Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_2:the_electrostatic_field [2022/03/21 01:55] – tfischer | electrical_engineering_2:the_electrostatic_field [2025/03/20 10:56] (aktuell) – mexleadmin | ||
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| Zeile 1: | Zeile 1: | ||
| - | ====== 1. The Electrostatic Field ====== | + | ====== 1 The Electrostatic Field ====== |
| < | < | ||
| - | For this chapter the online | + | The online |
| * Chapter [[https:// | * Chapter [[https:// | ||
| * Chapter [[https:// | * Chapter [[https:// | ||
| Zeile 9: | Zeile 9: | ||
| </ | </ | ||
| - | From everyday | + | Everyday |
| < | < | ||
| < | < | ||
| - | </ | + | </ |
| {{url> | {{url> | ||
| </ | </ | ||
| - | In the first chapter of the last semester we had already considered the charge as the central quantity of electricity and understood | + | We had already considered the charge as the central quantity of electricity |
| - | First, | + | First, |
| - | - **{{wp> | + | - **{{wp> |
| - | - **{{wp> | + | - **{{wp> |
| - | - **{{wp> | + | - **{{wp> |
| - | At this chapter, only electrostatics are considered. The magnetic fields are therefore | + | Only electrostatics is discussed in this chapter. |
| - | Also electrodynamics is not considered | + | Furthermore, |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 33: | Zeile 32: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | 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. | ||
| - | - be able to sketch the field lines of electric fields. | + | - sketch the field lines of electric fields. |
| - | - be able to represent the field vectors in a sketch when given several charges. | + | - represent the field vectors in a sketch when given several charges. |
| - | - be able to 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. |
| - | - Be able to determine the force on a charge in an electrostatic field by applying Coulomb' | + | - determine the force on a charge in an electrostatic field by applying Coulomb' |
| - the force vector in coordinate representation | - the force vector in coordinate representation | ||
| - the magnitude of the force vector | - the magnitude of the force vector | ||
| Zeile 52: | Zeile 51: | ||
| <panel type=" | <panel type=" | ||
| - | The simulation in <imgref ImgNr02> | + | The simulation in was already |
| - | In the simulation, please position | + | Place a negative charge $Q$ in the middle |
| - | For impact analysis, a sample charge $q$ is placed | + | A sample charge $q$ is placed |
| < | < | ||
| - | </ | + | </ |
| {{url> | {{url> | ||
| - | Take a charge ($+1nC$) and position it. \\ Measure the field across a sample charge (a sensor). | + | Take a charge ($+1~{ \rm nC}$) and position it. \\ Measure the field across a sample charge (a sensor). |
| </ | </ | ||
| - | The concept of a field shall now be briefly | + | The concept of a field will now be briefly |
| - | - The introduction of the field separates | + | - The introduction of the field distinguishes |
| - | - 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 | + | - This distinction |
| - | - As with physical quantities, there are different-dimensional fields: | + | -There are different-dimensional fields, just like physical quantities: |
| - | - In a **scalar field**, | + | - 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 |
| - | - 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' |
| - electric field $\vec{E}(\vec{x})$ | - electric field $\vec{E}(\vec{x})$ | ||
| - magnetic field $\vec{H}(\vec{x})$ | - 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 | + | - 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 |
| - | Vector fields | + | Vector fields |
| - | - Effects along spatial axes $x$,$y$ and $z$ (cartesian | + | - Effects along spatial axes $x$, $y$ and $z$ (Cartesian |
| - Effect in magnitude and direction vector (polar coordinate system) | - Effect in magnitude and direction vector (polar coordinate system) | ||
| Zeile 92: | Zeile 91: | ||
| ==== The Electric Field ==== | ==== The Electric Field ==== | ||
| - | Thus, to determine the electric field, a measure | + | To determine the electric field, a measurement |
| \begin{align*} | \begin{align*} | ||
| Zeile 98: | Zeile 97: | ||
| \end{align*} | \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 |
| \begin{align*} | \begin{align*} | ||
| Zeile 105: | Zeile 104: | ||
| \end{align*} | \end{align*} | ||
| - | The left part is therefore | + | 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> | <WRAP centeralign> | ||
| - | $E = {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \quad$ with $[E]={{[F]}\over{[q]}}=1 {{N}\over{As}}=1 {{N\cdot m}\over{As \cdot m}} = 1 {{V \cdot A \cdot s}\over{As \cdot m}} = 1 {{V}\over{m}}$ | + | $E = {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \quad$ with $[E]={{[F]}\over{[q]}}=1 |
| </ | </ | ||
| Zeile 119: | Zeile 118: | ||
| <callout icon=" | <callout icon=" | ||
| - | - 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. | - The sampled charge here is always a point charge. | ||
| </ | </ | ||
| Zeile 125: | Zeile 124: | ||
| <callout icon=" | <callout icon=" | ||
| - | A charge $Q$ generates | + | At a measuring point $P$, a charge $Q$ produces |
| - the magnitude $|\vec{E}|=\Bigl| {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \Bigl| $ and | - the magnitude $|\vec{E}|=\Bigl| {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \Bigl| $ and | ||
| - | - the direction of the force $\vec{F_C}$ | + | - the direction of the force $\vec{F_C}$ |
| - | Be arware, that in English courses and literature $\vec{E}$ is simply | + | Be aware, that in English courses and literature $\vec{E}, $ is simply |
| </ | </ | ||
| The direction of the electric field is switchable in <imgref ImgNr02> via the " | The direction of the electric field is switchable in <imgref ImgNr02> via the " | ||
| - | The electric field can also be viewed again in [[https:// | + | The electric field can also be viewed again in [[https:// |
| ==== Electric Field Lines ==== | ==== Electric Field Lines ==== | ||
| - | Electric field lines result | + | Electric field lines result |
| 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>: | + | The superposition is sketched in <imgref ImgNr032>: |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | 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 radialy | + | 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 |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 163: | Zeile 162: | ||
| * ... delete components with a right click onto it and '' | * ... delete components with a right click onto it and '' | ||
