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--- electrical_engineering_and_electronics_1:block15 [2025/11/02 22:07] – angelegt mexleadmin
+++ electrical_engineering_and_electronics_1:block15 [2025/11/02 22:49] (aktuell) – mexleadmin
@@ Zeile 139: / Zeile 139: @@
 ~~PAGEBREAK~~~~CLEARFIX~~
 ==== Superposition of the magnetostatic Field ====
+In the electric field, the field line density was a measure of the strength of the field. This is also used for the magnetic field. Looking at the simulations in Falstad (e.g. <imgref BildNr105>) with this understanding, one notices an inconsistency: contrary to the relationship just given, the field line density in the Falstad simulation __**not**__ indicates the strength of the field. A realistic simulation is shown in <imgref BildNr106> for comparison, which makes the difference clear: the field is stronger near the conductor. Thus the field line density must also be stronger there.
+<WRAP>
+<imgcaption BildNr106 | correct Picture of Magnetic Field Lines around a Conductor>
+</imgcaption> \\
+{{url>https://www.geogebra.org/material/iframe/id/fy0yrGYK/width/500/height/500/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/true/at/preferhtml5 700,350 noborder}}
+</WRAP>
+~~PAGEBREAK~~ ~~CLEARFIX~~
+<callout icon="fa fa-exclamation" color="red" title="Attention:">
+  * The density of the field lines is a measure of the field strength.
+  * The simulation in Falstad cannot represent this in this way. \\ Here the field strength is coded by the color intensity (dark green = low field strength, light green to white = high field strength).
+</callout>
 Before the magnetic field strength will be considered in more detail, the simulation and superposition of the magnetic field will be discussed in more detail here.
@@ Zeile 144: / Zeile 158: @@
 Magnetostatic fields can be superposed, just like electrostatic fields. This allows the fields of several current-carrying lines to be combined into a single one.
 This trick is used in the following chapter to examine the magnetic field in more detail.
+Below, the magnetic field of a single current-carrying conductor is shown. This was already derived in the previous chapter by symmetry considerations. The representation in the simulation can be simplified a bit here to see the conditions more clearly: Currently, the field lines are displayed in 3D, which is done by selecting ''Display: Field Lines'' and ''No Slicing''. If you change the selection to ''Show Z Slice'' instead of ''No Slicing'', you can switch to a 2D display. In this display, small compass needles can also show the magnetic field. To do this, select ''Display: Field Vectors'' instead of ''Display: Field Lines''. In addition, a "magnetic sample", i.e. a moving compass, can be found at the mouse pointer in the 2D display.
 ~~PAGEBREAK~~~~CLEARFIX~~
@@ Zeile 151: / Zeile 168: @@
 </WRAP>
-On the right side, the magnetic field of a single current-carrying conductor is shown. This was already derived in the previous chapter by symmetry considerations. The representation in the simulation can be simplified a bit here to see the conditions more clearly: Currently, the field lines are displayed in 3D, which is done by selecting ''Display: Field Lines'' and ''No Slicing''. If you change the selection to ''Show Z Slice'' instead of ''No Slicing'', you can switch to a 2D display. In this display, small compass needles can also show the magnetic field. To do this, select ''Display: Field Vectors'' instead of ''Display: Field Lines''. In addition, a "magnetic sample", i.e. a moving compass, can be found at the mouse pointer in the 2D display.
+If there is another current-carrying conductor near the first conductor, the fields overlap. In the simulation below, the current of both conductors is directed in the same direction. The field between the conductors overlaps just enough to weaken. This can also be deduced by previous knowledge if just the middle point between both conductors is considered: There, for the left conductor the right-hand rule results in a vector directed towards the observer. For the right conductor, it results in a vector that is directed away from the observer. These just cancel each other out. Further outward field lines go around both conductors. The North and south poles here are not fixed localized toward the outside.
 ~~PAGEBREAK~~~~CLEARFIX~~
@@ Zeile 159: / Zeile 176: @@
 </WRAP>
-If there is another current-carrying conductor near the first conductor, the fields overlap. In the simulation below, the current of both conductors is directed in the same direction. The field between the conductors overlaps just enough to weaken. This can also be deduced by previous knowledge if just the middle point between both conductors is considered: There, for the left conductor the right-hand rule results in a vector directed towards the observer. For the right conductor, it results in a vector that is directed away from the observer. These just cancel each other out. Further outward field lines go around both conductors. The North and south poles here are not fixed localized toward the outside.
+If, on the other hand, the current in the second conductor is directed in the opposite direction to the current in the first conductor, the picture changes: Here there is a reinforcing superposition between the two conductors.
+Using the nomenclature from the previous chapter, it is also possible to assign north and south poles locally. Towards the outside, one pole appears to be located in front of the two conductors and another one behind.
+in both simulations, the distances between the conductors can also be changed using the ''Line Separation'' slider. What do you notice in each case when the two lines are brought close together?
 ~~PAGEBREAK~~~~CLEARFIX~~
 <WRAP>
@@ Zeile 166: / Zeile 186: @@
 <WRAP>
-If, on the other hand, the current in the second conductor is directed in the opposite direction to the current in the first conductor, the picture changes: Here there is a reinforcing superposition between the two conductors.
