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electrical_engineering_and_electronics_1:block16 [2025/11/22 18:55] – mexleadminelectrical_engineering_and_electronics_1:block16 [2026/01/10 12:46] (aktuell) – mexleadmin
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 ====== Block 16 - Ampère's Law and Magnetomotive Force (MMF) ====== ====== Block 16 - Ampère's Law and Magnetomotive Force (MMF) ======
  
-===== Learning objectives =====+===== 16.0 Intro ===== 
 + 
 +==== 16.0.1 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 =====+==== 16.0.2 Preparation at Home ====
  
 Well, again  Well, again 
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   * ...   * ...
  
-===== 90-minute plan =====+==== 16.0.3 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 =====+==== 16.0.4  Conceptual overview ====
 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
   - ...   - ...
 </callout> </callout>
  
-===== Core content =====+===== 16.1 Core content =====
  
-===== Generalization of the Magnetic Field Strength =====+==== 16.1.1 Generalization of the Magnetic Field Strength ====
  
-So far, only the rotational symmetric problem on a single wire was considered in formula, when the current $I$ ant the length $s$ of a magnetic field line around it is given:+So far, only the rotational symmetric problem of a single wire was considered in formula. I.e a current $I$ and the length $s$ of a magnetic field line around the wire was given to calculate the magnetic field strength $H$:
  
 \begin{align*} \begin{align*}
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 \begin{align*} \begin{align*}
-U = E \cdot s \quad \quad | \quad \text{applies to capacitor only}+U = E \cdot s \quad \quad | \quad \text{applies to plate capacitor only}
 \end{align*} \end{align*}
  
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 |  \begin{align*} \boxed{\oint_{s} \vec{H} {\rm d} \vec{s} = \theta } \end{align*}| The magnetic voltage $\theta$ can be given as \\ (nbsp)(nbsp) •  $\theta = I \quad \quad \quad \ $ for a single conductor \\  (nbsp)(nbsp) • $\theta = N \cdot I \quad \:\; \, $ for a coil\\  (nbsp)(nbsp) • $\theta = \sum_n \cdot I_n \quad$ for multiple conductors\\  (nbsp)(nbsp) • $\theta = \iint_A \; \vec{S} {\rm d}\vec{A}$ for any spatial distribution (see [[block15]])| |  \begin{align*} \boxed{\oint_{s} \vec{H} {\rm d} \vec{s} = \theta } \end{align*}| The magnetic voltage $\theta$ can be given as \\ (nbsp)(nbsp) •  $\theta = I \quad \quad \quad \ $ for a single conductor \\  (nbsp)(nbsp) • $\theta = N \cdot I \quad \:\; \, $ for a coil\\  (nbsp)(nbsp) • $\theta = \sum_n \cdot I_n \quad$ for multiple conductors\\  (nbsp)(nbsp) • $\theta = \iint_A \; \vec{S} {\rm d}\vec{A}$ for any spatial distribution (see [[block15]])|
 +
 +The unit of the magnetic voltage $\theta$ is **Ampere** (or **Ampere-turns**). 
 +
 +In the english literature the magnetic voltage is called **{{wp>Magnetomotive force}}**
  
 </callout> </callout>
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 </callout> </callout>
  
 +~~PAGEBREAK~~ ~~CLEARFIX~~
 +==== 16.1.2  Recap of the fieldline images ====
  
 +<WRAP group><WRAP half column>
 +=== longitudinal coil ===
 +<WRAP>
 +<imgcaption BildNr04 | Magnetic field in a longitudinal coil></imgcaption> \\
 +{{url>https://www.falstad.com/vector3dm/vector3dm.html?f=SolenoidField&d=streamlines&sl=none&st=3&ld=5&a1=21&a2=30&a3=100&rx=63&ry=1&rz=2&zm=2.396 700,450 noborder}}
 +</WRAP>
  
 +A longitudinal coil can be seen in <imgref BildNr04>. \\ 
  
 +The created field density of the coil can be derived from Ampere's Circuital Law
  
 +\begin{align*} 
 +\theta(t) &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} \\ 
 +          &= \int & \vec{H}_{\rm inner}(t) \cdot {\rm d}\vec{s} & + & \int \vec{H}_{\rm outer}(t) \cdot {\rm d} \vec{s} \\ 
 +          &= \int & \vec{H}(t) \cdot {\rm d}\vec{s}             & + &   0 \\ 
 +          &=      & {H}(t) \cdot l \\ 
 +\end{align*}
  
