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--- electrical_engineering_2:the_electrostatic_field [2023/03/29 09:31] – mexleadmin
+++ electrical_engineering_2:the_electrostatic_field [2025/03/20 10:56] (aktuell) – mexleadmin
@@ Zeile 1: / Zeile 1: @@
-====== 1. The Electrostatic Field ======
+====== 1 The Electrostatic Field ======
 <callout>
@@ Zeile 9: / Zeile 9: @@
 </callout>
-From everyday life, it is known that there are different charges and effects of charge. <imgref ImgNr01> shows a chargeable body, which can be charged via charge separation between the sole and the floor. The movement of the foot creates a negative excess charge in the person, which is gradually distributed throughout the body. If a pointed part of the body (e.g. finger) is brought into the vicinity of a charge reservoir without excess charges, a current can flow even through the air.
+Everyday life teaches us that there are various charges and charges' effects. The image <imgref ImgNr01> depicts a chargeable body that can be charged through charge separation between the sole and the floor. The movement of the foot generates a negative surplus charge in the body, which progressively spreads throughout the body. A current can flow even through the air if a pointed portion of the body (e.g., a finger) is brought into close proximity to a charge reservoir with no extra charges.
 <WRAP>
 <imgcaption ImgNr01 | John Tra-Voltage >
@@ Zeile 17: / Zeile 16: @@
 </WRAP>
-In the first chapter of the last semester, we had already considered the charge as the central quantity of electricity and understood it as a multiple of the elementary charge. The mutual force action ([[electrical_engineering_1:preparation_properties_proportions#coulomb-force|the Coulomb-force]]) was already derived there. This is to be explained now nearer.
+We had already considered the charge as the central quantity of electricity in the first chapter of the previous semester and recognized it as a multiple of the elementary charge. There was already a mutual force action ([[electrical_engineering_1:preparation_properties_proportions#coulomb-force|the Coulomb-force]]) derived. This will be more fully explained.
-First, we will differentiate some terms:
+First, we shall define certain terms:
-  - **{{wp>Electricity}}** describes as an umbrella term all phenomena of moving and resting charges. \\ \\
+  - **{{wp>Electricity}}** is a catch-all term for any occurrences involving moving and resting charges.
-  - **{{wp>Electrostatics}}** describes the phenomena of charges at rest and thus of electric fields which do not change in time. Thus, there is no time dependence on the electrical quantities. \\ Mathematically, ${{{\rm d}  f}\over{{\rm d} t}}=0$ holds for any function of the electric quantities. \\ \\
+  - **{{wp>Electrostatics}}** is the study of charges at rest and consequently electric fields that do not vary over time. As a result, the electrical quantities have no temporal dependence. \\ For any function of the electric quantities,  ${{{\rm d}  f}\over{{\rm d} t}}=0$ holds mathematically.
-  - **{{wp>Electrodynamics}}** describes the phenomena of moving charges. Thus electrodynamics includes both electric fields that change with time and magnetic fields. \\ For the present state of the course, the simple explanation shall be, that magnetic fields are based on a current or a charge movement. \\ In electrodynamics, it is no longer valid for every function of the electric quantities, that the derivative is necessarily equal to zero. \\
+  - **{{wp>Electrodynamics}}** describes the behavior of moving charges. Hence, electrodynamics covers both changing electric fields and magnetic fields. \\ For the time being, the simple explanation will be that magnetic fields are dependent on current or charge flow. \\ It is no longer true in electrodynamics that the derivative is always necessary for any function of electric values.
-In this chapter, only electrostatics are considered. The magnetic fields are therefore excluded here for the time being.
+Only electrostatics is discussed in this chapter. For the time being, magnetic fields are thus excluded.
-Also, electrodynamics is not considered in this chapter and is introduced step by step in the following chapters.
+Furthermore, electrodynamics is not covered in this chapter and is provided in further detail in subsequent chapters.
 ~~PAGEBREAK~~ ~~CLEARFIX~~
@@ Zeile 52: / Zeile 51: @@
 <panel type="info" title="educational Task "> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
-The simulation in <imgref ImgNr02> was already briefly considered in the first chapter. Here, however, another point is to be dealt with.
+The simulation in was already mentioned briefly in the first chapter. However, another issue must be addressed here.
-In the simulation, please position a negative charge $Q$ in the middle and deactivate the electric field. The latter is done via the hook on the right. Now the situation is close to reality because a charge shows no effect at first sight.
+Place a negative charge $Q$ in the middle of the simulation and turn off the electric field. The latter is accomplished by using the hook on the right. The situation is now close to reality because a charge appears to have no effect at first glance.
-For impact analysis, a sample charge $q$ is placed in the vicinity of the existing charge $Q$ (in the simulation, the sample charge is called "sensors"). It is observed that the charge $Q$ causes a force on the sample charge. This force can be determined by magnitude and direction at any point in space. The force acts in space in a similar way to gravity. The description of the state in space changed by the charge $Q$ is defined with the help of a field.
+A sample charge $q$ is placed near the existing charge $Q$ for impact analysis (in the simulation, the sample charge is called "sensors"). The charge $Q$ is observed to effect a force on the sample charge. At any point in space, the magnitude and direction of this force can be determined. In space, the force behaves similarly to gravity. A field serves to describe the condition space changed by the charge $Q$.
