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--- electrical_engineering_2:the_electrostatic_field [2023/03/16 14:27] – mexleadmin
+++ electrical_engineering_2:the_electrostatic_field [2025/06/06 00:07] (aktuell) – mexleadmin
@@ Zeile 1: / Zeile 1: @@
-====== 1. The Electrostatic Field ======
+====== 1 The Electrostatic Field ======
 <callout>
-For this chapter the online Book 'University Physics II' is strongly recommended as reference. In detail this is:
+The online book 'University Physics II' is strongly recommended as a reference for this chapter. Especially the following chapters:
   * Chapter [[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/05%3A_Electric_Charges_and_Fields|5. Electric Charges and Fields]]
   * Chapter [[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/06%3A_Gauss's_Law|6. Gauss's Law]]
@@ 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 of the foot 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.
+Every day life teaches us that there are various charges and their 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 on 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 37: / Zeile 36: @@
 By the end of this section, you will be able to:
-  - know that an electric field is formed around a charge.
+  - Know that an electric field is formed around a charge.
-  - sketch the field lines of electric fields.
+  - Sketch the field lines of electric fields.
-  - represent the field vectors in a sketch when given several charges.
+  - Represent the field vectors in a sketch when given several charges.
-  - determine the resulting field vector by superimposing several field vectors using vector calculus.
+  - Determine the resulting field vector by superimposing several field vectors using vector calculus.
-  - determine the force on a charge in an electrostatic field by applying Coulomb's law. Specifically:
+  - Determine the force on a charge in an electrostatic field by applying Coulomb's law. Specifically:
-      - the force vector in coordinate representation
+      - The force vector in coordinate representation
-      - the magnitude of the force vector
+      - The magnitude of the force vector
-      - the angle of the force vector
+      - The angle of the force vector
 </callout>
@@ 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 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 affect 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*}
-In order 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 137: / Zeile 136: @@
-Electric field lines result from the (fictitious) path of a sample charge. Thus also electric field lines of several charges can be determined.
+Electric field lines result from the (fictitious) path of a sample charge. Thus, also electric field lines of several charges can be determined.
-However, these also result from a superposition of the individual effects - i.e. electric field - at a measuring point $P$.
+However, these also result from a superposition of the individual effects - i.e., electric field - at a measuring point $P$.
-The superposition is sketched in <imgref ImgNr032>: Two charges $Q_1$ and $Q_2$ act on the test charge $q$ with the forces $F_1$ and $F_2$. Depending on the positions and charges, the forces vary and so does the resulting force. The simulation also shows a single field line.
+The superposition is sketched in <imgref ImgNr032>: Two charges $Q_1$ and $Q_2$ act on the test charge $q$ with the forces $F_1$ and $F_2$. Depending on the positions and charges, the forces vary, and so does the resulting force. The simulation also shows a single field line.
 <WRAP>
@@ Zeile 148: / Zeile 147: @@
 </WRAP></WRAP>
-For a full picture of the field lines between charges, one has to start with a single charge. The in- and outgoing lines on this charge are drawn in equidistance on the charge. This is also true for the situation with multiple charges. However there, the lines are not necessarily run radially anymore. The test charge is influenced by all the single charges, and therefore the field lines can get bent.
+For a full picture of the field lines between charges, one has to start with a single charge. The in- and outgoing lines on this charge are drawn equidistant from the charge. This is also true for the situation with multiple charges. However, there, the lines are not necessarily run radially anymore. The test charge is influenced by all the single charges, and therefore, the field lines can get bent.
 <WRAP>
@@ Zeile 161: / Zeile 160: @@
     * ... move the charges by drag and drop.
     * ... add another Charge with ''Add'' >> ''Add Point Charge''.
-    * ... delete components with a right click onto it and ''delete''
+    * ... delete components with a right click on them and ''delete''
   * Where is the density of the field lines higher?
-  * How does the field between two positive charges look like? How does it look between two different charges?
+  * How does the field between two positive charges look? How does it look between two different charges?
 <WRAP>
@@ Zeile 186: / Zeile 185: @@
   * Electric field lines cannot cut each other: This is based on the fact that the direction of the force at a cutting point would not be unique.
   * The field lines are always perpendicular to conducting surfaces. This is also based on energy considerations; more details later.
-  * The inside of a conducting component is always field free. Also, this will be discussed in the following.
+  * The inside of a conducting component is always field-free. Also, this will be discussed in the following.
 </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 226: / Zeile 225: @@
 By the end of this section, you will be able to:
-  - determine the direction of the forces using given charges.
+  - Determine the direction of the forces using the given charges.
-  - represent the acting force vectors in a sketch.
+  - Represent the acting force vectors in a sketch.
-  - determine a force vector by superimposing several force vectors using vector calculus.
+  - Determine a force vector by superimposing several force vectors using vector calculus.
-  - state the following quantities for a force vector:
+  - State the following quantities for a force vector:
-      - the force vector in coordinate representation
+      - The force vector in coordinate representation
-      - the magnitude of the force vector
+      - The magnitude of the force vector
-      - the angle of the force vector
+      - The angle of the force vector
 </callout>
-The electric charge and Coulomb force have already been described last semester. However, some points are to be caught up here to it.
+The electric charge and Coulomb force have already been described last semester. However, some points are to be caught up here.
 ==== Direction of the Coulomb force and Superposition ====
-In the case of the force, only the direction has been considered so far, e.g. direction towards the sample charge. For future explanations, it is important to include the cause-effect in the naming. This is done by giving the correct labeling of the subscript of the force. In <imgref ImgNr06> (a) and (b) the convention is shown: A force $\vec{F}_{21}$ acts on charge $Q_2$ and is caused by charge $Q_1$. As a mnemonic, you can remember "tip-to-tail" (first the effect, then the cause).
+In the case of the force, only the direction has been considered so far, e.g., direction towards the sample charge. For future explanations, it is important to include the cause and effect in the naming. This is done by giving the correct labeling of the subscript of the force. In <imgref ImgNr06> (a) and (b), the convention is shown: A force $\vec{F}_{21}$ acts on charge $Q_2$ and is caused by charge $Q_1$. As a mnemonic, you can remember "tip-to-tail" (first the effect, then the cause).
-Furthermore, several forces on a charge can be superimposed resulting in a single, equivalent force. \\
+Furthermore, several forces on a charge can be superimposed, resulting in a single, equivalent force. \\
 Strictly speaking, it must hold that $\varepsilon$ is constant in the structure. For example, the resultant force in <imgref ImgNr06> Fig. (c) on $Q_3$ becomes equal to: $\vec{F_3}= \vec{F_{31}}+\vec{F_{32}}$.
@@ Zeile 254: / Zeile 253: @@
 ==== Geometric Distribution of Charges ====
-In previous chapters, only single charges (e.g. $Q_1$, $Q_2$) were considered.
+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 if there are two extensions that 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>
+  * It is spoken of as an **area charge** when the charge is distributed over an area. \\ Examples of this are the floor or the 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>
+  * 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>
 In the following, area charges and their interactions will be considered.
@@ Zeile 270: / Zeile 269: @@
 <WRAP group><WRAP column half>
 In **homogeneous fields**, magnitude and direction are constant throughout the field range.
-This field form is idealized to exist within plate capacitors. e.g., in the plate capacitor (<imgref ImgNr07>), or in the vicinity of widely extended bodies.
+This field form is idealized to exist within plate capacitors. e.g., in the plate capacitor (<imgref ImgNr07>), or the vicinity of widely extended bodies.
 <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 310: / Zeile 314: @@
 By the end of this section, you will be able to:
-  - know how work is defined in the electrostatic field.
+  - Know how work is defined in the electrostatic field.
-  - describe when work has to be performed and when it does not in the situation of a movement.
+  - Describe when work has to be performed and when it does not in the situation of a movement.
-  - know the definition of electric voltage and be able to calculate it in an electric field.
+  - Know the definition of electric voltage and be able to calculate it in an electric field.
-  - understand why the calculation of voltage is independent of displacement.
+  - Understand why the calculation of voltage is independent of displacement.
-  - know what a potential difference is and recognize or be able to state equipotential surfaces (lines).
+  - Know what a potential difference is and recognize or be able to state equipotential surfaces (lines).
-  - determine a potential curve for a given arrangement.
+  - Determine a potential curve for a given arrangement.
 </callout>
@@ Zeile 322: / Zeile 326: @@
 In the following, only a few brief illustrations of the concepts are given. \\
-A detailed explanation can be found in the online Book 'University Physics II'. It is recommended to work through this independently. \\ \\
+A detailed explanation can be found in the online book 'University Physics II'. It is recommended to work through this independently. \\ \\
 In particular, this applies to:
   * Chapter "[[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/07%3A_Electric_Potential|7. electric potential]]"
@@ Zeile 329: / Zeile 333: @@
 ==== Energy required to Displace a Charge in the electric Field ====
-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.