| * Where is the density of the field lines higher? | * Where is the density of the field lines higher? | ||
| - | * How does the field between two positive charges look like? How between two different charges? | + | * How does the field between two positive charges look like? How does it look between two different charges? |
| < | < | ||
| Zeile 174: | Zeile 173: | ||
| - The electrostatic field is a source field. This means there are sources and sinks. | - The electrostatic field is a source field. This means there are sources and sinks. | ||
| - From the field line diagrams, the following can be obtained: | - From the field line diagrams, the following can be obtained: | ||
| - | - Direction of the field ($\hat{=}$ | + | - Direction of the field ($\hat{=}$ |
| - Magnitude of the field ($\hat{=}$ number of field lines per unit area). | - Magnitude of the field ($\hat{=}$ number of field lines per unit area). | ||
| - The magnitude of the field along a field line is usually __not__ constant. | - The magnitude of the field along a field line is usually __not__ constant. | ||
| Zeile 183: | Zeile 182: | ||
| * The electric field lines have a beginning (at a positive charge) and an end (at a negative charge). | * The electric field lines have a beginning (at a positive charge) and an end (at a negative charge). | ||
| * The direction of the field lines represents the direction of a force onto a positive test charge. | * The direction of the field lines represents the direction of a force onto a positive test charge. | ||
| - | * There are no closed field lines in electrostatic fields. The reason for this can be explained considering the energy of the moved particle (see later subchapters). | + | * There are no closed field lines in electrostatic fields. The reason for this can be explained |
| * 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. | * 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; | * The field lines are always perpendicular to conducting surfaces. This is also based on energy considerations; | ||
| - | * 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. |
| </ | </ | ||
| Zeile 193: | Zeile 192: | ||
| ==== Tasks ==== | ==== Tasks ==== | ||
| - | <panel type=" | + | <panel type=" |
| - | {{youtube> | + | {{youtube> |
| - | ([[https:// | + | |
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| Sketch the field line plot for the charge configurations given in <imgref ImgNr04> | Sketch the field line plot for the charge configurations given in <imgref ImgNr04> | ||
| Zeile 212: | Zeile 210: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 219: | Zeile 217: | ||
| - | ===== 1.2 Electric | + | ===== 1.2 Electric |
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | By the end of this section, you will be able to: |
| - | - be able to determine the direction of the forces using given charges. | + | - determine the direction of the forces using given charges. |
| - | - be able to represent the acting force vectors in a sketch. | + | - represent the acting force vectors in a sketch. |
| - | - be able to determine a force vector by superimposing several force vectors using vector calculus. | + | - determine a force vector by superimposing several force vectors using vector calculus. |
| - | - be able to state the following quantities for a force vector: | + | - state the following quantities for a force vector: |
| - | - Force vector in coordinate representation | + | - the force vector in coordinate representation |
| - | - magnitude of the force vector | + | - the magnitude of the force vector |
| - | - Angle of the force vector | + | - the angle of the force vector |
| </ | </ | ||
| - | The electric charge and Coulomb force has already been described | + | The electric charge and Coulomb force have already been described last semester. However, some points are to be caught up here to it. |
| ==== Direction of the Coulomb force and Superposition ==== | ==== 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 donw by giving the correct labeling the subsript | + | 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 |
| - | Furthermore, | + | Furthermore, |
| 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: $\vec{F_3}= \vec{F_{31}}+\vec{F_{32}}$. | 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 249: | Zeile 247: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 255: | Zeile 253: | ||
| ==== Geometric Distribution of Charges ==== | ==== 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. | * 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 | + | * 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, |
| - | * It is spoken of an **area charge** when the charge distributed over an area. \\ Examples of this are the floor or a plate of a capacitor. Again, an extended charged object can be considered | + | * It is spoken of as an **area charge** when the charge |
| - | * 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, | + | * 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, |
| In the following, area charges and their interactions will be considered. | In the following, area charges and their interactions will be considered. | ||
| - | < | + | < |
| ==== Types of Fields depending on the Charge Distribution ==== | ==== Types of Fields depending on the Charge Distribution ==== | ||
| Zeile 270: | Zeile 268: | ||
| <WRAP group>< | <WRAP group>< | ||
| - | 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> | + | 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> | ||
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| </ | </ | ||
| - | For **inhomogeneous fields**, the magnitude and/or direction of the electic | + | For **inhomogeneous fields**, the magnitude and/or direction of the electric |
| + | This is the rule in real systems, even the field of a point charge is inhomogeneous (<imgref ImgNr08> | ||
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 300: | Zeile 300: | ||
| {{youtube> | {{youtube> | ||
| </ | </ | ||
| + | |||
| + | {{page> | ||
| + | {{page> | ||
| + | {{page> | ||
| + | |||
| =====1.3 Work and Potential ===== | =====1.3 Work and Potential ===== | ||
| Zeile 305: | Zeile 310: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | 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. | ||
| - | - be able to describe when work has to be performed and when it does not in the situation of a moving | + | - 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 recognise | + | - know what a potential difference is and recognize |
| - | - be able to determine a potential curve for a given arrangement. | + | - determine a potential curve for a given arrangement. |
| </ | </ | ||
| Zeile 321: | Zeile 326: | ||