 <imgcaption ImgNr02 | setup for own experiments >
@@ Zeile 65: / Zeile 64: @@
 </WRAP></WRAP></panel>
-The concept of a field shall now be briefly considered in a little more detail.
+The concept of a field will now be briefly discussed in more detail.
-  - The introduction of the field separates the cause from the effect.
+  - The introduction of the field distinguishes the cause from the effect.
-    - The charge $Q$ causes the field in space.
+    - The field in space is caused by the charge $Q$.
-    - The charge $q$ in space feels a force as an effect of the field.
+    - As a result of the field, the charge $q$ in space feels a force.
-    - This distinction becomes important again in this chapter. \\ Also in electrodynamics for high frequencies this distinction becomes clear: the field there corresponds to photons, i.e. to a transmission of effects with the finite (light)speed $c$.
+    - This distinction is brought up again in this chapter. \\ It is also fairly obvious in electrodynamics at high frequencies: the field corresponds to photons, i.e. to a transmission of effects with a finite (light)speed $c$.
-  - As with physical quantities, there are different-dimensional fields:
+  -There are different-dimensional fields, just like physical quantities:
-    - In a **scalar field**, a single number is assigned to each point in space. \\ E.g.
+    - In a **scalar field**, each point in space is assigned a single number. \\ For example,
-      - temperature field $T(\vec{x})$ on the weather map or in an object
+      - a temperature field $T(\vec{x})$ on a weather map or in an object
-      - pressure field $p(\vec{x})$
+      - a pressure field $p(\vec{x})$
-    - In a **vector field**, each point in space is assigned several numbers in the form of a vector. This reflects the action along the spatial coordinates. \\ For example.
+    - Each point in space in a **vector field** is assigned several numbers in the form of a vector. This reflects the action as it occurs along the spatial coordinates. \\ As an example.
-      - gravitational field $\vec{g}(\vec{x})$ pointing to the center of mass of the object.
+      - gravitational field $\vec{g}(\vec{x})$ pointing to the object's center of mass.
       - electric field $\vec{E}(\vec{x})$
       - magnetic field $\vec{H}(\vec{x})$
-  - If each point in space is associated with a two- or more-dimensional physical quantity - that is a tensor - then this field is called a tensor field. Tensor fields are relevant in mechanics (e.g., stress tensor) but are not necessary for electrical engineering.
+  - A tensor field is one in which each point in space is associated with a two- or more-dimensional physical quantity - that is, a tensor. Tensor fields are useful in mechanics (for example, the stress tensor), but they are not required in electrical engineering.
-Vector fields can be stated as:
+Vector fields are defined as follows:
   - Effects along spatial axes $x$, $y$ and $z$ (Cartesian coordinate system).
   - Effect in magnitude and direction vector (polar coordinate system)
@@ Zeile 92: / Zeile 91: @@
 ==== The Electric Field ====
-Thus, to determine the electric field, a measure of the magnitude and direction of the field is now needed. From the first chapter of the last semester, the Coulomb Force between two charges $Q_1$ and $Q_2$ is known:
+To determine the electric field, a measurement of its magnitude and direction is now required. The Coulomb force between two charges $Q_1$ and $Q_2$ is known from the first chapter of the previous semester:
 \begin{align*}
@@ Zeile 98: / Zeile 97: @@
 \end{align*}
-To obtain a measure of the magnitude of the electric field, the force on a (fictitious) sample charge $q$ is now considered.
+The force on a (fictitious) sample charge $q$ is now considered to obtain a measure of the magnitude of the electric field.
 \begin{align*}
@@ Zeile 105: / Zeile 104: @@
 \end{align*}
-The left part is therefore a measure of the magnitude of the field, i.e. independent of the size of the sample charge $q$. The magnitude of the electric field is thus given by
+As a result, the left part is a measure of the magnitude of the field, independent of the size of the sample charge $q$. Thus, the magnitude of the electric field is given by
 <WRAP centeralign>
@@ Zeile 119: / Zeile 118: @@
 <callout icon="fa fa-exclamation" color="red" title="Note:">
-  - The test charge $q$ is always considered to be posi**__t__**ive (mnemonic: t = +). It is used only as a thought experiment and has no retroactive effect on the sampled charge $Q$.
+  - The test charge $q$ is always considered to be posi**__t__**ive (mnemonic: t = +). It is only used as a thought experiment and has no retroactive effect on the sampled charge $Q$.
   - The sampled charge here is always a point charge.
 </callout>
@@ Zeile 125: / Zeile 124: @@
 <callout icon="fa fa-exclamation" color="red" title="Note:">
-A charge $Q$ generates an electric field $\vec{E}(Q)$ at a measuring point $P$. This electric field is given by
+At a measuring point $P$, a charge $Q$ produces an electric field $\vec{E}(Q)$. This electric field is given by
   - the magnitude $|\vec{E}|=\Bigl| {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \Bigl| $ and
-  - the direction of the force $\vec{F_C}$ which a sample charge on the measurement point $P$ experiences. This direction is given by the unit vector $\vec{e_{ \rm r}}={{\vec{F_C}}\over{|F_C|}}$ in that direction.
+  - the direction of the force $\vec{F_C}$ experienced by a sample charge on the measurement point $P$. This direction is indicated by the unit vector $\vec{e_{ \rm r}}={{\vec{F_C}}\over{|F_C|}}$ in that direction.