+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 in the 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 were free movable (e.g., an 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 339: / Zeile 343: @@
 <WRAP>
-It is known from mechanics, that the work done (thus energy needed) is defined by the force one needs to move along a path. \\
+It is known from mechanics that the work done (thus the energy needed) is defined by the force one needs to move along a path. \\
 In a homogeneous field, the following holds for a force-producing motion along a field line from ${ \rm A}$ to ${ \rm B}$ (see <imgref ImgNr09>):
 \begin{align*}
@@ Zeile 345: / Zeile 349: @@
 \end{align*}
-For a motion perpendicular to the field lines (i.e. from ${ \rm A}$ to ${ \rm C}$) no work is needed - so $W_{ \rm AC}=0$ results - because the formula above is only true for $F_C$ parallel to $s$. The motion perpendicular to the field lines is similar to the movement of weight in the gravitational field at the same height. Or more illustrative: It is similar to walking at the same floor of a house. There, too, no energy is released or absorbed with regard to the field.
+For a motion perpendicular to the field lines (i.e. from ${ \rm A}$ to ${ \rm C}$) no work is needed - so $W_{ \rm AC}=0$ results - because the formula above is only true for $F_C$ parallel to $s$. The motion perpendicular to the field lines is similar to the movement of weight in the gravitational field at the same height. Or more illustratively: It is similar to walking on the same floor of a house. There, too, no energy is released or absorbed concerning the field.
 For any direction through the field, the part of the path has to be considered, which is parallel to the field lines. This results from the angle $\alpha$ between $\vec{F}$ and $\vec{s}$:
 \begin{align*}
@@ Zeile 352: / Zeile 356: @@
 The work $W_{ \rm AB}$ here describes the energy difference experienced by the charge $q$. \\
-Similar to the electric field, we now look for a quantity that is independent of the (sample) charge $q$ in order to describe the energy component. This is done by the **voltage** $U$. The voltage of a movement from $A$ to $B$ in a homogeneous field is defined as:
+Similar to the electric field, we now look for a quantity that is independent of the (sample) charge $q$ to describe the energy component. This is done by the **voltage** $U$. The voltage of a movement from $A$ to $B$ in a homogeneous field is defined as:
 \begin{align}
@@ Zeile 387: / Zeile 391: @@
 \end{align*}
-Interestingly, it does not matter which way the integration takes place. So, it doesn't matter how the charge gets from ${ \rm A}$ to ${ \rm B}$: the energy needed and the voltage is always the same.
+Interestingly, it does not matter which way the integration takes place. So, it doesn't matter how the charge gets from ${ \rm A}$ to ${ \rm B}$: the energy needed and the voltage are always the same.
 This follows from the fact that a charge $q$ at a point ${ \rm A}$ in the field has a unique potential energy.
 No matter how this charge is moved to a point ${ \rm B}$ and back again: as soon as it gets back to point ${ \rm A}$, it has the same energy again.
@@ Zeile 398: / Zeile 402: @@
 <WRAP>
-This independency of the taken path leads to the closed path in <imgref ImgNr09b> from ${ \rm A}$ to ${ \rm B}$ and back to:
+This independence of the taken path leads to the closed path in <imgref ImgNr09b> from ${ \rm A}$ to ${ \rm B}$ and back to:
 \begin{align*}
@@ Zeile 419: / Zeile 423: @@
 <callout icon="fa fa-exclamation" color="red" title="Note:">
   - 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.
+  - 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 432: / Zeile 436: @@
 In the previous subchapter, the term voltage got a more general meaning.
-This shall be now applied to investigate the electric field a bit more.
+This shall now be applied to investigate the electric field a bit more.
 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 is referred to as:
   * equipotential lines for a 2-dimensional representation of the field.
   * equipotential surfaces for a 3-dimensional field
@@ Zeile 442: / Zeile 446: @@
 In <imgref ImgNr98>, the equipotential lines of a point charge are shown.
-  * The equipotential surfaces are drawn with a fixed step size, e.g. $1~{ \rm V}$, $2~{ \rm V}$, $3~{ \rm V}$, ... .
+  * The equipotential surfaces are drawn with a fixed step size, e.g. $1~{ \rm V}$, $2~{ \rm V}$, $3~{ \rm V}$, ....
   * Since the electric field is higher near charges, equipotential surfaces are also closer together there.
   * The angle between the field vectors (and therefore the field lines) and the equipotential lines is always $90°$
@@ Zeile 456: / Zeile 460: @@
 ==== Electric Potential ====
-So up to now, the voltage was investigated and also equipotential areas were found. But what is this potential anyway?
+So up to now, the voltage was investigated, and also equipotential areas were found. But what is this potential anyway?
 Since the voltage is independent of the path, one can conclude that the path integral can always be expressed as the difference between two scalar values:
@@ Zeile 472: / Zeile 476: @@
 Here, the **electric potential** $\varphi$ is introduced as the scalar local function of the electric field (see <imgref ImgNr10b>). This means: any point in space can either be connected to the two-dimensional value $\vec{E}$ or the one-dimensional value $\varphi$. Both fully and equally represent the electrostatic field.
-Similar to the reference or ground level for the altitude in the gravitational field, the **reference or ground potential** can be chosen arbitrarily for a single task. Often the ground potential $\varphi_{ \rm G}= \varphi_{ \rm GND}$ is chosen to be located at infinity (see <imgref ImgNr10c>). In this case, the potential at the point $\rm A$ can be calculated as follows:
+Similar to the reference or ground level for the altitude in the gravitational field, the **reference or ground potential** can be chosen arbitrarily for a single task. Often, the ground potential $\varphi_{ \rm G}= \varphi_{ \rm GND}$ is chosen to be located at infinity (see <imgref ImgNr10c>). In this case, the potential at the point $\rm A$ can be calculated as follows:
 <WRAP>
@@ Zeile 486: / Zeile 490: @@
 \end{align*}
-Alternatively, also the potential $\varphi_{ \rm B}$ could be considered as ground potential. This would lead to the following potentials for $\varphi_{ \rm A}$ and $\varphi_{ \rm C}$:
+Alternatively, also the potential $\varphi_{ \rm B}$ could also be considered as ground potential. This would lead to the following potentials for $\varphi_{ \rm A}$ and $\varphi_{ \rm C}$:
 \begin{align*}
@@ 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 537: / Zeile 535: @@
 By the end of this section, you will be able to:
-  - know that no current flows in a conductor in an electrostatic field.
+  - Know that no current flows in a conductor in an electrostatic field.
-  - know how charges in a conductor are distributed in the electrostatic field.
+  - Know how charges in a conductor are distributed in the electrostatic field.
-  - sketch the field lines at the surface of the conductor.
+  - Sketch the field lines at the surface of the conductor.
   - Understand the effect of the electrostatic induction of an external electric field.
 </callout>
-Up to now, charges were considered which were either rigid or not freely movable.
+Up to now, charges were considered that were either rigid or not freely movable.
-In the following, charges at an electric conductor are investigated.
+In the following, charges on an electric conductor are investigated.
 These charges are only free to move within the conductor.
 At first, an ideal conductor without resistance is considered.
-==== Stationary Situation of a charged Object without external Field ====
+==== Stationary Situation of a charged Object without an external Field ====
-In the first thought experiment, a conductor (e.g. a metal plate) is charged, see <imgref ImgNr10>.
+In the first thought experiment, a conductor (e.g., a metal plate) is charged, see <imgref ImgNr10>.
 The additional charges create an electric field. Thus, a resultant force acts on each charge.
 The causes of this force are the electric fields of the surrounding electric charges. So the charges repel and move apart. \\
@@ Zeile 567: / Zeile 565: @@
   * The charge carriers are distributed on the surface.
   * Due to the dispersion of the charges, the interior of the conductor is free of fields.
-  * All field lines are perpendicular to the surface. Because: if they were not, there would be a parallel component of the field, i.e. along the surface. Thus a force would act on charge carriers and they would move accordingly.
+  * All field lines are perpendicular to the surface. Because: if they were not, there would be a parallel component of the field, i.e., along the surface. Thus, a force would act on charge carriers, and they would move accordingly.
 <wrap #edu_task_1 />
 <panel type="info" title="Educational Task - Why is there a discharge at pointy ends of conductors?"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
-Point discharge is a well-known phenomenon, which can be seen as {{wp>corona discharge}} on power lines (where it also creates the summing sound) or is used in {{wp>spark plug}}s. The phenomenon addresses the effect, that there are much more charges at the corners and edges of a conductor. But why is that so? For this, it is feasible to try to calculate the charge density at different spots of a conductor.
+Point discharge is a well-known phenomenon, which can be seen as {{wp>corona discharge}} on power lines (where it also creates the summing sound) or is used in {{wp>spark plug}}s. The phenomenon addresses the effect that there are many more charges at the corners and edges of a conductor. But why is that so? For this, it is feasible to try to calculate the charge density at different spots of a conductor.
 <WRAP>
@@ Zeile 588: / Zeile 586: @@
 In the <imgref ImgNr194> an example of a "pointy" conductor is given in image (a). The surface of the conductor is always at the same potential.
-In order to cope with this complex shape and the wanted charge density, the following path shall be taken:
+To cope with this complex shape and the desired charge density, the following path shall be taken:
-  -  It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm AB} =  \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ with $\vec{E} ={{1} \over {4\pi\cdot\varepsilon}} \cdot {{q} \over {r^2}} \cdot \vec{e}_r $, where $\vec{e}_r$ is the unit vector pointing radially away, ${ \rm C}$ is a point at distance $r_0$ from the charge and ${ \rm G}$ is the ground potential at infinity.