| In the following, only a few brief illustrations of the concepts are given. \\ | In the following, only a few brief illustrations of the concepts are given. \\ | ||
| - | A detailed explanation can be found in the online | + | A detailed explanation can be found in the online |
| In particular, this applies to: | In particular, this applies to: | ||
| * Chapter " | * Chapter " | ||
| </ | </ | ||
| - | ==== Energy required to Displace a Charge in the electic | + | ==== Energy required to Displace a Charge in the electric |
| 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 extended conductor) it would be accelerated along field lines. Thus its kinetic energy increases. Because the whole system of plates (for field generation) and charge however does not change its energetic state - thermodynamically the system is closed. From this follows: if the kinetic energy increases, the potential energy must decrease. | + | If this charge would be free movable (e.g. electron in a vacuum or an extended conductor) it would be accelerated along field lines. Thus its kinetic energy increases. Because the whole system of plates (for field generation) and charge however does not change its energetic state - thermodynamically the system is closed. From this follows: if the kinetic energy increases, the potential energy must decrease. |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| < | < | ||
| - | It is known from mechanics, that the work done (thus energy needed) is defined by force one needs to move along a path. \\ | + | It is known from mechanics, that the work done (thus 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 $A$ to $B$ (see <imgref ImgNr09> | + | 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*} | \begin{align*} | ||
| - | W_{AB} = F_C \cdot s | + | W_{ \rm AB} = F_C \cdot s |
| \end{align*} | \end{align*} | ||
| - | For a motion perpendicular to the field lines (i.e. from $A$ to $C$) no work is needed - so $W_{AC}=0$ 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 a weight in the gravitational field at the same height. Or more illustrative: | + | 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: |
| - | 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 $\vec{s}$: | + | 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*} | \begin{align*} | ||
| - | W_{AB} = F_C \cdot s \cdot cos(\alpha) = \vec{F_C}\cdot \vec{s} | + | W_{\rm AB} = F_C \cdot s \cdot \cos(\alpha) = \vec{F_C}\cdot \vec{s} |
| \end{align*} | \end{align*} | ||
| - | The work $W_{AB}$ here describes the energy difference experienced by the charge $q$. \\ | + | 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} | \begin{align} | ||
| - | U_{AB} = {{W_{AB}}\over{q}} = {{F_C \cdot s}\over{q}} = {{E \cdot q \cdot s}\over{q}} = E \cdot s_{AB} | + | U_{ \rm AB} = {{W_{ \rm AB}}\over{q}} = {{F_C \cdot s}\over{q}} = {{E \cdot q \cdot s}\over{q}} = E \cdot s_{ \rm AB} |
| \end{align} | \end{align} | ||
| <callout icon=" | <callout icon=" | ||
| - | - The voltage $U_{AB}$ represents the work $W$ per charge needed to move a probe charge from point $A$ to point {B} in an $E$-field. | + | - The voltage $U_{ \rm AB}$ represents the work $W$ per charge needed to move a probe charge from point $A$ to point {B} in an $E$-field. |
| - | - The voltage is measured in Volts: $[U] = 1V$ | + | - The voltage is measured in Volts: $[U] = 1~{ \rm V}$ |
| </ | </ | ||
| - | To obtain a general approach to __in__homogeneous fields and arbitrary paths $s_{AB}$, it helps (as is so often the case) to decompose the problem into small parts. In the concrete case, these are small path segments on which the field can be assumed to be homogeneous. These are to be assumed to be infinitesimally small in the extreme case (i.e., from $s$ to $\Delta s$ to $ds$): | + | To obtain a general approach to __in__homogeneous fields and arbitrary paths $s_{ \rm AB}$, it helps (as is so often the case) to decompose the problem into small parts. |
| + | In the concrete case, these are small path segments on which the field can be assumed to be homogeneous. | ||
| + | These are to be assumed to be infinitesimally small in the extreme case (i.e., from $s$ to $\Delta s$ to $ds$): | ||
| \begin{align} | \begin{align} | ||
| - | W_{AB} = \vec{F_C}\cdot \vec{s} \quad \rightarrow \quad \Delta W = \vec{F_C}\cdot \Delta \vec{s}\quad \rightarrow \quad dW = \vec{F_C}\cdot d \vec{s} | + | W_{ \rm AB} = \vec{F_C}\cdot \vec{s} \quad \rightarrow \quad \Delta W = \vec{F_C}\cdot \Delta \vec{s}\quad \rightarrow \quad {\rm d}W = \vec{F_C}\cdot |
| \end{align} | \end{align} | ||
| Zeile 371: | Zeile 378: | ||
| \begin{align*} | \begin{align*} | ||
| - | W_{AB} &= \int_{W_A}^{W_B} dW \ | + | W_{ \rm AB} &= \int_{W_{ \rm A}}^{W_{ \rm B}} {\rm d} W \ |
| - | &= \int_{A}^{B} \vec{F_C}\cdot d \vec{s} \\ | + | &= \int_{ |
| - | &= \int_{A}^{B} q \cdot \vec{E} \cdot d \vec{s} | + | &= \int_{ |
| - | &= q \cdot \int_{A}^{B} \vec{E} \cdot d \vec{s} | + | &= q \cdot \int_{ |
| \end{align*} | \end{align*} | ||
| Zeile 380: | Zeile 387: | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{AB} & | + | U_{ \rm AB} & |
| - | &= \int_{A}^{B} \vec{E} \cdot d \vec{s} | + | &= \int_{ |
| \end{align*} | \end{align*} | ||
| - | Interestingly, | + | Interestingly, |
| + | 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. | ||
| + | So the voltage of the way there and back must be equal in magnitude. | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| < | < | ||
| - | This independency of the taken path leads for the closed path in <imgref ImgNr09b> | + | This independency of the taken path leads to the closed path in <imgref ImgNr09b> |
| \begin{align*} | \begin{align*} | ||
| - | \sum W &= W_{AB} &+ W_{BA} \\ | + | \sum W &= W_{ \rm AB} &+ W_{ \rm BA} \\ |
| - | & | + | & |
| - | & | + | & |
| \end{align*} | \end{align*} | ||
| Zeile 403: | Zeile 413: | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{AB} + U_{BA} &= 0 \\ | + | U_{ \rm AB} + U_{ \rm BA} &= 0 \\ |
| - | \int_{A}^{B} \vec{E} \cdot d \vec{s} + \int_{B}^{A} \vec{E} \cdot d \vec{s} &= 0 \\ | + | \int_{ |
| - | \rightarrow \boxed{ \oint \vec{E} \cdot d \vec{s} = 0} | + | \rightarrow \boxed{ \oint \vec{E} \cdot {\rm d} \vec{s} = 0} |
| \end{align*} | \end{align*} | ||
| Zeile 412: | Zeile 422: | ||
| <callout icon=" | <callout icon=" | ||
| - | - 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 d \vec{s} = 0 \end{align} | + | - 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: | - Or spoken differently: | ||
| - | - A field $\vec{X}$ which satisfies the condition $\oint \vec{X} \cdot d \vec{s}=0$ is called | + | - 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 |
| </ | </ | ||
| Zeile 425: | Zeile 435: | ||