-===== Common pitfalls =====+The magnetic field in a toroidal coil is often considered as homogenious in the inner volume, when the length $l$ is much larger than the diameter: $l \gg d$. \\ 
 +With a given number $N$ of windings, the magnetic field strength $H$ is 
 + 
 +\begin{align*} 
 +\theta = H \cdot l = N \cdot I 
 +\end{align*} 
 +\begin{align*} 
 +\boxed{H = {{N \cdot I}\over{l}}}  \biggr | _\text{longitudinal coil} 
 +\end{align*} 
 + 
 +</WRAP><WRAP half column> 
 +=== toroidal coil === 
 +<WRAP> 
 +<imgcaption BildNr05 | Magnetic field in a toroidal coil></imgcaption> \\ 
 +{{url>https://www.falstad.com/vector3dm/vector3dm.html?f=ToroidalSolenoidField&d=streamlines&sl=none&st=1&ld=8&a1=77&a2=26&a3=100&rx=0&ry=0&rz=0&zm=1.8 700,450 noborder}} 
 +</WRAP> 
 + 
 +A toroidal coil has a donut-like setup. This can be seen in <imgref BildNr05>. \\  
 +The toroidal coil is often defined by:  
 +  * The minor radius $r$: The radius  of the circular cross-section of the coil. 
 +  * The major radius $R$: The distance from the center of the entire toroid (the center of the hole) to the center of the circular cross-section of the coil. 
 +For reasons of symmetry, it shall get clear that the field lines form concentric circles. \\ 
 +Also the magnetic field strength $H$ in a toroidal coil is often considered as homogenious, when $R \gg r$. With a given number $N$ of windings, the magnetic field strength $H$ is 
 + 
 +\begin{align*} 
 +\theta = H \cdot 2\pi R = N \cdot I  
 +\end{align*} 
 +\begin{align*} 
 +\boxed{H = {{N \cdot I}\over{2\pi R}}} \biggr | _\text{toroidal coil} 
 +\end{align*} 
 + 
 +</WRAP></WRAP> 
 + 
 + 
 +===== 16.2 Common pitfalls =====
   * ...   * ...
  
-===== Exercises ===== +===== 16.3 Exercises ===== 
-==== Worked examples ====+ 
 +<panel type="info" title="Task 3.2.3 Magnetic Potential Difference"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> 
 + 
 +<WRAP> 
 +<imgcaption BildNr05 | different trajectories around current-carrying conductors> 
 +</imgcaption> 
 +{{drawio>Task3MagneticFieldCurrentFlowingConductor.svg}} \\ 
 +</WRAP> 
 + 
 +Given are the adjacent closed trajectories in the magnetic field of current-carrying conductors (see <imgref BildNr05>). Let $I_1 = 2~\rm A$ and $I_2 = 4.5~\rm A$ be valid. 
 + 
 +In each case, the magnetic potential difference $V_{\rm m}$ along the drawn path is sought. 
 + 
 + 
 +#@HiddenBegin_HTML~323100,Path~@# 
 + 
 +  * The magnetic potential difference is given as the **sum of the current through the area within a closed path**. 
 +  * The direction of the current and the path have to be considered with the righthand rule. 
 + 
 +#@HiddenEnd_HTML~323100,Path~@# 
 + 
 +#@HiddenBegin_HTML~323102,Result a)~@# 
 +a) $V_{\rm m,a} = - I_1 = - 2~\rm A$ \\ 
 +#@HiddenEnd_HTML~323102,Result~@# 
 + 
 +#@HiddenBegin_HTML~323103,Result b)~@# 
 +b) $V_{\rm m,b} = - I_2 = - 4.5~\rm A$ \\ 
 +#@HiddenEnd_HTML~323103,Result~@# 
 + 
 +#@HiddenBegin_HTML~323104,Result c)~@# 
 +c) $V_{\rm m,c} = 0 $ \\ 
 +#@HiddenEnd_HTML~323104,Result~@# 
 + 
 +#@HiddenBegin_HTML~323105,Result d)~@# 
 +d) $V_{\rm m,d} = + I_1 - I_2 = 2~\rm A - 4.5~\rm A = - 2.5~\rm A$ \\ 
 +#@HiddenEnd_HTML~323105,Result~@# 
 + 
 +#@HiddenBegin_HTML~323106,Result e)~@# 
 +e) $V_{\rm m,e} = + I_1 = + 2~\rm A$ \\ 
 +#@HiddenEnd_HTML~323106,Result~@# 
 + 
 +#@HiddenBegin_HTML~323107,Result f)~@# 
 +f) $V_{\rm m,f} = 2 \cdot (- I_1) = - 4~\rm A$ \\ 
 +#@HiddenEnd_HTML~323107,Result~@# 
 + 
 +</WRAP></WRAP></panel> 
 + 
 +{{page>electrical_engineering_and_electronics:task_jfzlmsucghsqvop5_with_calculation&nofooter}} 
 +{{page>electrical_engineering_and_electronics:task_kmp8r8y6lvwjnoc3_with_calculation&nofooter}} 
 +{{page>electrical_engineering_and_electronics:task_ddjurcpk494go2q1_with_calculation&nofooter}}
  
-... 
  
 ===== Embedded resources ===== ===== Embedded resources =====