-Be aware, that in English courses and literature $\vec{E}, $ is simply called the electric field and the electric field strength is the magnitude $|\vec{E}|$. In German notation, the //Elektrische Feldstärke// refers to $\vec{E}$ (magnitude and direction), and the //Elektrische Feld// denotes the general presence of an electrostatic interaction (often without considering exact magnitude).
+Be aware, that in English courses and literature $\vec{E}, $ is simply referred to as the electric field and the electric field strength is the magnitude $|\vec{E}|$. In German notation, the //Elektrische Feldstärke// refers to $\vec{E}$ (magnitude and direction), and the //Elektrische Feld// denotes the general presence of an electrostatic interaction (often without considering exact magnitude).
 </callout>
@@ Zeile 199: / Zeile 198: @@
 </WRAP></WRAP></panel>
-<panel type="info" title="Task 1.1.5 Field lines"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+<panel type="info" title="Task 1.1.2 Field lines"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
 Sketch the field line plot for the charge configurations given in <imgref ImgNr04>. \\
@@ Zeile 256: / Zeile 255: @@
 In previous chapters, only single charges (e.g. $Q_1$, $Q_2$) were considered.
   * The charge $Q$ was previously reduced to a **point charge**. \\ This can be used, for example, for the elementary charge or for extended charged objects from a large distance. The distance is sufficiently large if the ratio between the largest object extent and the distance to the measurement point $P$ is small.
-  * If the charges are lined up along a line, this is called a **line charge**. \\ Examples of this are a straight trace on a circuit board or a piece of wire. Furthermore, this also applies to an extended charged object, which has exactly an extension that is no longer small in relation to the distance. For this purpose, the charge $Q$ is considered to be distributed over the line. Thus, a (line) charge density $\rho_l$ can be determined: <WRAP centeralign>$\rho_l = {{Q}\over{l}}$</WRAP> or, in the case of different charge densities on subsections: <WRAP centeralign>$\rho_l = {{\Delta Q}\over{\Delta l}} \rightarrow \rho_l(l)={{\rm d}\over{{\rm d}l}} Q(l)$</WRAP>
+  * If the charges are lined up along a line, this is referred to as a **line charge**. \\ Examples of this are a straight trace on a circuit board or a piece of wire. Furthermore, this also applies to an extended charged object, which has exactly an extension that is no longer small in relation to the distance. For this purpose, the charge $Q$ is considered to be distributed over the line. Thus, a (line) charge density $\rho_l$ can be determined: <WRAP centeralign>$\rho_l = {{Q}\over{l}}$</WRAP> or, in the case of different charge densities on subsections: <WRAP centeralign>$\rho_l = {{\Delta Q}\over{\Delta l}} \rightarrow \rho_l(l)={{\rm d}\over{{\rm d}l}} Q(l)$</WRAP>
   * It is spoken of as an **area charge** when the charge is distributed over an area. \\ Examples of this are the floor or a plate of a capacitor. Again, an extended charged object can be considered when two dimensions are no longer small in relation to the distance (e.g. surface of the earth). Again, a (surface) charge density $\rho_A$ can be determined: <WRAP centeralign>$\rho_A = {{Q}\over{A}}$</WRAP> or if there are different charge densities on partial surfaces: <WRAP centeralign>$\rho_A = {{\Delta Q}\over{\Delta A}} \rightarrow \rho_A(A) ={{\rm d}\over{{\rm d}A}} Q(A)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}} Q(A)$</WRAP>
   * Finally, a **space charge** is the term for charges that span a volume. \\ Here examples are plasmas or charges in extended objects (e.g. the doped volumes in a semiconductor). As with the other charge distributions, a (space) charge density $\rho_V$ can be calculated here: <WRAP centeralign>$\rho_V = {{Q}\over{V}}$</WRAP> or for different charge density in partial volumes: <WRAP centeralign>$\rho_V = {{\Delta Q}\over{\Delta V}} \rightarrow \rho_V(V) ={{\rm d}\over{{\rm d}V}} Q(V)={{\rm d}\over{{\rm d}x}}{{\rm d}\over{{\rm d}y}}{{\rm d}\over{{\rm d}z}} Q(V)$</WRAP>
@@ Zeile 301: / Zeile 300: @@
 {{youtube>QWOwK-zyEnE}}
 </WRAP></WRAP></panel>
+{{page>task_1.1.3_with_calc&nofooter}}
+{{page>task_1.1.4&nofooter}}
+{{page>task_1.1.5&nofooter}}
 =====1.3 Work and Potential =====
@@ Zeile 331: / Zeile 335: @@
 First, the situation of a charge in a homogeneous electric field shall be considered. As we have seen so far, the magnitude of $E$ is constant and the field lines are parallel. Now a positive charge $q$ is to be brought into this field.
-If this charge would be free movable (e.g. electron in vacuum or an extended conductor) it would be accelerated along field lines. Thus its kinetic energy increases. Because the whole system of plates (for field generation) and charge however does not change its energetic state - thermodynamically the system is closed. From this follows: if the kinetic energy increases, the potential energy must decrease.
+If this charge would be free movable (e.g. electron in a vacuum or an extended conductor) it would be accelerated along field lines. Thus its kinetic energy increases. Because the whole system of plates (for field generation) and charge however does not change its energetic state - thermodynamically the system is closed. From this follows: if the kinetic energy increases, the potential energy must decrease.