-Using the nomenclature from the previous chapter, it is also possible to assign north and south poles locally. Towards the outside, one pole appears to be located in front of the two conductors and another one behind.
-in both simulations, the distances between the conductors can also be changed using the ''Line Separation'' slider. What do you notice in each case when the two lines are brought close together?
 ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== Magnetic Field Strength part 1: toroidal Coil ====
+=== Complex Geometry: toroidal Coil ===
 <WRAP>
@@ Zeile 179: / Zeile 195: @@
 <WRAP>
-So far the magnetic field was defined quite pragmatically by the effect on a compass.
+A toroidal coil has a donut-like setup. This can be seen in <imgref BildNr04>. \\
-For a deeper analysis of the magnetic field, the field is now to be considered again - as with the electric field - from __two__ directions.
+For reasons of symmetry, it shall get clear that the field lines form concentric circles. \\
-The magnetic field will also be considered a "causer field" (a field caused by magnets) and an "acting field" (a field acting on a magnet).
+The magnetic field in a toroidal coil is often considered as homogenious.
-This chapter will first discuss the acting magnetic field. For this, it is convenient to consider the effects inside a toroidal coil (= donut-like setup).
-This can be seen in <imgref BildNr04>. For reasons of symmetry, it is also clear here that the field lines form concentric circles.
-In an experiment, a magnetic needle inside the toroidal coil is now to be aligned perpendicular to the field lines.
-Then, the magnetic field will generate a torque $M$ which tries to align the magnetic needle in the field direction.
-<WRAP>
-<imgcaption BildNr21 | Toroidal Coil>
-</imgcaption>
-{{drawio>toroidalcoil.svg}} \\
-</WRAP>
-It now follows:
-  - $M = {\rm const.} \neq f(\varphi)$: For the same distance from the axis of symmetry, the torque $M$ is independent of the angle $\phi$.
-  - $M \sim I$: The stronger the current flowing through a winding, the stronger the effect, i.e. the stronger the torque.
-  - $M \sim N$: The greater the number $N$ of windings, the stronger the torque $M$.
-  - $M \sim {1 \over l}$ : The smaller the average coil circumference $l$ the greater the torque. The average coil circumference $l$ is equal to the **mean magnetic path length** (=average field line length).
-To summarize:
-\begin{align*}
-M \sim {{I \cdot N}\over{l}}
-\end{align*}
-The **magnetic field strength** $H$ inside the toroidal coil is given as:
-\begin{align*}
-\boxed{H ={{I \cdot N}\over{l}}} \quad \quad | \quad \text{applies to toroidal coil only}
-\end{align*}
-For the unit of the magnetic field strength $H$ we get $[H] = {{[I]}\over{[l]}}= \rm 1~{{A}\over{m}}$
-==== Magnetic Field Strength part 2: straight conductor ====
-The previous derivation from the toroidal coil is now used to derive the field strength around a long, straight conductor. For a single conductor the part $N \cdot I$ of the formula can be reduced to $ N \cdot I = 1 \cdot I = I$ since there is only one conductor. For the toroidal coil, the magnetic field strength was given by this current(s) divided by the (average) field line length. Because of the (same rotational) symmetry, this is also true for the single conductor. Also here the field line length has to be taken into account.
-The length of a field line around the conductor is given by the distance $r$ of the field line from the conductor: $l = l(r) = 2 \cdot \pi \cdot r$. \\ For the magnetic field strength of the single conductor we then get:
-\begin{align*}
-\boxed{H ={I\over{l}} = {{I}\over{2 \cdot \pi \cdot r}}} \quad \quad | \quad \text{applies only to the long, straight conductor}
-\end{align*}
-<WRAP>
-<imgcaption BildNr105 | magnetic Field Lines around a Conductor>
-</imgcaption> \\
-{{url>https://www.falstad.com/vector3dm/vector3dm.html?f=InverseRotational&d=streamlines&sl=z&st=19&ld=12&rx=0&ry=0&rz=0&zm=1.8 700,350 noborder}}
-</WRAP>
-In the electric field, the field line density was a measure of the strength of the field. This is also used for the magnetic field. Looking at the simulations in Falstad (e.g. <imgref BildNr105>) with this understanding, one notices an inconsistency: contrary to the relationship just given, the field line density in the Falstad simulation __**not**__ indicates the strength of the field. A realistic simulation is shown in <imgref BildNr106> for comparison, which makes the difference clear: the field is stronger near the conductor. Thus the field line density must also be stronger there.
-<WRAP>
-<imgcaption BildNr106 | correct Picture of Magnetic Field Lines around a Conductor>
-</imgcaption> \\
-{{url>https://www.geogebra.org/material/iframe/id/fy0yrGYK/width/500/height/500/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/true/at/preferhtml5 700,350 noborder}}
-</WRAP>
-~~PAGEBREAK~~ ~~CLEARFIX~~
-<callout icon="fa fa-exclamation" color="red" title="Attention:">
-  * The density of the field lines is a measure of the field strength.
-  * The simulation in Falstad cannot represent this in this way. Here the field strength is coded by the color intensity (dark green = low field strength, light green to white = high field strength).
-</callout>
+The magnetic field will also be considered a "causer field" (a field caused by magnets) and an "acting field" (a field acting on a magnet).