+  -  It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm CG} =  \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ with $\vec{E} ={{1} \over {4\pi\cdot\varepsilon}} \cdot {{q} \over {r^2}} \cdot \vec{e}_r $, where $\vec{e}_r$ is the unit vector pointing radially away, ${ \rm C}$ is a point at distance $r_0$ from the charge and ${ \rm G}$ is the ground potential at infinity.
   - Compare the field and the potentials of the different spherical conductors in <imgref ImgNr194>, image (b).
     - Are there differences for the electric field $\vec{E}$ outside the spherical conductors? Are the potentials on the surface the same?
     - What can be conducted for the field of the three situations in (b) and (d), when the total charge on the surface is considered to be always the same?
-  - For spherical conductors the surface charge density is constant. Given that this charge density leads to the overall charge $q$, how is $\varrho_A$ depending on the radius $r$ of a sphere?
+  - For spherical conductors, the surface charge density is constant. Given that this charge density leads to the overall charge $q$, how does $\varrho_A$ depend on the radius $r$ of a sphere?
-  - Now, the situation in (c) shall be considered. Here, all components are conducting, i.e. the potentials on the surface are similar. Both spheres shall be considered to be as far away from each other, that they show an undisturbed field nearby their surfaces. In this case, charges on the surface of the curvature to the left and to the right represent the same situation as in (a). For the next step, it is important that by this, the potentials of the left sphere with $q_1$ and $r_1$ and the right sphere with $q_2$ and $r_2$ are the same.
+  - Now, the situation in (c) shall be considered. Here, all components are conducting, i.e., the potentials on the surface are similar. Both spheres shall be considered to be as far away from each other, so that they show an undisturbed field near their surfaces. In this case, charges on the surface of the curvature to the left and the right represent the same situation as in (a). For the next step, it is important that by this, the potentials of the left sphere with $q_1$ and $r_1$ and the right sphere with $q_2$ and $r_2$ are the same.
     - Set up this equality formula based on the formula for the potential from question 1.
     - Insert the relationship for the overall charges $q_1$ and $q_2$ based on the surface charge densities $\varrho_{A1}$ and $\varrho_{A2}$ of a sphere and their radii $r_1$ and $r_2$.
@@ Zeile 603: / Zeile 601: @@
 ==== Electrostatic Induction ====
-In the second thought experiment, an uncharged conductor (e.g. a metal plate) is brought into an electrostatic field (<imgref ImgNr11>).
+In the second thought experiment, an uncharged conductor (e.g., a metal plate) is brought into an electrostatic field (<imgref ImgNr11>).
 The external field or the resulting Coulomb force causes the moving charge carriers to be displaced. \\
@@ Zeile 620: / Zeile 618: @@
 Results:
   * The charge carriers are still distributed on the surface.
-  * Now equilibrium is reached when just so many charges have moved, that the electric field inside the conductor disappears (again).
+  * Now, equilibrium is reached when just so many charges have moved, that the electric field inside the conductor disappears (again).
   * 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:">
-  - The location of an induced charge is always the conductor surface. This results in a surface charge density $\varrho_A = {{\Delta Q}\over{\Delta A}}$
+  - The location of an induced charge is always on the conductor surface. This results in a surface charge density $\varrho_A = {{\Delta Q}\over{\Delta A}}$
   - The conductor surface in the electrostatic field is always an equipotential surface. Thus, the field lines always originate and terminate perpendicularly on conductor surfaces.
   - The interior of the conductor is always field-free (Faraday effect: metallic enclosures shield electric fields).
@@ Zeile 652: / Zeile 650: @@
 </WRAP></WRAP>
-In the simulation in <imgref ImgNr198> the equipotential lines and electric field at different objects can be represented.
+In the simulation in <imgref ImgNr198>, the equipotential lines and electric field at different objects can be represented.
 In the beginning, the situation of an infinitely long cylinder in a homogeneous electric field is shown.
 The solid lines show the equipotential surfaces. The small arrows show the electric field.
-  - What is the angle between of the field on the surface of the cylinder?
+  - What is the angle between the field on the surface of the cylinder?
-  - Once the option ''Flat View'' is deactivated, an alternative view of this situation can be seen. Additionally, charged test particles can be added with ''Display: Particles (Vel.)''. This alternative view looks similar to which other physical field?
+  - Once the option ''Flat View'' is deactivated, an alternative view of this situation can be seen. Additionally, charged test particles can be added with ''Display: Particles (Vel.)''. This alternative view looks similar to what other physical fields?
   - What can be said about the potential distribution on the cylinder?
-  - On the left half the field lines enter the body, on the right half they leave the body. What can be said about the charge carrier distribution at the surface? Check also the representation ''Floor: charge''!
+  - On the left half of the field lines enter the body, on the right half, they leave the body. What can be said about the charge carrier distribution at the surface? Check also the representation ''Floor: charge''!
   - Is there an electric field inside the body?
   - Is this cylinder metallic, semiconducting, or insulating?
@@ 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 703: / Zeile 706: @@
 By the end of this section, you will be able to:
-  - know how to get the electric displacement field from single charges
+  - Know how to get the electric displacement field from single charges
-  - state for a given area the electric displacement field of an arrangement
+  - State for a given area the electric displacement field of an arrangement
-  - know the general meaning of Gauss's law of electrostatics
+  - Know the general meaning of Gauss's law of electrostatics
-  - choose a closed surface appropriately and apply Gauss's law
+  - Choose a closed surface appropriately and apply Gauss's law
 </callout>
@@ Zeile 716: / Zeile 719: @@
 Up to now, ...
   * ... we investigated the __effect__ of the electric field onto a (probe) charge, which can be calculated by $\vec{F}= \vec{E}\cdot q$.
-  * ... the field $\vec{E}$ is principally a property of the space and the charges inside of it.
+  * ... the field $\vec{E}$ is principally a property of the space and the charges inside it.
   * ... we also only had a look at "empty space" containing charges and/or ideally conducting components
 The following introduced **electric displacement flux density $\vec{D}$** is only focusing on the __cause__ of the electric fields.
-The effect can differ since the space can also "hinder" the electric field to an effect. This is especially true when the situation within a material and not a vacuum has to be analyzed.
+The effect can differ since the space can also "hinder" the electric field in an effect. This is especially true when the situation within a material and not a vacuum has to be analyzed.
-In order to investigate this situation, we want to consider two conductive plates (X) and (Y) with the area $\Delta A$ in the electrostatic field $\vec{E}$ in a vacuum a little more exactly. For this purpose, the plates shall first be brought into the field separately.
+To investigate this situation, we want to consider two conductive plates (X) and (Y) with the area $\Delta A$ in the electrostatic field $\vec{E}$ in a vacuum, a little more exactly. For this purpose, the plates shall first be brought into the field separately.
 <WRAP>
@@ Zeile 730: / Zeile 733: @@
 <WRAP>
-As written in <imgref ImgNr12> a), the electrostatic induction in a single plate is not considered. Rather, we are now interested in what happens based on the electrostatic induction __when the plates are brought together__. The electrostatic induction will again move charges inside the conductors. Near the negative outer plate (1) positive charges get induced on (X). Equally, near to positive outer plate (2) negative charges get induced on (Y). Graphically speaking, for each field line ending on the pair of plates, a single charge must move from one plate to the other. The direction of the movement is in similar to the direction of $\vec{E}$. This ability to separate charges (i.e. to generate electrostatic induction) is another property of space. This property is independent of any matter inside of the space.
+As written in <imgref ImgNr12> a), the electrostatic induction in a single plate is not considered. Rather, we are now interested in what happens based on the electrostatic induction __when the plates are brought together__. The electrostatic induction will again move charges inside the conductors. Near the negative outer plate (1), positive charges get induced on (X). Equally, near to positive outer plate (2), negative charges get induced on (Y). Graphically speaking, for each field line ending on the pair of plates, a single charge must move from one plate to the other. The direction of the movement is similar to the direction of $\vec{E}$. This ability to separate charges (i.e,. to generate electrostatic induction) is another property of space. This property is independent of any matter inside the space.
-This movement is represented with the **displacement flux $\Psi$**. The displacement flux is given by the amount of moved charge $\Psi = n \cdot e = Q$, with the unit $[\Psi]= [Q] = 1~{ \rm C}$. When looking at <imgref ImgNr12> b) and c), it is evident, that for larger plates (X) and (Y) more charges get displaced. So, in order to get a constant value by dividing displacement flux by the corresponding area. This leads to the **electric displacement field $D$** (sometimes also displacement flux density), which is defined as:
+This movement is represented by the **displacement flux $\Psi$**. The displacement flux is given by the amount of moved charge $\Psi = n \cdot e = Q$, with the unit $[\Psi]= [Q] = 1~{ \rm C}$. When looking at <imgref ImgNr12> b) and c), it is evident that for larger plates (X) and (Y) more charges get displaced. So, to get a constant value by dividing the displacement flux by the corresponding area. This leads to the **electric displacement field $D$** (sometimes also displacement flux density), which is defined as:
 \begin{align*}
@@ Zeile 754: / Zeile 757: @@
   * Similar to the electric field $\vec{E}$ also the flux density is a field.
   * It can be interpreted as a vector field. pointing in the same direction as the electric field $\vec{E}$.