 <WRAP>
@@ Zeile 420: / Zeile 424: @@
   - Returning to the starting point from any point $A$ after a closed circuit, the __circuit voltage__ along the closed path is 0. \\ A closed path is mathematically expressed as a ring integral: \begin{align} U = \oint \vec{E} \cdot {\rm d} \vec{s} = 0 \end{align}
   - Or spoken differently: In the electrostatic field there are no self-contained field lines.
-  - A field $\vec{X}$ which satisfies the condition $\oint \vec{X} \cdot {\rm d} \vec{s}=0$ is called __vortex-free__ or __potential field__. \\ From the potential difference, or the voltage, the work in the electrostatic field results as: \begin{align*} \boxed{W_{ \rm AB}= q \cdot U_{ \rm AB}} \end{align*}
+  - A field $\vec{X}$ which satisfies the condition $\oint \vec{X} \cdot {\rm d} \vec{s}=0$ is referred to as __vortex-free__ or __potential field__. \\ From the potential difference, or the voltage, the work in the electrostatic field results as: \begin{align*} \boxed{W_{ \rm AB}= q \cdot U_{ \rm AB}} \end{align*}
 </callout>
@@ Zeile 435: / Zeile 439: @@
 Once a charge $q$ moves perpendicular to the field lines, it experiences neither energy gain nor loss.
 The voltage along this path is $0~{ \rm V}$. All points where the voltage of $0~{ \rm V}$ is applied are at the same potential level.
-The connection of these points is called:
+The connection of these points are referred to as:
   * equipotential lines for a 2-dimensional representation of the field.
   * equipotential surfaces for a 3-dimensional field
@@ Zeile 516: / Zeile 520: @@
 \begin{align*}
 U_{ \rm AB} &=  \int_{ \rm A}^{ \rm B} \vec{E} \cdot {\rm d} \vec{s} \quad && | \vec{E} \text{ and } {\rm d}\vec{s} \text{ run parallel } \\
-U_{ \rm AB} &=  \int_{ \rm A}^{ \rm B} E \cdot {\rm d}s  \quad && | \text{E = \rm const.} \\
+U_{ \rm AB} &=  \int_{ \rm A}^{ \rm B} E \cdot {\rm d}s  \quad && | \text{E = const.} \\
 U_{ \rm AB} &=  E \cdot \int_{0}^{d} {\rm d}s  \quad && | s \text{ starts at the negative plate. } d \text{ denotes the distance between the two plates }\\
 U &=  E \cdot d  \quad && | U_{ \rm AB} \text{ corresponds to the voltage applied to the capacitor } U \\
@@ Zeile 522: / Zeile 526: @@
 </callout>
-==== Tasks ====
-{{page>task_1.1.3_with_calc&nofooter}}
-{{page>task_1.1.4&nofooter}}
-{{page>task_1.1.5&nofooter}}
 =====1.4 Conductors in the Electrostatic Field =====
@@ Zeile 623: / Zeile 621: @@
   * The field lines leave the surface again at right angles. Again, a parallel component would cause a charge shift in the metal.
-This effect of charge displacement in conductive objects by an electrostatic field is called **electrostatic induction** (in German: //Influenz//).
+This effect of charge displacement in conductive objects by an electrostatic field is referred to as **electrostatic induction** (in German: //Influenz//).
 Induced charges can be separated (<imgref ImgNr11> right).
-If we look at the separated induced charges without the external field, their field is again just as strong in magnitude as the external field only in opposite direction.
+If we look at the separated induced charges without the external field, their field is again just as strong in magnitude as the external field only in the opposite direction.
 <callout icon="fa fa-exclamation" color="red" title="Note:">
@@ Zeile 669: / Zeile 667: @@
 {{page>task_1.4.4&nofooter}}
+<wrap anchor #exercise_1_4_5 />
 <panel type="info" title="Task 1.4.5 Simulation"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
@@ Zeile 683: / Zeile 682: @@
---> Answer #
+#@HiddenBegin_HTML~1,Result~@#
 $\varrho_2 < \varrho_3 < \varrho_1 < \varrho_4$
@@ Zeile 690: / Zeile 689: @@
 <imgcaption ImgNr031 | examples of field lines>
 </imgcaption> <WRAP>
-{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+256+128+0+7+421+0.048828125+364%0Ae+1+2+0+256+256+276+276+0%0Ae+1+2+0+256+256+276+276+0%0Ae+1+2+100+64+84+124+144+0%0Ae+0+2+0+256+256+276+276+0%0Ae+0+2+100+69+81+146+126+0%0Ae+0+2+100+147+96+162+100+0%0Ae+0+2+100+92+88+158+109+0%0Aw+0+2+100+112+138+157+99%0Aw+0+2+100+146+105+169+89%0Ae+0+2+100+104+91+165+96+0%0AE+1+2+100+45+55+135+145+55+65+129+139+0%0AE+1+2+100+62+63+93+94+68+69+87+88+0%0Ae+0+2+100+86+60+125+107+0%0Ae+0+2+100+54+93+92+132+0%0Ae+0+2+100+85+56+106+74+0%0Ae+0+2+100+51+79+73+113+0%0Ae+0+2+100+110+84+140+103+0%0A 600,600 noborder}}
+{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+409+209+0+10+322+0.5+397%0Ae+1+2+100+127+165+245+284+0%0Ae+0+2+100+135+159+288+248+0%0Ae+0+2+100+285+187+315+195+0%0Ae+0+2+100+174+174+305+216+0%0Aw+0+2+100+221+271+309+194%0Aw+0+2+100+291+206+339+174%0Ae+0+2+100+199+182+319+191+0%0AE+1+2+100+88+104+265+284+107+124+255+271+0%0AE+1+2+100+121+123+184+185+133+135+171+173+0%0Ae+0+2+100+170+118+247+210+0%0Ae+0+2+100+106+184+180+261+0%0Ae+0+2+100+166+110+209+145+0%0Ae+0+2+100+100+154+144+222+0%0Ae+0+2+100+217+165+276+203+0%0A 600,600 noborder}}
 </WRAP></WRAP>
-<--
+#@HiddenEnd_HTML~1,Result~@#
 </WRAP></WRAP></panel>
-=====1.5 The Electric Displacement Field and Gauss's law of electrostatics =====
+=====1.5 The Electric Displacement Field and Gauss's Law of electrostatics =====
 <callout>
@@ Zeile 821: / Zeile 824: @@
 The "sum" of the $D$-field emanating over the surface is thus just as large as the sum of the charges contained therein since the charges are just the sources of this field.