| </ | </ | ||
| - | In the previous subchapter the term voltage got a more general meaning. This shall be now applied to investigate the electic | + | In the previous subchapter, the term voltage got a more general meaning. |
| + | This shall be now applied to investigate the electric | ||
| + | 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 | ||
| * equipotential lines for a 2-dimensional representation of the field. | * equipotential lines for a 2-dimensional representation of the field. | ||
| * equipotential surfaces for a 3-dimensional field | * equipotential surfaces for a 3-dimensional field | ||
| Zeile 432: | Zeile 446: | ||
| In <imgref ImgNr98>, | In <imgref ImgNr98>, | ||
| - | * The equipotential surfaces are drawn with a fixed step size, e.g. $1V$, $2V$, $3V$, ... . | + | * 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. | * 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° | + | * The angle between the field vectors (and therefore the field lines) and the equipotential lines is always |
| < | < | ||
| Zeile 447: | Zeile 461: | ||
| 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 independence | + | Since the voltage is independent |
| \begin{align*} | \begin{align*} | ||
| - | U_{AB} &= \int_{A}^{B} \vec{E} \cdot d \vec{s} \\ | + | U_{ \rm AB} &= \int_{ |
| - | & | + | & |
| \end{align*} | \end{align*} | ||
| Zeile 457: | Zeile 471: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| < | < | ||
| Here, the **electric potential** $\varphi$ is introduced as the scalar local function of the electric field (see <imgref ImgNr10b> | Here, the **electric potential** $\varphi$ is introduced as the scalar local function of the electric field (see <imgref ImgNr10b> | ||
| - | Similar to the 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 $\varphi_G$ is chosen to be located at infinity (see <imgref ImgNr10c> | + | 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> |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| < | < | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{AB} &= \int_{A}^{B} \vec{E} \cdot d \vec{s} &= \varphi_A | + | U_{ \rm AB} &= \int_{ |
| - | \rightarrow U_{AZ} &= \int_{A}^{Z} \vec{E} \cdot d \vec{s} &= \varphi_A | + | \rightarrow U_{ \rm AZ} &= \int_{ |
| - | \rightarrow \varphi_A | + | \rightarrow \varphi_{ \rm A} &= \int_{ |
| \end{align*} | \end{align*} | ||
| - | Alternatively, | + | Alternatively, |
| \begin{align*} | \begin{align*} | ||
| - | \varphi_A | + | \varphi_{ \rm A} &= \varphi_{ \rm A} - \underbrace{\varphi_{ \rm B}}_\text{=0} \\ |
| - | &= \int_{A}^{B} \vec{E} \cdot d \vec{s} | + | |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \varphi_C | + | \varphi_{ \rm C} &= \varphi_{ \rm C} - \underbrace{\varphi_{ \rm B}}_\text{=0} \\ |
| - | &= \int_{C}^{B} \vec{E} \cdot d \vec{s} \\ | + | |
| - | &= - \int_{B}^{C} \vec{E} \cdot d \vec{s} \\ | + | |
| \end{align*} | \end{align*} | ||
| - | For a positive charge the potential nearby the charge is positive and increasing, the closer one gets (see <imgref ImgNr197> | + | For a positive charge the potential nearby, the charge is positive and increasing, the closer one gets (see <imgref ImgNr197> |
| < | < | ||
| Zeile 499: | Zeile 513: | ||
| - | <callout title=" | + | <callout title=" |
| - | The equation $U_{AB} = \int_{A}^{B} \vec{E} \cdot d \vec{s}$ can be used and applied depending on the geometry present. | + | The equation $U_{ \rm AB} = \int_{ |
| As an example, consider the situation of a charge moving from one electrode to another inside a capacitor: | As an example, consider the situation of a charge moving from one electrode to another inside a capacitor: | ||
| \begin{align*} | \begin{align*} | ||
| - | U_{AB} & | + | U_{ \rm AB} & |
| - | U_{AB} & | + | U_{ \rm AB} & |
| - | U_{AB} & | + | U_{ \rm AB} & |
| - | U & | + | U & |
| \end{align*} | \end{align*} | ||
| </ | </ | ||
| - | |||
| - | ==== tasks ==== | ||
| - | |||
| - | {{page> | ||
| - | {{page> | ||
| - | {{page> | ||
| =====1.4 Conductors in the Electrostatic Field ===== | =====1.4 Conductors in the Electrostatic Field ===== | ||
| Zeile 523: | Zeile 531: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | 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. | ||
| - | - Be able to 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. | - Understand the effect of the electrostatic induction of an external electric field. | ||
| </ | </ | ||
| - | Up to now, charges were considered which were either rigid and not freely movable. | + | Up to now, charges were considered which were either rigid or not freely movable. |
| + | In the following, charges at 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 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 | ||
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| < | < | ||
| - | The movement of the charge continues until a force equilibrium is reached. In this steady state, there is no longer a resultant force acting on the single charge. In <imgref ImgNr10> this can be seen on the right: the repulsive forces of the charges are counteracted by the attractive forces of the atomic shells. \\ | + | The movement of the charge continues until a force equilibrium is reached. |
| + | In this steady state, there is no longer a resultant force acting on the single charge. | ||
| + | In <imgref ImgNr10> this can be seen on the right: the repulsive forces of the charges are counteracted by the attractive forces of the atomic shells. \\ | ||
| Results: | Results: | ||
| * The charge carriers are distributed on the surface. | * The charge carriers are distributed on the surface. | ||
| * Due to the dispersion of the charges, the interior of the conductor is free of fields. | * 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 tangential | + | * All field lines are perpendicular to the surface. Because: if they were not, there would be a parallel |
| - | <panel type=" | + | <wrap #edu_task_1 /> |
| + | <panel type=" | ||
| + | |||
| + | Point discharge is a well-known phenomenon, which can be seen as {{wp> | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | | ||
| + | </ | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| + | In the <imgref ImgNr194> | ||
| + | To cope with this complex shape and the wanted charge density, the following path shall be taken: | ||
| + | - It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm CG} = \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ 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>, | ||
| + | - 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? | ||
| + | - 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 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 $r_2$. | ||