   * The electric displacement field has the unit "charge per area", i.e. ${ \rm As/m^2}$.
-Why is now a second field introduced? This shall become clearer in the following, but first it shall be considered again how the electric field $\vec{E}$ was defined. This resulted from the Coulomb force, i.e. the __action on a sample charge__. The electric displacement field, on the other hand, is not described by an action, but __caused by charges__.
+Why is a second field introduced? This shall become clearer in the following, but first, it shall be considered again how the electric field $\vec{E}$ was defined. This resulted from the Coulomb force, i.e., the __action on a sample charge__. The electric displacement field, on the other hand, is not described by an action, but __caused by charges__.
 The two are related by the above equation.
 It will be shown in later sub-chapters that the different influences from the same cause of the field can produce different effects on other charges.
-The **permittivity** (or dielectric conductivity) $\varepsilon$ thus results as a constant of proportionality between $D$-field and $E$-field. The inverse ${{1}\over{\varepsilon}}$ is a measure of how much effect ($E$-field) is available from the cause ($D$-field) at a point. In vacuum, $\varepsilon$ is $\varepsilon_0$, the electric field constant.
+The **permittivity** (or dielectric conductivity) $\varepsilon$ thus results as a constant of proportionality between $D$-field and $E$-field. The inverse ${{1}\over{\varepsilon}}$ is a measure of how much effect ($E$-field) is available from the cause ($D$-field) at a point. In a vacuum, $\varepsilon$ is $\varepsilon_0$, the electric field constant.
 ~~PAGEBREAK~~ ~~CLEARFIX~~
 ==== General relationship between Charge Q and electric Displacement Field D ====
-Up to now, only a homogeneous field was considered and only a surface perpendicular to the field lines. Thus only equipotential surfaces (e.g. a metal foil) were investigated.
+Up to now, only a homogeneous field was considered, and only a surface perpendicular to the field lines. Thus, only equipotential surfaces (e.g., a metal foil) were investigated.
-In that case it was found that the charge is equal to the electric displacement field on the surface: $\Delta Q = D\cdot \Delta A$.
+In that case, it was found that the charge is equal to the electric displacement field on the surface: $\Delta Q = D\cdot \Delta A$.
 This formula is now to be extended to arbitrary surfaces and inhomogeneous fields.
-As with the potential and other physical problems, the problem is to be broken down into smaller sub-problems, solved and then summed up. For this purpose a small area element $\Delta A = \Delta x \cdot \Delta y$ is needed. In addition, the position of the area in space should be taken into account. This is possible if the cross product is chosen: $\Delta \vec{A} = \Delta \vec{x} \times \Delta \vec{y}$, since so is the surface normal. In what follows, the cross product will be relevant to the calculation, but the consequences of the cross product will be:
+As with the potential and other physical problems, the problem is to be broken down into smaller sub-problems, solved, and then summed up. For this purpose, a small area element $\Delta A = \Delta x \cdot \Delta y$ is needed. In addition, the position of the area in space should be taken into account. This is possible if the cross product is chosen: $\Delta \vec{A} = \Delta \vec{x} \times \Delta \vec{y}$, since so is the surface normal. In what follows, the cross-product will be relevant to the calculation, but the consequences of the cross-product will be:
   * The magnitude of $\Delta \vec{A}$ is equal to the area $\Delta A$.
   * The direction of $\Delta \vec{A}$ is perpendicular to the area.
@@ Zeile 775: / Zeile 778: @@
 === 1. Problem: Inhomogenity → Solution: infinitesimal Area ===
-First, we shall still assume an observation surface perpendicular to the field lines, but an inhomogeneous field. In the inhomogeneous field, the magnitude of $D$ is no longer constant. In order to correct this, ${\rm d}A$ is chosen so small that just "only one field line" passes through the surface. In this case $D$ is homogeneous again. Thus holds:
+First, we shall still assume an observation surface perpendicular to the field lines, but an inhomogeneous field. In the inhomogeneous field, the magnitude of $D$ is no longer constant. To correct this, ${\rm d}A$ is chosen so small that just "only one field line" passes through the surface. In this case, $D$ is homogeneous again. Thus holds:
 $Q = D\cdot A$
@@ Zeile 790: / Zeile 793: @@
 === 2nd problem: arbitrary surface → solution: vectors ===
-Now assume an arbitrary surface. Thus the $\vec{D}$-field no longer penetrates through the surface at right angles. But for the electrostatic induction only the rectangular part was relevant. So only this part has to be considered. This results from consideration of the cosine of the angle between (right-angled) area vector and $\vec{D}$-field:
+Now, assume an arbitrary surface. Thus, the $\vec{D}$-field no longer penetrates through the surface at right angles. But for the electrostatic induction, only the rectangular part was relevant. So only this part has to be considered. This results from consideration of the cosine of the angle between (right-angled) area vector and $\vec{D}$-field:
 \begin{align*}
@@ Zeile 811: / Zeile 814: @@
 === 3. Summing up ===
-Since so far only infinitesimally small surface pieces were considered must now be integrated again to a total surface. If a closed enveloping surface around a body is chosen, the result is:
+Since so far only infinitesimally small surface pieces have been considered must now be integrated again into a total surface. If a closed enveloping surface around a body is chosen, the result is:
 \begin{align}
-\boxed{\int {\rm d}Q = \iint_{\text{closed surf.}} \vec{D} \cdot {\rm d} \vec{A} = \iiint_V \varrho_V {\rm d}\vec{V} = Q}
+\boxed{\int {\rm d}Q = {\rlap{\rlap{\int_A} \int} \: \LARGE \circ} \vec{D} \cdot {\rm d} \vec{A} = \iiint_V \varrho_V {\rm d}\vec{V} = Q}
 \end{align}
+The symbol ${\rlap{\Large \rlap{\int} \int} \, \LARGE \circ}$ denotes that there is a closed surface used for the integration.
 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, 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.
+  * The sum of all sources and sinks equals, in this case, just the water stepping over the edge.
 <WRAP>
@@ Zeile 830: / Zeile 835: @@
 ==== Applications ====
-Are calculated in the course.
+They are calculated in the course.
 === Spherical Capacitor ===
-Spherical capacitors are now rarely found in practical applications. In the {{wp>Van-de-Graaff generator}}, spherical capacitors are used to store the high DC voltages. The earth also represents a spherical capacitor. In this context, the electric field of $100...300~{ \rm V/m}$ in the atmosphere is remarkable, since several hundred volts would have to be present between head and foot (for resolution, see the article [[https://www.en-former.com/en/electricity-from-the-air/|Electricity from the air]] in //Bild der Wissenschaft//).
+Spherical capacitors are now rarely found in practical applications. In the {{wp>Van-de-Graaff generator}}, spherical capacitors are used to store the high DC voltages. The earth also represents a spherical capacitor. In this context, the electric field of $100...300~{ \rm V/m}$ in the atmosphere is remarkable since several hundred volts would have to be present between head and foot (for resolution, see the article [[https://www.en-former.com/en/electricity-from-the-air/|Electricity from the air]] in //Bild der Wissenschaft//).
 === Plate Capacitor ===
@@ Zeile 840: / Zeile 845: @@
 The relation between the $E$-field and the voltage $U$ on the ideal plate capacitor is to be derived from the integral of displacement flux density $\vec{D}$:
 \begin{align*}
-Q = \iint_{\text{closed surf.}} \vec{D} \cdot {\rm d} \vec{A}
+Q = {\rlap{\rlap{\int_A} \int} \: \LARGE \circ} \vec{D} \cdot {\rm d} \vec{A}
 \end{align*}
 <callout icon="fa fa-info" color="grey" title="Outlook">
-The consideration of the displacement flux density also solved a problem, which arose quite for at electric circuits: From considerations about magnetic fields the following quite obvious sounding fact can be led: In a series-connected, switched circuit, the current at each point is the same. But if this series circuit contains a capacitor, no electric current can flow inside! The solution is to understand a temporal change of the displacement flux also as a current, which can be generated a magnetic field (thus vortex). Mathematically, vortices are described via the {{wp>Curl_(mathematics)|Curl}} (in German: //Rotation//) - a multidimensional differential operator. A deeper {{wp>Displacement_current#Generalizing_Ampère's_circuital_law|derivation and solution}} is not considered in the first semester. However, the application will show that the above equation plays a central role in electrical engineering. It is part of the so-called {{wp>Maxwell's equations}}.
+The consideration of the displacement flux density also solved a problem that arose for electric series circuits. We know that the current at each point of a series circuit is the same. But what if there is a capacitor in this series circuit? There is no electric current flowing inside the dielectric material. This problem can be solved considering the connection of magnetic fields and current flow: any magnetic field is based on a moving charge, and any moving charge creates a magnetic field. By this, the solution is that the temporal change of the displacement flux is interpreted as a current, which is generated by a magnetic field (thus a magnetic "vortex" around the circuit). Mathematically, vortices are described via the {{wp>Curl_(mathematics)|Curl}} (in German: //Rotation//) - a multidimensional differential operator. A deeper {{wp>Displacement_current#Generalizing_Ampère's_circuital_law|derivation and solution}} is not considered in the first semester. However, the application will show that the equation above plays a central role in electrical engineering. It is part of the so-called {{wp>Maxwell's equations}}.
