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electrical_engineering_and_electronics_1:block18 [2025/12/02 18:51] – mexleadminelectrical_engineering_and_electronics_1:block18 [2026/01/24 15:38] (aktuell) – mexleadmin
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-====== Block 18 — Magnetic Flux and Inductivity ======+====== Block 18 — Magnetic Flux and Induction ======
  
-===== Learning objectives =====+===== 18.0 Intro ===== 
 + 
 +==== 18.0.1 Learning objectives ====
 <callout> <callout>
 After this 90-minute block, you can After this 90-minute block, you can
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 </callout> </callout>
  
-===== Preparation at Home =====+==== 18.0.2 Preparation at Home ====
  
 Well, again  Well, again 
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 For checking your understanding please do the following exercises: 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):    - Warm-up (x min): 
     - ....      - .... 
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   - Wrap-up (x min): Summary box; common pitfalls checklist.   - 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 icon="fa fa-lightbulb-o" color="blue">
   - ...   - ...
 </callout> </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. 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.
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 <WRAP> <imgcaption ImgNr01 | a Credit Card as an magnetic Application> </imgcaption> {{drawio>Creditcard.svg}} </WRAP> <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. 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.
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 \end{align*} \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: 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:
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 An animation of this situation can be seen [[https://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/22-Faraday-magnet|here]]. 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. \\ Instead of a current in the magnetic field, we will now have a look on a charge moving in the magnetic field. \\
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 </callout> </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. \\ Coming from a single free charge, let us have a look onto free charges in a conductor, when the conductor is moving. \\
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 For constant $|\vec{v}|$ and $|\vec{B}|$ this leads to:  For constant $|\vec{v}|$ and $|\vec{B}|$ this leads to: 
 \begin{align*}  \begin{align*} 
-u_{\rm ind} &= - v \cdot B \cdot l \\ +u_{\rm ind}&= - \vec{v} \times \vec{B} \cdot \vec{l} \\ 
 +|u_{\rm ind} | &= v \cdot B \cdot l \\ 
 \end{align*} \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. 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.
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 which is identical to the potential difference between the ends of the rod that we determined earlier. 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).  When looking at the magnetic field in a coil multiple windings capture the passing flux, see <imgref ImgNr14> (a). 
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 </callout> </callout>
  
-===== Common pitfalls ===== +===== 18.2 Common pitfalls ===== 
-  * ...+  * Confusing the orientation of force → current and velocity → induced voltage on the right hand rule: \\ always have the cause on thumb, always have the effect on the middle finger: 
 +    * (Lorentz) Force on a current: $\vec{F}_L = I \cdot \vec{l} \times \vec{B}$. \\ The current (thumb) causes a force (middle finger) on the conductor. \\ Without the current there would be no force. 
 +    * A conductor gets moved through a magnetic field: $u_{\rm ind}=  - \vec{v} \times \vec{B} \cdot \vec{l} $. \\ A possible induced current is based the induced voltage. \\ The movement (thumb) causes a voltage and current (middle finger) on the conductor. \\ Without the movement there would be no voltage / current. 
 + 
 +===== 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> 
  
-===== Exercises ===== 
  
 <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%> <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%>
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 </collapse> </WRAP></WRAP></panel> </collapse> </WRAP></WRAP></panel>
  
-{{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 4.1.6 Coil in magnetic Field I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <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%>
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 ===== Embedded resources ===== ===== Embedded resources =====
 <WRAP column half> <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> </WRAP>