 This can be compared with a bordered swamp area with water sources and sinks:
-  * The sources in the marsh correspond to the positive charges, and the sinks to the negative charges. The formed water corresponds to the $D$-field.
+  * The sources in the marsh correspond to the positive charges and the sinks to the negative charges. The formed water corresponds to the $D$-field.
   * The sum of all sources and sinks equals in this case just the water stepping over the edge.
@@ Zeile 877: / Zeile 880: @@
 <panel type="info" title="Task 1.5.2 Manipulating a Capacitor I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
-An ideal plate capacitor with a distance of $d_0 = 7 ~{ \rm mm}$ between the plates get charged up to $U_0 = 190~{ \rm V}$ by an external source.
+An ideal plate capacitor with a distance of $d_0 = 7 ~{ \rm mm}$ between the plates gets charged up to $U_0 = 190~{ \rm V}$ by an external source.
 The source gets disconnected. After this, the distance between the plates gets enlarged to $d_1 = 7 ~{ \rm cm}$.
@@ Zeile 898: / Zeile 901: @@
 An ideal plate capacitor with a distance of $d_0 = 6 ~{ \rm mm}$ between the plates and with air as dielectric ($\varepsilon_0=1$) is charged to a voltage of $U_0 = 5~{ \rm kV}$.
-The source remains connected to the capacitor. In the air gap between the plates, a glass plate with $d_{ \rm g} = 2 ~{ \rm mm}$ and $\varepsilon_{ \rm r} = 8$ is introduced parallel to the capacitor plates.
+The source remains connected to the capacitor. In the air gap between the plates, a glass plate with $d_{ \rm g} = 4 ~{ \rm mm}$ and $\varepsilon_{ \rm r} = 8$ is introduced parallel to the capacitor plates.
-  - Calculate the partial voltages on the glas $U_{ \rm g}$ and on the air gap $U_{ \rm a}$.
+. Calculate the partial voltages on the glas $U_{ \rm g}$ and on the air gap $U_{ \rm a}$.
-  - What would be the maximum allowed thickness of a glass plate, when the electric field in the air-gap shall not exceed $E_{ \rm max}=12~{ \rm kV/cm}$?
-<button size="xs" type="link" collapse="Loesung_1_5_3_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_1_5_3_Tipps" collapsed="true">
+#@HiddenBegin_HTML~1531T,Tipps~@#
   * build a formula for the sum of the voltages first
   * How is the voltage related to the electric field of a capacitor?
-</collapse>
+#@HiddenEnd_HTML~1531T,Tipps~@#
-<button size="xs" type="link" collapse="Loesung_1_5_3_Endergebnis">{{icon>eye}} Result</button><collapse id="Loesung_1_5_3_Endergebnis" collapsed="true">
+#@HiddenBegin_HTML~1531P,Path~@#
-  - $U_{ \rm a} = 4~{ \rm kV}$, $U_{ \rm g} = 1 ~{ \rm kV}$
-  - $d_{ \rm g} = 5.96~{ \rm mm}$
+The sum of the voltages across the glass and the air gap gives the total voltage $U_0$ and each individual voltage is given by the $E$-field in the individual material by $E = {{U}\over{d}}$:
-</collapse>
+\begin{align*}
+U_0 &= U_{\rm g}               + U_{\rm a} \\
+    &= E_{\rm g} \cdot d_{\rm g} + E_{\rm a} \cdot d_{\rm a}
+\end{align*}
+The displacement field $D$ must be continuous across the different materials since it is only based on the charge $Q$ on the plates.