 </callout>
@@ Zeile 856: / Zeile 861: @@
 A plate capacitor with a distance of $d = 2 ~{ \rm cm}$ between the plates and with air as dielectric ($\varepsilon_{ \rm r}=1$) gets charged up to $U = 5~{ \rm kV}$.
-In between the plates a thin metal foil with the area $A = 45~{ \rm cm^2}$ is introduced parallel to the plates.
+In between the plates, a thin metal foil with the area $A = 45~{ \rm cm^2}$ is introduced parallel to the plates.
 Calculate the amount of the displaced charges in the thin metal foil.
@@ Zeile 879: / Zeile 884: @@
   - What happens to the electric field and the voltage?
-  - How does the situation change (electric field / voltage), when the source is not disconnected?
+  - How does the situation change (electric field/voltage), when the source is not disconnected?
 <button size="xs" type="link" collapse="Loesung_1_5_2_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_1_5_2_Tipps" collapsed="true">
-  * consider the displacement flux through a surface around a plate
+  * Consider the displacement flux through a surface around a plate
 </collapse>
@@ Zeile 896: / 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 glas 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 glas 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
+  * 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 rearrange 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 916: / Zeile 991: @@
 <panel type="info" title="Task 1.5.4 Spherical capacitor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
-Two two concentric spherical conducting plates setup a spherical capacitor.
+Two concentric spherical conducting plates set up a spherical capacitor.
-The radius of the inner sphere is $r_{ \rm i} = 3~{ \rm mm}$, the inner radius from the outer sphere is $r_{ \rm o} = 9~{ \rm mm}$.
+The radius of the inner sphere is $r_{ \rm i} = 3~{ \rm mm}$, and the inner radius from the outer sphere is $r_{ \rm o} = 9~{ \rm mm}$.
-  - What is the capacity of this capacitor, given that air is used as dielectric?
+  - What is the capacity of this capacitor, given that air is used as a dielectric?
-  - What would be the limit value of the capacity, when the inner radius of the outer sphere is going to infinity ($r_{ \rm o} \rightarrow \infty$)?
+  - What would be the limit value of the capacity when the inner radius of the outer sphere goes to infinity ($r_{ \rm o} \rightarrow \infty$)?
 <button size="xs" type="link" collapse="Loesung_1_5_4_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_1_5_4_Tipps" collapsed="true">
   * What is the displacement flux density of the inner sphere?
   * Out of this derive the strength of the electric field $E$
-  * What ist the general relationship between $U$ and $\vec{E}$? Derive out of this the voltage between the spheres.
+  * What ist the general relationship between $U$ and $\vec{E}$? Derive from this the voltage between the spheres.
 </collapse>
 <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 935: / Zeile 1010: @@
-<panel type="info" title="Task 1.5.5 Applying Gauss's law: electric Field of a line charge"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+<panel type="info" title="Task 1.5.5 Applying Gauss's law: Electric Field of a line charge"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
 {{youtube>NyRjHj2uy6k}}
@@ Zeile 953: / Zeile 1028: @@
 By the end of this section, you will be able to:
-  - know the two field-describing quantities of the electrostatic field,
+  - Know the two field-describing quantities of the electrostatic field,
-  - describe and apply the relationship between these two quantities via the material law,
+  - Describe and apply the relationship between these two quantities via the material law,
-  - understand the effect of an electrostatic field on an insulator,
+  - Understand the effect of an electrostatic field on an insulator,
-  - know what the effect of dielectric polarization does,
+  - Know what the effect of dielectric polarization does,
-  - relate the term dielectric strength to a property of insulators and know what it means
+  - Relate the term dielectric strength to a property of insulators and know what it means
 </callout>
@@ Zeile 964: / Zeile 1039: @@
 First of all, a thought experiment is to be carried out again (see <imgref ImgNr13>):
-  - First a charged plate capacitor in vacuum is assumed, which is separated from the voltage source after charging.
+  - First, a charged plate capacitor in a vacuum is assumed, which is separated from the voltage source after charging.
   - Next, the intermediate region is to be filled with a material.
@@ Zeile 974: / Zeile 1049: @@
 Think about how $E$ and $D$ would change before you unfold the subsection.\\
-Why might which of the two quantities change?
+Why might one of the two quantities change?
-You may have considered what happens to the charge $Q$ on the plates. This charge cannot escape the plates. So $Q = {\rlap{\Large \rlap{\int} \int} \, \LARGE \circ} \vec{D} \cdot {\rm d} \vec{A}$ cannot change. ((The symbol ${\rlap{\Large \rlap{\int} \int} \, \LARGE \circ}$ denotes, that there is a closed surface used for the integration.)) \\
+You may have considered what happens to the charge $Q$ on the plates. This charge cannot escape the plates. So $Q = {\rlap{\Large \rlap{\int_A} \int} \, \LARGE \circ} \enspace \vec{D} \cdot {\rm d} \vec{A}$ cannot change. \\
 Since the fictitious surface around an electrode does not change either, $\vec{D}$ cannot change either.
@@ Zeile 1005: / 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.
+  - Relative permittivity is unitless and indicates how much the electric field decreases in the presence of a 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 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 analysed on the basis of 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 1020: / Zeile 1095: @@
 <tabcaption tab01| relative permittivity>
-^ material             ^ relative permittivity \\ $\varepsilon_{ \rm r}$ for low frequencies ^
+^ material               ^ relative permittivity \\ $\varepsilon_{ \rm r}$ for low frequencies ^
-| air                  | 1.0006  |
+| air                    | $\rm 1.0006$  |
-| paper                | 2       |
+| paper                  | $\rm 2$       |
-| hard paper           | 5       |
+| PE, PP                 | $\rm 2.3$     |
-| glass                | 6...8   |
+| PS                     | $\rm 2.5$     |
-| PE, PP               | 2.3     |
+| hard paper             | $\rm 5$       |
-| PS                   | 2.5     |
+| glass                  | $\rm 6...8$   |
-| water ($20~°{ \rm C}$) | 80      |
+| water ($20~°{ \rm C}$) | $\rm 80$      |
 </tabcaption>
 </WRAP>
@@ Zeile 1039: / Zeile 1114: @@
   * The dielectrics act as insulators. The flow of current is therefore prevented
   * The ability to insulate is dependent on the material.
-  * If a maximum electric field $E_0$ is exceeded, the insulating ability is eliminated
+  * If a maximum electric field $E_0$ is exceeded, the insulating ability is eliminated.
-    * One says: The insulator breaks down. This means that above this electric field a current can flow through the insulator.
+    * 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 1048: / Zeile 1123: @@
 <tabcaption tab02| Dielectric strength>
 ^ Material        ^ Dielectric strength $E_0$ in ${ \rm kV/mm}$ ^
-| air             | 0.1...0.3  |
+| air             | $\rm 0.1...0.3$  |
-| SF6 gas         | 8          |
+| SF6 gas         | $\rm 8$          |
-| vacuum          | 20...30    |
+| insulating oils | $\rm 5...30$     |
-| insulating oils | 5...30     |
+| vacuum          | $\rm 20...30$    |
-| quartz          | 30...40    |
+| quartz          | $\rm 30...40$    |
-| PP, PE          | 50         |
+| PP, PE          | $\rm 50$         |
-| PS              | 100        |
+| PS              | $\rm 100$        |
-| distilled water | 70         |
+| distilled water | $\rm 70$         |
 </tabcaption>
 </WRAP>
@@ Zeile 1076: / Zeile 1151: @@
 By the end of this section, you will be able to:
-  - know what a capacitor is and how capacitance is defined,
+  - Know what a capacitor is and how capacitance is defined,
-  - know the basic equations for calculating a capacitance and be able to apply them,
+  - Know the basic equations for calculating capacitance and be able to apply them,
-  - imagine a plate capacitor and know examples of its use You also have an idea of what a cylindrical or spherical capacitor looks like and what examples of its use there are,
+  - Imagine a plate capacitor and give examples of its use. You also have an idea of what a cylindrical or spherical capacitor looks like and what examples of its use there are,
-  - know the characteristics of the E-field, D-field and electric potential in the three types of capacitors presented here
+  - Know the characteristics of the E-field, D-field, and electric potential in the three types of capacitors presented here
 </callout>
@@ Zeile 1089: / 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 inductance. However, both of these are usually very small.
   * Examples are
     * the electrical component "capacitor",
@@ Zeile 1099: / Zeile 1174: @@
 The capacitance $C$ can be derived as follows:
   - It is known that $U = \int \vec{E} {\rm d} \vec{s} = E \cdot l$ and hence $E= {{U}\over{l}}$ or $D= \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{U}\over{l}}$.
-  - Furthermore, $\iint_{\text{closed surf.}} \vec{D} \cdot {\rm d} \vec{A} = Q$ by the idealized form of the plate capacitor: $Q=D \cdot A$.
+  - Furthermore, ${\rlap{\Large \rlap{\int_A} \int} \, \LARGE \circ} \; \vec{D} \cdot {\rm d} \vec{A} = Q$ by the idealized form of the plate capacitor: $Q=D \cdot A$.
   - Thus, the charge $Q$ is given by: \begin{align*} Q = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{U}\over{l}} \cdot A \end{align*}
   - This means that $Q \sim U$, given the geometry (i.e., $A$ and $d$) and the dielectric ($\varepsilon_{ \rm r} $).