| + | - What is the relationship between the bending of the surface and the charge density? | ||
| </ | </ | ||
| Zeile 565: | Zeile 601: | ||
| ==== Electrostatic Induction ==== | ==== 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. \\ | ||
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| < | < | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| Results: | Results: | ||
| * The charge carriers are still distributed on the surface. | * The charge carriers are still distributed on the surface. | ||
| - | * Now an 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 tangential | + | * The field lines leave the surface again at right angles. Again, a parallel |
| - | This effect of charge displacement in conductive objects by an electrostatic field is called | + | This effect of charge displacement in conductive objects by an electrostatic field is referred to as **electrostatic induction** (in German: |
| + | 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 the opposite direction. | ||
| <callout icon=" | <callout icon=" | ||
| Zeile 586: | Zeile 631: | ||
| </ | </ | ||
| - | How can the conductor surface be an equipotential surface despite different | + | How can the conductor surface be an equipotential surface despite different |
| + | Equipotential surfaces are defined only by the fact that the movement of a charge along such a surface does not require/ | ||
| + | Since the interior of the conductor is field-free, movement there can occur without a change in energy. | ||
| + | As the potential between two points is independent of the path between them, a path along the surface is also possible without energy expenditure. | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 592: | Zeile 640: | ||
| ==== Tasks ==== | ==== Tasks ==== | ||
| - | Application of electrostatic induction: Protective bag against electrostatic charge / discharge (cf. [[https:// | + | Application of electrostatic induction: Protective bag against electrostatic charge/ |
| <panel type=" | <panel type=" | ||
| Zeile 599: | Zeile 647: | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| In the simulation in <imgref ImgNr198> | In the simulation in <imgref ImgNr198> | ||
| - | 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. | + | 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 | + | - What is the angle between the field on the surface of the cylinder? |
| - | - Once the option '' | + | - Once the option '' |
| - What can be said about the potential distribution on the cylinder? | - 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 '' | - 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 '' | ||
| - Is there an electric field inside the body? | - Is there an electric field inside the body? | ||
| - | - Is this cylinder metallic, semiconducting or insulating? | + | - Is this cylinder metallic, semiconducting, or insulating? |
| </ | </ | ||
| Zeile 618: | Zeile 667: | ||
| {{page> | {{page> | ||
| + | <wrap anchor # | ||
| + | <panel type=" | ||
| - | =====1.5 The Electric Displacement Field and Gaussian theorem of electrostatics ===== | + | Given is the two-dimensional component shown in <imgref ImgNr148> |
| + | In the picture, there are 4 positions marked with numbers. \\ \\ | ||
| + | |||
| + | Order the numbered positions by increasing charge density! | ||
| < | < | ||
| - | The electric displacement Field or electric (flux) density | + | < |
| - | {{youtube>UqzXWU6TsQY}} | + | </ |
| - | </ | + | {{drawio>electrical_engineering_2: |
| + | </ | ||
| + | |||
| + | |||
| + | # | ||
| + | |||
| + | $\varrho_2 < \varrho_3 < \varrho_1 < \varrho_4$ | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | |||
| + | </ | ||
| + | |||
| + | # | ||
| + | |||
| + | |||
| + | </ | ||
| + | |||
| + | =====1.5 The Electric Displacement Field and Gauss' | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | 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 | ||
| - | - be able to 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' | + | - know the general meaning of Gauss's law of electrostatics |
| - | - be able to choose a closed surface appropriately and apply Gauss' | + | - choose a closed surface appropriately and apply Gauss's law |
| </ | </ | ||
| - | Now we want to consider | + | For a detailed description please see the chapters [[https:// |
| + | |||
| + | ==== Electric Displacement Flux Density D ==== | ||
| + | |||
| + | Up to now, ... | ||
| + | * ... we investigated | ||
| + | * ... the field $\vec{E}$ is principally a property of the space and the charges inside of 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 " | ||
| + | |||
| + | To investigate this situation, we want to consider | ||
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| < | < | ||
| - | In the previous | + | 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, 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: | ||
| + | |||
| + | \begin{align*} | ||
| + | \boxed{ D = {{\Psi}\over{A}} } | ||
| + | \end{align*} | ||
| + | |||
| + | On the other hand one could also only focus on the induced charges on the surfaces: | ||
| \begin{align*} | \begin{align*} | ||
| Zeile 654: | Zeile 748: | ||
| \end{align*} | \end{align*} | ||
| - | The **electric displacement field** | + | Since the induced charges $\Delta Q$ are equal to the flux $\Psi$ the **electric displacement field** |
| \begin{align} | \begin{align} | ||
| Zeile 660: | Zeile 755: | ||
| \end{align} | \end{align} | ||
| - | The electric | + | * Similar to the electric field $\vec{E}$ also the flux density is a field. |
| + | * It can be interpreted as a vector field. pointing | ||
| + | * The electric displacement field has the unit " | ||
| - | Why is now a second field introduced? This shall become clearer in the following, but first it shall be considered again how the electrostatic | + | Why is now a second field introduced? This shall become clearer in the following, but first, it shall be considered again how the electric |
| The two are related by the above equation. | 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. | 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, $\varepsilon= \varepsilon_0$, | + | 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 $\varepsilon_0$, |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== General relationship between | + | ==== General relationship between |
| - | Up to now, only a homogeneous field and an observation | + | Up to now, only a homogeneous field was considered |
| - | 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. | 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 | + | As with the potential and other physical problems, the problem is to be broken down into smaller |
