicular to a homogeneous magnetic field ($B=1.3 ~\rm T$ on a length of $l=5 ~\rm cm$).
+The area of the coil is tilted with regard to the field in $\alpha=60°$ and enters the field from the left side (see <imgref ImgNrEx04>).
+  * Determine the function $u_{\rm ind}(t)$ on the coil along the given path. Sketch of the $u_{\rm ind}(t)$ diagram.
+  * What is the maximum induced voltage $u_{\rm ind,Max}$?
+<WRAP> <imgcaption ImgNrEx04| Winding between Pole Pieces> </imgcaption> <WRAP> {{drawio>WindingPolePieces2.svg}} \\ </WRAP></WRAP>
+<button size="xs" type="link" collapse="Solution_4_1_7_1_Solution">{{icon>eye}} Solution</button><collapse id="Solution_4_1_7_1_Solution" collapsed="true">
+Let assume, that $l$ is in the $x$-axis, $\vec{B}$ in the $y$-axis and $a$.
+\\ \\
+**Step 1**: Calculate the effective area, perpendicular to the $\vec{B}$-field (independent from whether the area is in the $\vec{B}$-field or not).
+For this $b$ has to be projected onto the plane perpendicular to the $\vec{B}$-field: $b_{\rm eff}= b \cdot \cos \alpha$
+\begin{align*}
+A_{\rm eff} &= a \cdot b \cdot \cos \alpha
+\end{align*}
+**Step 2**: Focus on entering and exiting the $\vec{B}$-field. \\
+Induction only occurs for ${{\rm d}\over{{\rm d}t}}(A\cdot B)\neq 0$, so here: when the area $A_{\rm eff}$ enters and leave the constant $\vec{B}$-field.
+When entering the $\vec{B}$-field the area $A$ with $0<A<A_{eff}$ is in the field.
+The area moves with $v$. Therefore, after $\Delta t = b_{\rm eff} \cdot v$ the full $\vec{B}$-field is provided onto the area $A_{\rm eff}$:
+\begin{align*}
+u_{\rm ind} &= -        {{{\rm d}\Psi}\over{{\rm d}t}} \\
+            &= -N \cdot       {{\rm d}\over{{\rm d}t}} B \cdot A \\
+            &= -N \cdot           {{1}\over{\Delta t}} B \cdot A_{\rm eff} \\
+            &= -N \cdot {{1}\over{b \cdot \cos \alpha \cdot v}} B \cdot a \cdot b \cdot \cos \alpha \\
+            &= -N \cdot B \cdot {{a}\over{v}}\\
+\end{align*}
+The following diagram shows ...
+  * ... how one can derive the effective width $b_{\rm eff}$, which is projected onto the plane perpendicular to the $\vec{B}$-field: $b_{\rm eff}= b \cdot \cos \alpha$
+  * ... what happens on the effective area $A_{\rm eff}$: there is a change of the field lines in the area only for entering and leaving the $\vec{B}$-field.
+  * ... how the $u_{\rm ind}(t)$ looks as a graph: the part of $A_{\rm eff}$, where the $\vec{B}$-field passes through increase linearly due to the constant speed $v$
+Be aware, that the task did not provide a clue for the direction of windings and therefore it provides no clue for the polarization of the induced voltage. \\
+So, the course of the voltage when entering or exiting is not uniquely given.
+<WRAP> <imgcaption ImgNrEx04s| Solution> </imgcaption> <WRAP>{{drawio>WindingPolePieces2solution.svg}}  \\ </WRAP></WRAP>
+</collapse> </WRAP></WRAP></panel>
 ===== Embedded resources =====
 <WRAP column half>
-Explanation (video): ...
+How magnetism really works \\
+{{youtube>1TKSfAkWWN0}}
+</WRAP>
+<WRAP column half>
+Application of Eddy currents \\
+{{youtube>Yu1uRvErM80?start=35}}
+</WRAP>
+ \\
+<WRAP column half>
+Application of Eddy currents \\
+{{youtube>sENgdSF8ppA}}
+</WRAP>
+<WRAP column half>
+Magnet in a copper Tube \\
+{{youtube>TRihrPnLt78?start=453}}
+</WRAP>
+ \\
+<WRAP column half>
+Hall Sensor \\
+{{url>https://upload.wikimedia.org/wikipedia/commons/transcoded/7/77/Hall_Sensor.webm/Hall_Sensor.webm.480p.vp9.webm}}
 </WRAP>