@@ Zeile 1115: / Zeile 1190: @@
   * As larger the area as higher the capacity will be.
-The beckground behind the dielectric constant $\varepsilon_{ \rm r} $ and the field ist explained in the followinf video
+The background behind the dielectric constant $\varepsilon_{ \rm r} $ and the field is explained in the following video
 {{youtube>rkntp3_cZl4}}
@@ Zeile 1121: / 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 1129: / Zeile 1204: @@
 <--
-The <imgref ImgNr171> shows the topology of the electric field inside of an plate capacitor.
+The <imgref ImgNr171> shows the topology of the electric field inside a plate capacitor.
 <WRAP>
-<imgcaption ImgNr171 | Topological situation inside of an plate capacitor>
+<imgcaption ImgNr171 | Topological situation inside a plate capacitor>
 </imgcaption> <WRAP>
 {{url>https://www.falstad.com/vector2de/vector2de.html?f=ChargedPlateDipole&fc=Floor%3A%20field%20magnitude&fl=Overlay%3A%20equipotentials&d=vectors&m=Mouse%20%3D%20Adjust%20Angle&st=20&vd=32&a1=63&a2=16&rx=77&ry=5&rz=107&zm=1.165 600,400 noborder}}
@@ Zeile 1139: / Zeile 1214: @@
 ==== Designs and types of capacitors ====
-To calculate the capacitance of different designs, the definition equations of $\vec{D}$ and $\vec{E}$ are used. This can be viewed in detail e.g. in [[https://www.youtube.com/watch?v=kAXg1xMkR_4&ab_channel=PatrickKaploo|this video]]. \\ Based on the geometry, different equations result (see also <imgref ImgNr17>).
+To calculate the capacitance of different designs, the definition equations of $\vec{D}$ and $\vec{E}$ are used. This can be viewed in detail, e.g., in [[https://www.youtube.com/watch?v=kAXg1xMkR_4&ab_channel=PatrickKaploo|this video]]. \\ Based on the geometry, different equations result (see also <imgref ImgNr17>).
 <WRAP>
@@ Zeile 1149: / Zeile 1224: @@
 ^Shape of the Capacitor^  Parameter                                                                            ^  Equation for the Capacity  ^
 |plate capacitor       | area $A$ of plate \\ distance $l$ between plates                                      | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{A}\over{l}} \end{align*}|
-|cylinder capacitor    |radius of outer conductor $R_{ \rm o}$ \\ radius of inner conductor $R_{ \rm i}$ \\ length $l$       | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot 2\pi {{l}\over{ln \left({{R_{ \rm o}}\over{R_{ \rm i}}}\right)}} \end{align*}|
+|cylinder capacitor    |radius of outer conductor $R_{ \rm o}$ \\ radius of inner conductor $R_{ \rm i}$ \\ length $l$       | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot 2\pi {{l}\over{{\rm ln} \left({{R_{ \rm o}}\over{R_{ \rm i}}}\right)}} \end{align*}|
 |spherical capacitor   |radius of outer spherical conductor $R_{ \rm o}$ \\ radius of inner spherical conductor $R_{ \rm i}$ | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot 4 \pi {{R_{ \rm i} \cdot R_{ \rm o}}\over{R_{ \rm o} - R_{ \rm i}}} \end{align*} |
@@ Zeile 1163: / Zeile 1238: @@
   - **{{wp>variable_capacitor|rotary variable capacitor}}** (also variable capacitor or trim capacitor).
     - A variable capacitor consists of two sets of plates: a fixed set and a movable set (stator and rotor). These represent the two electrodes.
-    - The movable set can be rotated radially into the fixed set. This covers a certain area $A$.
+    - The movable set can be rotated radially into the fixed set. This covers a certain area of $A$.
     - The size of the area is increased by the number of plates. Nevertheless, only small capacities are possible because of the necessary distance.
-    - Air is usually used as the dielectric, occasionally small plastic or ceramic plates are used to increase the dielectric constant.
+    - Air is usually used as the dielectric; occasionally, small plastic or ceramic plates are used to increase the dielectric constant.
   - **{{wp>Ceramic_capacitor#Multi-layer_ceramic_capacitors_(MLCC)|multilayer capacitor}}**
     - 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 dielectric for the disk capacitor. This is positioned as a round disc between two electrodes.
+    - 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.
+    - 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. aluminium, tantalum or niobium.
+    - Different metals can be used as the oxidized electrode, e.g., aluminum, tantalum, or niobium.
     - Because the oxide layer is very thin, a very high capacitance results (depending on the size: up to a few millifarads).
-    - 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.
+    - 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 wound and often has a cross-shaped predetermined breaking point at the top for gas leakage.
+    - 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 cancelled again.
+    - In the event of an 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.
     - The charges are not only stored in the electrode, but - similar to a battery - the charges are transferred into the electrolyte. Due to the polarization of the charges, they surround themselves with a thin (atomic) electrolyte layer. The charges then accumulate at the other electrode.
-    - Supercapacitors can achieve very large capacitance values (up to the kilofarad range), but only have a low maximum voltage
+    - Supercapacitors can achieve very large capacitance values (up to the Kilofarad range), but only have a low maximum voltage
 ~~PAGEBREAK~~ ~~CLEARFIX~~
@@ Zeile 1199: / Zeile 1274: @@
 In <imgref ImgNr17> are shown different capacitors:
-  - Above two SMD capacitors
+  - The above two SMD capacitors
     - On the left a $100~{ \rm µF}$ electrolytic capacitor
     - On the right a $100~{ \rm nF}$ MLCC in the commonly used {{wp>Surface-mount_technology#Packages}} 0603 ($1.6~{ \rm mm}$ x $0.8~{ \rm mm}$)
   - below different THT capacitors (__T__hrough __H__ole __T__echnology)
-    - a big electrolytic capacitor with $10~{ \rm mF}$ in blue, the positive terminal is marked with $+$
+    - A big electrolytic capacitor with $10~{ \rm mF}$ in blue, the positive terminal is marked with $+$
-    - in the second row is a Kerko with $33~{ \rm pF}$ and two Folkos with $1,5~{ \rm µF}$ each
+    - In the second row is a Kerko with $33~{ \rm pF}$ and two Folkos with $1,5~{ \rm µF}$ each
-    - in the bottom row you can see a trim capacitor with about $30~{ \rm pF}$ and a tantalum electrolytic capacitor and another electrolytic capacitor
+    - In the bottom row, you can see a trim capacitor with about $30~{ \rm pF}$ and a tantalum electrolytic capacitor and another electrolytic capacitor
 Various conventions have been established for designating the capacitance value of a capacitor [[https://www.eit.edu.au/resources/different-types-of-capacitors/|various conventions]].
@@ Zeile 1216: / 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.
+  - 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.
-  - The capacitance value often varies by more than $\pm 10~{ \rm \%}$. I.e. a calculation accurate to several decimal places is rarely necessary/possible.
+  - The capacitance value often varies by more than $\pm 10~{ \rm \%}$. I.e., a calculation accurate to several decimal places is rarely necessary/possible.
   - The charge current seems to be able to flow through the capacitor because the charges added to one side induce correspondingly opposite charges on the other side.
@@ Zeile 1236: / Zeile 1311: @@
 By the end of this section, you will be able to:
-  - recognise a series connection of capacitors and distinguish it from a parallel connection,
+  - Recognize a series connection of capacitors and distinguish it from a parallel connection,
-  - calculate the resulting total capacitance of a series or parallel circuit,
+  - Calculate the resulting total capacitance of a series or parallel circuit,
-  - know how the total charge is distributed among the individual capacitors in a parallel circuit,
+  - Know how the total charge is distributed among the individual capacitors in a parallel circuit,
-  - determine the voltage across a single capacitor in a series circuit.
+  - Determine the voltage across a single capacitor in a series circuit.
 </callout>
@@ Zeile 1251: / Zeile 1326: @@
 \end{align*}
-Furthermore, after charging, a voltage is formed across the series circuit which corresponds to the source voltage $U_q$. This results from the addition of the partial voltages across the individual capacitors.
+Furthermore, after charging, a voltage is formed across the series circuit, which corresponds to the source voltage $U_q$. This results from the addition of partial voltages across the individual capacitors.
 \begin{align*}
 U_q = U_1 + U_2 + ... + U_n = \sum_{k=1}^n U_k
@@ Zeile 1282: / Zeile 1357: @@
 \end{align*}
-In the simulation below, besides the parallel connected capacitors $C_1$, $C_2$,$C_3$, an ideal voltage source $U_q$, a resistor $R$, a switch $S$ and a lamp are installed.
+In the simulation below, besides the parallel connected capacitors $C_1$, $C_2$,$C_3$, an ideal voltage source $U_q$, a resistor $R$, a switch $S$, and a lamp are installed.
   * The switch $S$ allows the voltage source to charge the capacitors.
   * The resistor $R$ is necessary because the simulation cannot represent instantaneous charging. The resistor limits the charging current to a maximum value. \\ This leads to the DC circuit transients, explained in the  [[electrical_engineering_1:dc_circuit_transients#time_course_of_the_charging_and_discharging_process|last semester]].