| * The magnitude of $\Delta \vec{A}$ is equal to the area $\Delta A$. | * The magnitude of $\Delta \vec{A}$ is equal to the area $\Delta A$. | ||
| * The direction of $\Delta \vec{A}$ is perpendicular to the area. | * The direction of $\Delta \vec{A}$ is perpendicular to the area. | ||
| - | In addition, let $\Delta A$ now become infinitesimally small, that is, $dA = dx \cdot dy$. | + | In addition, let $\Delta A$ now become infinitesimally small, that is, ${\rm d}A = {\rm d}x \cdot {\rm d}y$. |
| - | === 1. problem: inhomogeneity | + | === 1. Problem: Inhomogenity |
| - | 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. | + | 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. |
| $Q = D\cdot A$ | $Q = D\cdot A$ | ||
| \begin{align*} | \begin{align*} | ||
| - | Q = D\cdot A \quad \rightarrow \quad dQ = D\cdot | + | Q = D\cdot A \quad \rightarrow \quad {\rm d}Q = D\cdot |
| \end{align*} | \end{align*} | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| === 2nd problem: arbitrary surface → solution: vectors === | === 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) | + | 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) |
| \begin{align*} | \begin{align*} | ||
| - | dQ = D\cdot | + | {\rm d}Q = D\cdot |
| \end{align*} | \end{align*} | ||
| - | === 3. summing | + | < |
| - | 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: | + | < |
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | The area vector and the surface {{wp> | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | === 3. Summing | ||
| + | Since so far only infinitesimally small surface pieces were 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} | \begin{align} | ||
| - | \boxed{\int | + | \boxed{\int |
| \end{align} | \end{align} | ||
| - | The " | + | The symbol ${\rlap{\Large \rlap{\int} \int} \, \LARGE \circ}$ denotes, that there is a closed surface used for the integration. |
| - | * 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 " | ||
| + | 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 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. | ||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| ==== Applications ==== | ==== Applications ==== | ||
| Zeile 714: | Zeile 839: | ||
| === Spherical Capacitor === | === Spherical Capacitor === | ||
| - | Spherical capacitors are now rarely found in practical applications. In the {{wp> | + | Spherical capacitors are now rarely found in practical applications. In the {{wp> |
| === Plate Capacitor === | === Plate Capacitor === | ||
| Zeile 720: | 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 $\vec{D}$: | 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*} | \begin{align*} | ||
| - | Q = \iint_{\text{closed surf.}} \vec{D} \cdot d \vec{A} | + | Q = {\rlap{\rlap{\int_A} \int} \: \LARGE \circ} \vec{D} \cdot {\rm d} \vec{A} |
| \end{align*} | \end{align*} | ||
| <callout icon=" | <callout icon=" | ||
| - | 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, | + | The consideration of the displacement flux density also solved a problem, which arose for electric |
| </ | </ | ||
| + | |||
| + | |||
| + | ==== tasks==== | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | 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. | ||
| + | |||
| + | Calculate the amount of the displaced charges in the thin metal foil. | ||
| + | |||
| + | <button size=" | ||
| + | * What is the strength of the electric field $E$ in the capacitor? | ||
| + | * Calculate the displacement flux density $D$ | ||
| + | * How can the charge $Q$ be derived from $D$? | ||
| + | </ | ||
| + | |||
| + | <button size=" | ||
| + | $Q = 10 ~{ \rm nC}$ | ||
| + | </ | ||
| + | |||
| + | </ | ||
| + | |||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | An ideal plate capacitor with a distance of $d_0 = 7 ~{ \rm mm}$ between the plates gets charged up to $U_0 = 190~{ \rm V}$ by an external source. | ||
| + | The source gets disconnected. After this, the distance between the plates gets enlarged to $d_1 = 7 ~{ \rm cm}$. | ||
| + | |||
| + | - What happens to the electric field and the voltage? | ||
| + | - How does the situation change (electric field/ | ||
| + | |||
| + | <button size=" | ||
| + | * Consider the displacement flux through a surface around a plate | ||
| + | </ | ||
| + | |||
| + | <button size=" | ||
| + | - $U_1 = 1.9~{ \rm kV}$, $E_1 = 27~{ \rm kV/ | ||
| + | - $U_1 = 190~{ \rm V}$, $E_1 = 2.7~{ \rm kV/ | ||
| + | </ | ||
| + | |||
| + | </ | ||
| + | |||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | An ideal plate capacitor with a distance of $d_0 = 6 ~{ \rm mm}$ between the plates and with air as dielectric ($\varepsilon_0=1$) is charged to a voltage of $U_0 = 5~{ \rm kV}$. | ||
| + | The source remains connected to the capacitor. In the air gap between the plates, a glass plate with $d_{ \rm g} = 4 ~{ \rm mm}$ and $\varepsilon_{ \rm r} = 8$ is introduced parallel to the capacitor plates. | ||
| + | |||
| + | 1. Calculate the partial voltages on the glas $U_{ \rm g}$ and on the air gap $U_{ \rm a}$. | ||
| + | |||
| + | # | ||
| + | * build a formula for the sum of the voltages first | ||
| + | * How is the voltage related to the electric field of a capacitor? | ||
| + | # | ||
| + | |||
| + | # | ||
| + | |||
| + | 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 $E = {{U}\over{d}}$: | ||
| + | \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 | ||
| + | \end{align*} | ||
| + | |||
| + | Therefore, we can put $E_\rm a= \varepsilon_{\rm r, g} \cdot E_\rm g $ into the formula of the total voltage and re-arrange to get $E_\rm g$: | ||
| + | \begin{align*} | ||
| + | U_0 &= E_{\rm g} \cdot d_{\rm g} + \varepsilon_{\rm r, g} \cdot E_{\rm g} \cdot d_{\rm a} \\ | ||
| + | &= E_{\rm g} \cdot ( d_{\rm g} + \varepsilon_{\rm r, g} \cdot d_{\rm a}) \\ | ||
| + | |||
| + | \rightarrow E_{\rm g} &= {{U_0}\over{d_{\rm g} + \varepsilon_{\rm r, g} \cdot d_{\rm a}}} | ||
| + | \end{align*} | ||
| + | |||
| + | Since we know that the distance of the air gap is $d_{\rm a} = d_0 - d_{\rm a}$ we can calculate: | ||
| + | \begin{align*} | ||
| + | E_{\rm g} &= {{5' | ||
| + | & | ||
| + | \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} & | ||
| + | |||
| + | \end{align*} | ||
| + | # | ||
| + | |||
| + | |||
| + | # | ||
| + | $U_{ \rm a} = 4~{ \rm kV}$, $U_{ \rm g} = 1 ~{ \rm kV}$ | ||
| + | # | ||
| + | |||
| + | |||
| + | 2. What would be the maximum allowed thickness of a glass plate, when the electric field in the air-gap shall not exceed $E_{ \rm max}=12~{ \rm kV/cm}$? | ||
| + | |||
| + | # | ||
| + | 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 | ||