+\begin{align*}
+D_{\rm g}                                            &= D_{\rm a} \\
+\varepsilon_0 \varepsilon_{\rm r, g} \cdot E_{\rm g} &= \varepsilon_0  \cdot E_{\rm a}
+\end{align*}
+Therefore, we can put $E_\rm a=  \varepsilon_{\rm r, g} \cdot E_\rm g $ into the formula of the total voltage and re-arrange to get $E_\rm g$:
+\begin{align*}
+U_0 &= E_{\rm g} \cdot d_{\rm g} + \varepsilon_{\rm r, g} \cdot E_{\rm g}  \cdot d_{\rm a} \\
+    &= E_{\rm g} \cdot ( d_{\rm g} + \varepsilon_{\rm r, g} \cdot d_{\rm a}) \\
+\rightarrow E_{\rm g}  &= {{U_0}\over{d_{\rm g} + \varepsilon_{\rm r, g} \cdot d_{\rm a}}}
+\end{align*}
+Since we know that the distance of the air gap is $d_{\rm a} = d_0 - d_{\rm a}$ we can calculate:
+\begin{align*}
+E_{\rm g}  &= {{5'000 ~\rm V}\over{0.004 ~{\rm m} + 8 \cdot 0.002 ~{\rm m}}} \\
+         &= 250 ~\rm{{kV}\over{m}}
+\end{align*}
+By this, the individual voltages can be calculated:
+\begin{align*}
+U_{ \rm g} &= E_{\rm g} \cdot d_\rm g &&= 250 ~\rm{{kV}\over{m}} \cdot 0.004~\rm m &= 1 ~{\rm kV}\\
+U_{ \rm a} &= U_0 - U_{ \rm g}      &&= 5  ~{\rm kV} - 1 ~{\rm kV}               &= 4 ~{\rm kV}\\
+\end{align*}
+#@HiddenEnd_HTML~1531P,Path~@#
+#@HiddenBegin_HTML~1531R,Results~@#
+$U_{ \rm a} = 4~{ \rm kV}$, $U_{ \rm g} = 1 ~{ \rm kV}$
+#@HiddenEnd_HTML~1531R,Results~@#
+. What would be the maximum allowed thickness of a glass plate, when the electric field in the air-gap shall not exceed $E_{ \rm max}=12~{ \rm kV/cm}$?
+#@HiddenBegin_HTML~1532P,Path~@#
+Again, we can start with the sum of the voltages across the glass and the air gap, such as the formula we got from the displacement field: $D = \varepsilon_0 \varepsilon_{\rm r, g} \cdot E_{\rm g} = \varepsilon_0  \cdot E_\rm a $. \\
+Now we shall eliminate $E_\rm g$, since $E_\rm a$ is given in the question.
+\begin{align*}
+U_0 &=   E_{\rm g}                               \cdot d_{\rm g} + E_{\rm a} \cdot d_{\rm a} \\
+    &= {{E_\rm a}\over{\varepsilon_{\rm r,g}}}   \cdot d_{\rm g} + E_{\rm a} \cdot d_{\rm a} \\
+\end{align*}
+The distance $d_\rm a$ for the air is given by the overall distance $d_0$ and the distance for glass $d_\rm g$:
+\begin{align*}
+d_{\rm a} = d_0 - d_{\rm g}
+\end{align*}
+This results in:
+\begin{align*}
+U_0                      &= {{E_{\rm a}}\over{\varepsilon_{\rm r,g}}} \cdot d_{\rm g} + E_{\rm a} \cdot (d_0 - d_{\rm g}) \\
+{{U_0}\over{E_{\rm a} }} &= {{1}\over{\varepsilon_{\rm r,g}}} \cdot d_{\rm g} + d_0 - d_{\rm g} \\
+                         &= d_{\rm g} \cdot ({{1}\over{\varepsilon_{\rm r,g}}} - 1) + d_0 \\
+d_{\rm g}                &= { { {{U_0}\over{E_{\rm a} }} - d_0 } \over { {{1}\over{\varepsilon_{\rm r,g}}} - 1 } }     &= { { d_0 - {{U_0}\over{E_{\rm a} }} } \over { 1 - {{1}\over{\varepsilon_{\rm r,g}}} } }
+\end{align*}
+With the given values:
+\begin{align*}
+d_{\rm g}                &= { { 0.006 {~\rm m} - {{5 {~\rm kV} }\over{ 12 {~\rm kV/cm}}} } \over { 1 - {{1}\over{8}} } }    &= {  {{8}\over{7}} } \left( { 0.006 - {{5 }\over{ 1200}} }  \right)  {~\rm m}
+\end{align*}
+#@HiddenEnd_HTML~1532P,Path~@#
+#@HiddenBegin_HTML~1532R,Results~@#
+$d_{ \rm g} = 2.10~{ \rm mm}$
+#@HiddenEnd_HTML~1532R,Results~@#
 </WRAP></WRAP></panel>
@@ Zeile 930: / Zeile 1003: @@
 <button size="xs" type="link" collapse="Loesung_1_5_4_Endergebnis">{{icon>eye}} Result</button><collapse id="Loesung_1_5_4_Endergebnis" collapsed="true">
-  - $C = 0.5~pF$
+  - $C = 0.5~{ \rm pF}$
   - $C_{\infty} = 0.33~{ \rm pF}$
 </collapse>
@@ Zeile 1007: / Zeile 1080: @@
 ~~PAGEBREAK~~ ~~CLEARFIX~~
 <callout icon="fa fa-exclamation" color="red" title="Note:">
-  - The material constant $\varepsilon_{ \rm r}$ is called relative permittivity, relative permittivity, or dielectric constant.
+  - The material constant $\varepsilon_{ \rm r}$ is referred to as relative permittivity, relative permittivity, or dielectric constant.
   - Relative permittivity is unitless and indicates how much the electric field decreases with the presence of material for the same charge.
   - The relative permittivity $\varepsilon_{ \rm r}$ is always greater than or equal to 1 for dielectrics (i.e., nonconductors).
-  - The relative permittivity depends on the polarizability of the material, i.e. the possibility to align the molecules in the field. Correspondingly, relative permittivity depends on the frequency and often direction and temperature.
+  - The relative permittivity depends on the polarizability of the material, i.e. the possibility of aligning the molecules in the field. Correspondingly, relative permittivity depends on the frequency and often direction and temperature.
 </callout>
 <callout icon="fa fa-info" color="grey" title="Outlook">
-If now the relative permittivity $\varepsilon_{ \rm r}$ depends on the possibility to align the molecules in the field, the following interesting relation arises: if frequencies are "caught", at which the oscillation of the molecule can build up, the energy of the external field is absorbed by the molecule. This build-up is similar to the shattering of a wine glass at a suitable irradiated frequency and is called resonance. Materials can be analyzed based on the resonance frequencies. These resonance frequencies are enormously high ($1 ~{ \rm GHz}$ to $1'000'000 ~{ \rm GHz}$) and in these frequencies, the $E$-field detaches from the conductor. This may sound strange, but it becomes a bit more illustrative with the resonant circuits in the next chapters. For here it is more than sufficient that in the range of $1'000'000 ~{ \rm GHz}$ is the visual light, which is obviously not bound to a conductor. But this also makes clear that the relative permittivity $\varepsilon_{ \rm r}$ for high frequencies also has to do with the absorption (and reflection) of electromagnetic waves.
+Suppose now the relative permittivity $\varepsilon_{ \rm r}$ depends on the possibility of aligning the molecules in the field. In that case, the following interesting relation arises: if frequencies are "caught", at which the oscillation of the molecule can build up, the energy of the external field is absorbed by the molecule. This build-up is similar to the shattering of a wine glass at a suitable irradiated frequency and is referred to as resonance. Materials can be analyzed based on the resonance frequencies. These resonance frequencies are enormously high ($1 ~{ \rm GHz}$ to $1'000'000 ~{ \rm GHz}$) and in these frequencies, the $E$-field detaches from the conductor. This may sound strange, but it becomes a bit more illustrative with the resonant circuits in the next chapters. For here it is more than sufficient that in the range of $1'000'000 ~{ \rm GHz}$ is the visual light, which is obviously not bound to a conductor. But this also makes clear that the relative permittivity $\varepsilon_{ \rm r}$ for high frequencies also has to do with the absorption (and reflection) of electromagnetic waves.
 </callout>
@@ Zeile 1044: / Zeile 1117: @@
     * One says: The insulator breaks down. This means that above this electric field, a current can flow through the insulator.
     * Examples are: Lightning in a thunderstorm, ignition spark, glow lamp in a {{wp>Test_light#One-contact_neon_test_lights|phase tester}}
-    * The maximum electric field $E_0$ is called ** dielectric strength** (in German: //Durchschlagfestigkeit// or //Durchbruchfeldstärke//).
+    * The maximum electric field $E_0$ is referred to as ** dielectric strength** (in German: //Durchschlagfestigkeit// or //Durchbruchfeldstärke//).
     * $E_0$ depends on the material (see <tabref tab02>), but also on other factors (temperature, humidity, ...).
@@ Zeile 1091: / Zeile 1164: @@
   * This makes it possible to build up an electric field in the capacitor without charge carriers moving through the dielectric.
   * The characteristic of the capacitor is the capacitance $C$.
-  * In addition to the capacitance, every capacitor also has a resistance and an inductance. However, both of these are usually very small.
+  * In addition to the capacitance, every capacitor also has resistance and an inductance. However, both of these are usually very small.
   * Examples are
     * the electrical component "capacitor",
@@ Zeile 1123: / Zeile 1196: @@
 This relationship can be examined in more detail in the following simulation:
--->capacitor lab#
+--> Capacitor lab#
 If the simulation is not displayed optimally, [[https://phet.colorado.edu/sims/cheerpj/capacitor-lab/latest/capacitor-lab.html?simulation=capacitor-lab&locale=de|this link]] can be used.
@@ Zeile 1171: / Zeile 1244: @@
     - In the multilayer capacitor, there are again two electrodes. Here, too, the area $A$ (and thus the capacitance $C$) is multiplied by the finger-shaped interlocking.
     - Ceramic is used here as the dielectric.
-    - The multilayer ceramic capacitor is also called KerKo or MLCC.
+    - The multilayer ceramic capacitor is also referred to as KerKo or MLCC.
     - The variant shown in (2) is an SMD variant (surface mound device).
   - Disk capacitor
     - A ceramic is also used as a dielectric for the disk capacitor. This is positioned as a round disc between two electrodes.
     - Disc capacitors are designed for higher voltages, but have a low capacitance (in the microfarad range).
-  - **{{wp>Electrolytic capacitor}}**, in German also called //Elko// for //__El__ektrolyt__ko__ndensator//
+  - **{{wp>Electrolytic capacitor}}**, in German also referred to as  //Elko// for //__El__ektrolyt__ko__ndensator//
     - In electrolytic capacitors, the dielectric is an oxide layer formed on the metallic electrode. the second electrode is the liquid or solid electrolyte.
     - Different metals can be used as the oxidized electrode, e.g. aluminum, tantalum or niobium.