@@ Zeile 1322: / Zeile 1397: @@
 \end{align*}
 \begin{align*}
-\boxed{ U_k = const}
+\boxed{ U_k = {\rm const.}}
 \end{align*}
 </WRAP>
@@ Zeile 1335: / Zeile 1410: @@
 \end{align*}
-In the simulation below, again besides the parallel connected capacitors $C_1$, $C_2$,$C_3$, an ideal voltage source $U_q$, a resistor $R$, a switch $S$ and a lamp are installed.
+In the simulation below, again, besides the parallel connected capacitors $C_1$, $C_2$,$C_3$, an ideal voltage source $U_q$, a resistor $R$, a switch $S$, and a lamp are installed.
 This derivation is also well explained, for example, in [[https://www.youtube.com/watch?v=fH-9pUeEpZU|this video]].
@@ Zeile 1359: / Zeile 1434: @@
 By the end of this section, you will be able to:
-  - recognize the different layering of dielectrics and distinguish between a perpendicular and a lateral layering
+  - Recognize the different layering of dielectrics and distinguish between a normal (perpendicular) and a tangential (lateral) layering
-  - know which quantity remains constant in the case of perpendicular layers
+  - Know which quantity remains constant for the different layerings
-  - know the constant quantity for a lateral layers as well
+  - Be familiar with the equivalent circuits for normal and tangential layering
-  - be familiar with the equivalent circuits for perpendicular and lateral layering
+  - Calculate the total capacitance of a capacitor with layering
-  - calculate the total capacitance of a capacitor with layering
+  - Know the law of refraction at interfaces for the field lines in the electrostatic field.
-  - know the law of refraction at interfaces for the field lines in the electrostatic field.
 </callout>
-Up to now was assumed only one dielectric resp. only vacuum within capacitor. Now is looked at more detailed, 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. Different variants can be distinguished (<imgref ImgNr18>):
+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.
-  - **perpendicular layering**: There are different dielectrics perpendicular to the field lines. \\ Thus, the boundary layers are parallel to the capacitor plates.
+It is possible to tell the following variations apart  (<imgref ImgNr18>). \\
-  - **lateral layering**: There are different dielectrics parallel to the field lines. \\ So the boundary layers are perpendicular to the capacitor plates.
-  - **arbitrary configuration**: The boundary layers are neither parallel nor perpendicular to the capacitor plates.
+  - **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.  \\ \\
+  - **arbitrary configuration**: \\ The boundary layers are neither parallel nor perpendicular to the capacitor plates.
 <WRAP>
@@ Zeile 1383: / Zeile 1459: @@
 ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== Lateral Configuration ====
+==== Dielectrics in Series ====
-First, the situation is considered that the boundary layers are parallel to the electrode surfaces. A voltage $U$ is applied to the structure from the outside. \\
+First, the situation is considered where the boundary layers are parallel to the electrode surfaces. A voltage $U$ is applied to the structure from the outside. \\
 <WRAP 40em>
-<imgcaption ImgNr19 | lateral layered capacitor>
+<imgcaption ImgNr19 | Dielectrics in Series - Layers parallel to Capacitor Plates>
 </imgcaption>
 {{drawio>crosslayeredcapacitor.svg}}
 </WRAP>
-The layering is now parallel to equipotential surfaces. In particular, the boundary layers are then also equipotential surfaces. \\
+The layering is here parallel to the equipotential surfaces of the plate capacitor. In particular, the boundary layers are then also equipotential surfaces. \\
 The boundary layers can be replaced by an infinitesimally thin conductor layer (metal foil). The voltage $U$ can then be divided into several partial areas:
 \begin{align*}
-U = \int \limits_{total \, inner \\ volume} \! \! \vec{E} \cdot {\rm d} \vec{s} = E_1 \cdot d_1 + E_2 \cdot d_2 + E_3 \cdot d_3
+U = \int \limits_{\rm path \, inside \\ the \, capacitor} \! \! \vec{E} \cdot {\rm d} \vec{s} = E_1 \cdot d_1 + E_2 \cdot d_2 + E_3 \cdot d_3
 \tag{1.9.1}
 \end{align*}
@@ Zeile 1437: / Zeile 1513: @@
 <WRAP>
-<imgcaption ImgNr431 | Simulation of lateral layering>
+<imgcaption ImgNr431 | Simulation of Dielectrics in series>
 </imgcaption> <WRAP>
 {{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+64+32+0+0+159+0.078125+284%0Ab+0+3+4+-93+15+78+27+0%0Ab+0+2+-1000+-156+49+96+66+0%0Ab+0+2+1000+-153+-3+99+14+0%0Ab+0+3+1.5+-92+28+79+40+0%0A 600,600 noborder}}
 </WRAP></WRAP>
-The situation can also be transferred to a coaxial structure of a cylindrical capacitor or concentric structure of spherical capacitors.
+The situation can also be transferred to a coaxial structure of a cylindrical capacitor or the concentric structure of spherical capacitors.
 <callout icon="fa fa-exclamation" color="red" title="Note:">
-Lateral configuration results in:
+Conclusions:
-  - A perpendicular layering can be considered as a series connection of partial capacitors with respective thicknesses $d_k$ and dielectric constant $\varepsilon_{{ \rm r}k}$.
+  - The layering parallel to the capacitor plates can be considered as a series connection of partial capacitors with respective thicknesses $d_k$ and dielectric constants $\varepsilon_{{ \rm r}k}$.
-  - The flux density is constant in the capacitor
+  - The flux density for dielectrics in series is constant everywhere in the capacitor
-  - Considering the fields __along the field line__ - that is, perpendicular to the interface, or the normal components $E_{ \rm n}$ and $D_{ \rm n}$ of the fields - the following holds:
+  - We also found some results for the $E$ and $D$ fields __along the field line__. These parts of the fields - which are perpendicular to the capacitor plates - are the normal components $E_{ \rm n}$ and $D_{ \rm n}$.
     - The normal component of the electric field $E_{ \rm n}$ changes abruptly at the interface.
     - The normal component of the flux density $D_{ \rm n}$ is continuous at the interface: $D_{ \rm n1} = D_{ \rm n2}$
@@ Zeile 1455: / Zeile 1531: @@
 ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== Perpendicular Configuration ====
+==== Dielectrics in Parallel ====
-Now the boundary layers should be perpendicular to the electrode surfaces. Again a voltage $U$ is applied to the structure from the outside.
+Now the boundary layers should be perpendicular to the equipotential surfaces of the plate capacitor. Again, a voltage $U$ is applied to the structure from the outside.
 <WRAP 40em>
-<imgcaption ImgNr20 | longitudinal layered capacitor>
+<imgcaption ImgNr20 | Dielectrics in parallel - Layers perpendicular to Capacitor Plates>
 </imgcaption>
 {{drawio>longitudinallayeredcapacitor.svg}}
 </WRAP>
-The layering is now perpendicular to equipotential surfaces. However, the same voltage is applied to each dielectric. Thus it is valid:
+The layering is now perpendicular to equipotential surfaces. However, the same voltage is applied to each dielectric. Thus, it is valid:
 \begin{align*}
-U = \int \limits_{ \rm total \, inner \\ volume} \! \! \vec{E} \cdot {\rm d} \vec{s} = E_1 \cdot d = E_2 \cdot d = E_3 \cdot d
+U = \int \limits_{\rm path \, inside \\ the \, capacitor} \! \! \vec{E} \cdot {\rm d} \vec{s} = E_1 \cdot d = E_2 \cdot d = E_3 \cdot d
 \end{align*}
@@ Zeile 1481: / Zeile 1557: @@
 Since the electric flux density is just equal to the local surface charge density, the charge will no longer be uniformly distributed over the electrodes. \\
 Where a stronger polarization is possible, the $E$-field is damped in the dielectric. For a constant $E$-field, more charges must accumulate there. \\
-Concretely, more charges accumulate just around the dielectric constant $\varepsilon_{{ \rm r}k}$.
+Therefore, as more charges accumulate as higher the dielectric constant $\varepsilon_{{ \rm r}k}$.
@@ Zeile 1490: / Zeile 1566: @@
 </WRAP></WRAP>
-This situation can also be transferred to a coaxial structure of a cylindrical capacitor or concentric structure of spherical capacitors.
+This situation can also be transferred to a coaxial structure of a cylindrical capacitor or the concentric structure of spherical capacitors.
 <callout icon="fa fa-exclamation" color="red" title="Note:">
-In the case of perpendicular configuration, the result is:
-  - A perpendicular configuration can be viewed as a parallel connection of partial capacitors with respective areas $A_k$ and dielectric constant $\varepsilon_{{ \rm r}k}$.
+Conclusions:
-  - The electric field in the capacitor is constant.
+  - The layering perpendicular to the capacitor plates can be considered as a parallel connection of partial capacitors with respective areas $A_k$ and dielectric constant $\varepsilon_{{ \rm r}k}$.
-  - Considering the fields __transverse to the field lines__ - that is, perpendicular to the interface, or the parallel components $E_{ \rm p}$ and $D_{ \rm p}$ of the fields - the following holds:
+  - The electric field for dielectrics in parallel is constant everywhere in the capacitor.
-    - The parallel components of the flux density $D_{ \rm p}$ changes abruptly at the interface.