| + | Now we shall eliminate $E_\rm g$, since $E_\rm a$ is given in the question. | ||
| + | \begin{align*} | ||
| + | U_0 & | ||
| + | &= {{E_\rm a}\over{\varepsilon_{\rm r, | ||
| + | \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} &= { { {{U_0}\over{E_{\rm a} }} - d_0 } \over { {{1}\over{\varepsilon_{\rm r,g}}} - 1 } } & | ||
| + | \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) | ||
| + | \end{align*} | ||
| + | # | ||
| + | |||
| + | |||
| + | # | ||
| + | $d_{ \rm g} = 2.10~{ \rm mm}$ | ||
| + | # | ||
| + | |||
| + | </ | ||
| + | |||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | Two concentric spherical conducting plates set up a spherical capacitor. | ||
| + | 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 a dielectric? | ||
| + | - What would be the limit value of the capacity, when the inner radius of the outer sphere is going to infinity ($r_{ \rm o} \rightarrow \infty$)? | ||
| + | <button size=" | ||
| + | * 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. | ||
| + | </ | ||
| + | |||
| + | <button size=" | ||
| + | - $C = 0.5~{ \rm pF}$ | ||
| + | - $C_{\infty} = 0.33~{ \rm pF}$ | ||
| + | </ | ||
| + | |||
| + | </ | ||
| + | |||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| Zeile 738: | Zeile 1024: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | 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, |
| - | - be able to 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 | + | - know what the effect of dielectric |
| - | - be able to 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 |
| </ | </ | ||
| Zeile 753: | Zeile 1039: | ||
| First of all, a thought experiment is to be carried out again (see <imgref ImgNr13> | 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. | - Next, the intermediate region is to be filled with a material. | ||
| Zeile 759: | Zeile 1045: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 765: | Zeile 1051: | ||
| Why might which of the two quantities change? | Why might which of the two quantities change? | ||
| - | You may have considered what happens to the charge $Q$ on the plates. This charge cannot | + | You may have considered what happens to the charge $Q$ on the plates. This charge cannot |
| Since the fictitious surface around an electrode does not change either, $\vec{D}$ cannot change either. | Since the fictitious surface around an electrode does not change either, $\vec{D}$ cannot change either. | ||
| Zeile 771: | Zeile 1057: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 782: | Zeile 1068: | ||
| \end{align*} | \end{align*} | ||
| - | The determined change is packed into the material constant $\varepsilon_r$. This gives the **material law of electrostatics**: | + | The determined change is packed into the material constant $\varepsilon_{ \rm r}$. This gives the **material law of electrostatics**: |
| \begin{align*} | \begin{align*} | ||
| - | \boxed{D = \varepsilon_r | + | \boxed{D = \varepsilon_{ \rm r} \cdot \varepsilon_0 \cdot E} |
| \end{align*} | \end{align*} | ||
| - | Since the charge $Q$ cannot vanish from the capacitor in this experimental setup and thus $D$ remains constant, the $E$ field must become smaller for $\varepsilon_r>1$. | + | Since the charge $Q$ cannot vanish from the capacitor in this experimental setup and thus $D$ remains constant, the $E$ field must become smaller for $\varepsilon_{ \rm r} > 1$. |
| <imgref ImgNr14> is drawn here in a simplified way: the alignable molecules are evenly distributed over the material and are thus also evenly aligned. Accordingly, | <imgref ImgNr14> is drawn here in a simplified way: the alignable molecules are evenly distributed over the material and are thus also evenly aligned. Accordingly, | ||
| Zeile 794: | Zeile 1080: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| <callout icon=" | <callout icon=" | ||
| - | - The material constant $\varepsilon_r$ is called | + | - The material constant $\varepsilon_{ \rm r}$ is referred to as relative permittivity, |
| - 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 with the presence of material for the same charge. | ||
| - | - The relative permittivity $\varepsilon_r$ is always greater than or equal to 1 for dielectrics (i.e., nonconductors). | + | - 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 | + | - The relative permittivity depends on the polarizability of the material, i.e. the possibility |
| </ | </ | ||
| <callout icon=" | <callout icon=" | ||
| - | If now the relative permittivity $\varepsilon_r$ depends on the possibility | + | Suppose |
| </ | </ | ||
| Zeile 809: | Zeile 1095: | ||
| < | < | ||
| - | ^ material^ relative permittivity \\ $\varepsilon_r$ for low frequencies ^ | + | ^ material |
| - | | 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 |
| - | | water (20°C) | 80 | | + | | water ($20~°{ \rm C}$) | $\rm 80$ |
| </ | </ | ||
| </ | </ | ||
| - | Some values of the relative permittivity $\varepsilon_r$ for dielectrics are given in <tabref tab01>. | + | Some values of the relative permittivity $\varepsilon_{ \rm r}$ for dielectrics are given in <tabref tab01>. |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 828: | Zeile 1114: | ||
| * The dielectrics act as insulators. The flow of current is therefore prevented | * The dielectrics act as insulators. The flow of current is therefore prevented | ||
| * The ability to insulate is dependent on the material. | * 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, | * Examples are: Lightning in a thunderstorm, | ||
| - | * The maximum electric field $E_0$ is called | + | * The maximum electric field $E_0$ is referred to as ** dielectric strength** (in German: |
| * $E_0$ depends on the material (see <tabref tab02>), but also on other factors (temperature, | * $E_0$ depends on the material (see <tabref tab02>), but also on other factors (temperature, | ||
| <WRAP 30em> | <WRAP 30em> | ||
| < | < | ||
| - | ^ Material^ Dielectric strength $E_0$ in kV/mm ^ | + | ^ Material |
| - | | air | 0.1...0.3 | | + | | air |
| - | | SF6 gas| 8 | | + | | SF6 gas |
| - | | vacuum| 20...30| | + | | insulating oils | $\rm 5...30$ | |
| - | | insulating oils| 5...30| | + | | vacuum |
| - | | quartz| 30...40 | | + | | quartz |
| - | | PP,PE | 50 | | + | | PP, PE | $\rm 50$ | |
| - | | PS | 100 | | + | | PS | $\rm 100$ |
| - | | distilled water | 70 | | + | | distilled water | $\rm 70$ | |
| </ | </ | ||
| </ | </ | ||
| Zeile 861: | Zeile 1147: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | 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 a capacitance and be able to apply them, |
| - | - be able to 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 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, |