@@ Zeile 1182: / Zeile 1255: @@
     - Important for the application is that it is a polarized capacitor. I.e. it may only be operated in one direction with DC voltage. Otherwise, a current can flow through the capacitor, which destroys it and is usually accompanied by an explosive expansion of the electrolyte. To avoid reverse polarity, the negative pole is marked with a dash.
     - The electrolytic capacitor is built up wrapped and often has a cross-shaped predetermined breaking point at the top for gas leakage.
-  - **{{wp>film capacitor}}**, in German also called //Folko//, for //__Fol__ien__ko__ndensator//.
+  - **{{wp>film capacitor}}**, in German also referred to as  //Folko//, for //__Fol__ien__ko__ndensator//.
     - A material similar to a "chip bag" is used as an insulator: a plastic film with a thin, metalized layer.
     - The construction shows a high pulse load capacitance and low internal ohmic losses.
     - In the event of electrical breakdown, the foil enables "self-healing": the metal coating evaporates locally around the breakdown. Thus the short-circuit is canceled again.
-    - With some manufacturers, this type is called MKS (__M__metallized foil__c__capacitor, Polye__s__ter).
+    - With some manufacturers, this type is referred to as  MKS (__M__metallized foil__c__capacitor, Polye__s__ter).
   - **{{wp>Supercapacitor}}** (engl. Super-Caps)
     - As a dielectric is - similar to the electrolytic capacitor - very thin. In the actual sense, there is no dielectric at all.
@@ Zeile 1218: / Zeile 1291: @@
 <callout icon="fa fa-exclamation" color="red" title="Note:">
-  - There are polarized capacitors. With these, the installation direction and current flow must be observed, as otherwise an explosion can occur.
+  - There are polarized capacitors. With these, the installation direction and current flow must be observed, as otherwise, an explosion can occur.
   - Depending on the application - and the required size, dielectric strength, and capacitance - different types of capacitors are used.
   - The calculation of the capacitance is usually __not__ via $C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{A}\over{l}} $ . The capacitance value is given.
@@ Zeile 1324: / Zeile 1397: @@
 \end{align*}
 \begin{align*}
-\boxed{ U_k = const}
+\boxed{ U_k = {\rm const.}}
 \end{align*}
 </WRAP>
@@ Zeile 1370: / Zeile 1443: @@
-Up to now was assumed only one dielectric resp. only vacuum within the capacitor. Now we look at it in more detail, how multi-layered construction between sheets affects capacity.
+Up until this point, it was assumed that the capacitor contained only vacuum and one dielectric. We now examine the impact of multi-layered construction between sheets on capacity in more detail.
-Thereby several dielectrics build boundary layers between each other. Since the terminology here is sometimes misleading, it shall be discussed in more detail.
+By doing this, various dielectrics create boundary layers between one another. This terminology will be covered in more detail because it can occasionally be misleading.
-The following variants can be distinguished (<imgref ImgNr18>). \\
+It is possible to tell the following variations apart  (<imgref ImgNr18>). \\
-  - **layers are parallel to capacitor plates - dielectrics in series**: The boundary layers are __parallel__ to the capacitor plates. \\ So, the different dielectrics are _perpendicular__ to the field lines.
+  - **layers are parallel to capacitor plates - dielectrics in series**: \\ The boundary layers are __parallel__ to the capacitor plates. \\ So, the different dielectrics are __perpendicular__ to the field lines. \\ \\
-  - **layers are perpendicular to capacitor plates - dielectrics in parallel**: The boundary layers are __perpendicular__ to the capacitor plates. \\ So, the different dielectrics are __parallel__ to the field lines.
+  - **layers are perpendicular to capacitor plates - dielectrics in parallel**: \\ The boundary layers are __perpendicular__ to the capacitor plates. \\ So, the different dielectrics are __parallel__ to the field lines.  \\ \\
-  - **arbitrary configuration**: The boundary layers are neither parallel nor perpendicular to the capacitor plates.
+  - **arbitrary configuration**: \\ The boundary layers are neither parallel nor perpendicular to the capacitor plates.
 <WRAP>
@@ Zeile 1530: / Zeile 1603: @@
 </WRAP></WRAP>
-Since $\vec{D} = \varepsilon_{0} \varepsilon_{ \rm r} \cdot \vec{E}$ the direction of the fields must be the same. \\
+Since $\vec{D} = \varepsilon_{0} \varepsilon_{ \rm r} \cdot \vec{E} \;$ the direction of the fields must be the same. \\
 Using the fields, we can now derive the change in the angle:
@@ Zeile 1550: / Zeile 1623: @@
 </WRAP></WRAP></panel>
+<panel type="info" title="Exercise 1.9.2 Further capacitor charging/discharging practice Exercise "> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+{{youtube>a-gPuw6JsxQ}}
+</WRAP></WRAP></panel>
+<panel type="info" title="Exercise 1.9.3 Further practice charging the capacitor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+{{youtube>L0S_Aw8pBto}}
+</WRAP></WRAP></panel>
+<panel type="info" title="Exercise 1.9.4 Charge balance of two capacitors"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+{{youtube>EMdpkDoMXXE}}
+</WRAP></WRAP></panel>
-<panel type="info" title="Task 1.9.2 Capacitor with glass plate"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+<panel type="info" title="Exercise 1.9.5 Capacitor with glass plate"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
 <WRAP>