+  - We also found some results for the $E$ and $D$ fields __perpendicular to the field line__. These parts of the fields - which are parallel to the capacitor plates - are the tangential components $E_{ \rm t}$ and $D_{ \rm t}$.
-    - The parallel components of the electric field $E_{ \rm p}$ is continuous at the interface: $E_{\rm p1} = E_{\rm p2}$
+    - The tangential component of the flux density $D_{ \rm t}$ changes abruptly at the interface.
+    - The tangential component of the electric field $E_{ \rm t}$ is continuous at the interface: $E_{ \rm t1} = E_{ \rm t2}$
 </callout>
 ~~PAGEBREAK~~ ~~CLEARFIX~~
@@ Zeile 1508: / Zeile 1586: @@
   * Electric field $\vec{E}$:
     * The normal component $E_{ \rm n}$ is discontinuous at the interface: $\varepsilon_{ \rm r1} \cdot E_{ \rm n1} = \varepsilon_{ \rm r2} \cdot E_{ \rm n2}$
-    * The parallel component $E_{ \rm p}$ is continuous at the interface: $ E_{ \rm p1} = E_{ \rm p2}$
+    * The tangential component $E_{ \rm t}$ is continuous at the interface: $ E_{ \rm t1} = E_{ \rm t2}$
   * Electric displacement flux density $\vec{D}$:
     * The normal component $D_{ \rm n}$ is continuous at the interface: $ D_{ \rm n1} = D_{ \rm n2}$
-    * The parallel component $D_{ \rm p}$ is discontinuous at the interface: $  {{1}\over \Large{\varepsilon_{ \rm r1}}}\cdot D_{ \rm p1} =  {{1}\over \Large{\varepsilon_{ \rm r2}}} \cdot D_{ \rm p1} $
+    * The tangential component $D_{ \rm t}$ is discontinuous at the interface: $  {{1}\over \Large{\varepsilon_{ \rm r1}}}\cdot D_{ \rm t1} =  {{1}\over \Large{\varepsilon_{ \rm r2}}} \cdot D_{ \rm t2} $
 <WRAP 30em>
@@ Zeile 1522: / Zeile 1600: @@
 <imgcaption ImgNr731 | arbitrary configuration>
 </imgcaption> <WRAP>
-{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+64+32+0+6+575+0.078125+393%0Ab+0+2+-5+-156+49+96+66+0%0Ab+0+2+5+-153+-4+99+13+0%0Ab+0+3+4+31+37+105+43+0%0Ab+0+3+4+25+43+99+49+0%0Ab+0+3+4+43+25+117+31+0%0Ab+0+3+4+37+31+111+37+0%0Ab+0+3+4+49+19+121+25+0%0Ab+0+3+4+54+13+130+19+0%0Ab+0+3+4+47+21+51+26+0%0Ab+0+3+4+45+23+49+26+0%0Ab+0+3+4+35+33+39+38+0%0Ab+0+3+4+41+27+45+32+0%0Ab+0+3+4+23+45+27+49+0%0Ab+0+3+4+53+15+57+20+0%0Ab+0+3+4+51+17+55+21+0%0Ab+0+3+4+39+29+43+32+0%0Ab+0+3+4+33+35+37+38+0%0Ab+0+3+4+29+39+33+44+0%0Ab+0+3+4+27+41+31+44+0%0Ab+0+3+4+21+47+24+48+0%0Ab+0+3+4+50+18+51+20+0%0Ab+0+3+4+22+46+24+48+0%0Ab+0+3+4+52+16+54+18+0%0Ab+0+3+4+40+28+42+30+0%0Ab+0+3+4+38+30+40+32+0%0Ab+0+3+4+42+26+44+28+0%0Ab+0+3+4+46+22+48+24+0%0Ab+0+3+4+44+24+46+26+0%0Ab+0+3+4+48+20+50+22+0%0Ab+0+3+4+28+40+30+42+0%0Ab+0+3+4+32+36+34+38+0%0Ab+0+3+4+30+38+32+40+0%0Ab+0+3+4+36+32+38+34+0%0Ab+0+3+4+34+34+36+36+0%0Ab+0+3+4+26+42+28+44+0%0Ab+0+3+4+24+44+26+46+0%0Ab+0+3+4+20+48+23+49+0%0A 600,600 noborder}}
+{{url>https://www.falstad.com/emstatic/EMStatic.html?rol=$+1+64+32+0+10+575+0.078125+393%0Ab+0+2+-5+-156+49+96+66+0%0Ab+0+2+5+-153+-4+99+13+0%0Ab+0+3+4+-20+14+91+20+0%0Ab+0+3+4+1+20+75+21+0%0Ab+0+3+4+5+20+75+22+0%0Ab+0+3+4+9+20+75+23+0%0Ab+0+3+4+13+20+75+24+0%0Ab+0+3+4+17+20+75+25+0%0Ab+0+3+4+21+20+75+26+0%0Ab+0+3+4+25+20+75+27+0%0Ab+0+3+4+29+20+75+28+0%0Ab+0+3+4+33+20+75+29+0%0Ab+0+3+4+37+20+75+30+0%0Ab+0+3+4+41+20+75+31+0%0Ab+0+3+4+45+20+75+32+0%0Ab+0+3+4+49+20+75+33+0%0Ab+0+3+4+53+20+75+34+0%0Ab+0+3+4+57+20+75+35+0%0Ab+0+3+4+61+20+75+36+0%0Ab+0+3+4+65+20+75+37+0%0Ab+0+3+4+69+20+75+38+0%0Ab+0+3+4+73+20+75+39+0%0Ab+0+3+4+77+20+75+40+0%0Ab+0+3+4+81+20+75+41+0%0Ab+0+3+4+85+20+75+42+0%0Ab+0+3+4+89+20+75+43+0%0A 600,600 noborder}}
 </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 1532: / Zeile 1610: @@
 \end{align*}
-The formula obtained represents the law of refraction of the field line at interfaces. There is also a hint that for electromagnetic waves (like visible light) the refractive index might depend on the dielectric constant. In fact, this is the case. However, in the calculation presented here, electrostatic fields were assumed. In the case of electromagnetic waves, the distribution of energy between the two fields must be taken into account. This is not considered in detail in this course but is explained shortly in the following video.
+The formula obtained represents the law of refraction of the field line at interfaces. There is also a hint that for electromagnetic waves (like visible light) the refractive index might depend on the dielectric constant. This is the case. However, in the calculation presented here, electrostatic fields were assumed. In the case of electromagnetic waves, the distribution of energy between the two fields must be taken into account. This is not considered in detail in this course but is explained shortly in task 1.9.1.
-{{youtube>ATXnPRXXDi4&start=528}}
-~~PAGEBREAK~~ ~~CLEARFIX~~
@@ Zeile 1549: / 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>
@@ Zeile 1559: / Zeile 1650: @@
 Two parallel capacitor plates face each other with a distance $d_{ \rm K} = 10~{ \rm mm}$. A voltage of $U = 3'000~{ \rm V}$ is applied to the capacitor.
-Parallel to the capacitor plates there is a glass plate ($\varepsilon_{ \rm r,G}=8$) with a thickness $d_{ \rm G} = 3~{ \rm mm}$ in the capacitor.
+Parallel to the capacitor plates, there is a glass plate ($\varepsilon_{ \rm r, G}=8$) with a thickness $d_{ \rm G} = 3~{ \rm mm}$ in the capacitor.
   - Calculate the partial voltages $U_{ \rm G}$ in the glass and $U_{ \rm A}$ in the air gap.
-  - What is the maximum thickness of the glass pane if the electric field $E_{ \rm 0,G} =12 ~{ \rm kV/cm}$ must not exceed.
+  - What is the maximum thickness of the glass pane if the electric field $E_{ \rm 0, G} =12 ~{ \rm kV/cm}$ must not exceed?
 ~~PAGEBREAK~~ ~~CLEARFIX~~
@@ Zeile 1582: / Zeile 1673: @@
 ====== Further links ======
-  * [[https://lx3.mint-kolleg.kit.edu/onlinekursphysik/html/1.4.1/modstart.html|Online Bridge Course Physics KIT]]: This semi-interactive course contains some of the information from my course. Furthermore, videos, exercises and more can be found there
+  * [[https://obkp.mint-kolleg.kit.edu/#LABEL_ELEKTROMAGNETISMUS_LADUNG|Online Bridge Course Physics KIT]]: This semi-interactive course contains some of the information from my course. Furthermore, videos, exercises, and more can be found here.
 ====== additional Links ======
@@ Zeile 1593: / Zeile 1684: @@
-A really great introduction in electric and magnetic fields (but a bit too deep for this course) can be found in the [[https://www.youtube.com/watch?v=rtlJoXxlSFE&list=PLyQSN7X0ro2314mKyUiOILaOC2hk6Pc3j&ab_channel=LecturesbyWalterLewin.Theywillmakeyou%E2%99%A5Physics.|physics lecture of Walter Lewin]]
+A great introduction to electric and magnetic fields (but a bit too deep for this course) can be found in the [[https://www.youtube.com/watch?v=rtlJoXxlSFE&list=PLyQSN7X0ro2314mKyUiOILaOC2hk6Pc3j&ab_channel=LecturesbyWalterLewin.Theywillmakeyou%E2%99%A5Physics.|physics lecture of Walter Lewin]]
 examples: