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electrical_engineering_and_electronics_1:block16 [2025/11/22 19:33] – [Exercises] mexleadminelectrical_engineering_and_electronics_1:block16 [2025/11/23 12:21] (aktuell) – mexleadmin
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 ===== Generalization of the Magnetic Field Strength ===== ===== 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**).+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~~
 +==== 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 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>
  
  
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