| - | - 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 |
| </ | </ | ||
| Zeile 878: | Zeile 1164: | ||
| * This makes it possible to build up an electric field in the capacitor without charge carriers moving through the dielectric. | * 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 $C$. | * The characteristic of the capacitor is the capacitance $C$. | ||
| - | * In addition to the capacitance, | + | * In addition to the capacitance, |
| * Examples are | * Examples are | ||
| * the electrical component " | * the electrical component " | ||
| Zeile 887: | Zeile 1173: | ||
| The capacitance $C$ can be derived as follows: | The capacitance $C$ can be derived as follows: | ||
| - | - It is known that $U = \int \vec{E} d \vec{s} = E \cdot l$ and hence $E= {{U}\over{l}}$ or $D= \varepsilon_0 \cdot \varepsilon_r | + | - It is known that $U = \int \vec{E} |
| - | - Furthermore, | + | - Furthermore, |
| - | - Thus, the charge $Q$ is given by: \begin{align*} Q = \varepsilon_0 \cdot \varepsilon_r | + | - 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_r | + | - This means that $Q \sim U$, given the geometry (i.e., $A$ and $d$) and the dielectric ($\varepsilon_{ \rm r} $). |
| - So it is reasonable to determine a proportionality factor ${{Q}\over{U}}$. | - So it is reasonable to determine a proportionality factor ${{Q}\over{U}}$. | ||
| Zeile 896: | Zeile 1182: | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{C = \varepsilon_0 \cdot \varepsilon_r | + | \boxed{C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{A}\over{l}} = {{Q}\over{U}}} |
| \end{align*} | \end{align*} | ||
| + | |||
| + | Some of the main results here are: | ||
| + | * The capacity can be increased by increasing the dielectric constant $\varepsilon_{ \rm r} $, given the the same geometry. | ||
| + | * As near together the plates are as higher the capacity will be. | ||
| + | * As larger the area as higher the capacity will be. | ||
| + | |||
| + | The background behind the dielectric constant $\varepsilon_{ \rm r} $ and the field is explained in the following video | ||
| + | {{youtube> | ||
| + | |||
| This relationship can be examined in more detail in the following simulation: | This relationship can be examined in more detail in the following simulation: | ||
| - | -->capacitor | + | --> |
| If the simulation is not displayed optimally, [[https:// | If the simulation is not displayed optimally, [[https:// | ||
| Zeile 909: | Zeile 1204: | ||
| <-- | <-- | ||
| - | https://www.falstad.com/ | + | The <imgref ImgNr171> |
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| ==== Designs and types of capacitors ==== | ==== 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:// | + | 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:// |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | ^Shape of the Capacitor^Parameter^Equation for the Capacity^ | + | ^Shape of the Capacitor^ |
| - | |plate capacitor| area $A$ of plate \\ distance $l$ between plates | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_r | + | |plate capacitor |
| - | |cylinder capacitor |radius of outer conductor $R_o$ \\ radius of inner conductor $R_i$ \\ length $l$| \begin{align*}C = \varepsilon_0 \cdot \varepsilon_r | + | |cylinder capacitor |
| - | |spherical capacitor |radius of outer spherical conductor $R_o$ \\ radius of inner spherical conductor $R_i$| \begin{align*}C = \varepsilon_0 \cdot \varepsilon_r | + | |spherical capacitor |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 932: | Zeile 1232: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 938: | Zeile 1238: | ||
| - **{{wp> | - **{{wp> | ||
| - A variable capacitor consists of two sets of plates: a fixed set and a movable set (stator and rotor). These represent the two electrodes. | - 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, | - The size of the area is increased by the number of plates. Nevertheless, | ||
| - 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. | ||
| Zeile 944: | Zeile 1244: | ||
| - In the multilayer capacitor, there are again two electrodes. Here, too, the area $A$ (and thus the capacitance $C$) is multiplied by the finger-shaped interlocking. | - 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. | - Ceramic is used here as the dielectric. | ||
| - | - The multilayer ceramic capacitor is also called | + | - The multilayer ceramic capacitor is also referred to as KerKo or MLCC. |
| - The variant shown in (2) is an SMD variant (surface mound device). | - The variant shown in (2) is an SMD variant (surface mound device). | ||
| - Disk capacitor | - 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). | - Disc capacitors are designed for higher voltages, but have a low capacitance (in the microfarad range). | ||
| - | - **{{wp> | + | - **{{wp> |
| - 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). | - 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 |
| - | - **{{wp> | + | - **{{wp> |
| - | - A material similar to a "chip bag" is used as an insulator: a plastic film with a thin, metallized | + | - A material similar to a "chip bag" is used as an insulator: a plastic film with a thin, metalized |
| - The construction shows a high pulse load capacitance and low internal ohmic losses. | - The construction shows a high pulse load capacitance and low internal ohmic losses. | ||
| - | - In the event of electrical breakdown, the foil enables " | + | - In the event of electrical breakdown, the foil enables " |
| - | - With some manufacturers this type is called | + | - With some manufacturers, this type is referred to as |
| - **{{wp> | - **{{wp> | ||
| - As a dielectric is - similar to the electrolytic capacitor - very thin. In the actual sense, there is no dielectric at all. | - 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. | - 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 | + | - Supercapacitors can achieve very large capacitance values (up to the Kilofarad |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 975: | Zeile 1275: | ||
| In <imgref ImgNr17> are shown different capacitors: | In <imgref ImgNr17> are shown different capacitors: | ||
| - Above two SMD capacitors | - Above two SMD capacitors | ||
| - | - On the left a $100\mu F$ electrolytic capacitor | + | - On the left a $100~{ \rm µF}$ electrolytic capacitor |
| - | - On the right a $100nF$ MLCC in the commonly used {{wp> | + | - On the right a $100~{ \rm nF}$ MLCC in the commonly used {{wp> |
| - below different THT capacitors (__T__hrough __H__ole __T__echnology) | - below different THT capacitors (__T__hrough __H__ole __T__echnology) | ||
| - | - a big electrolytic capacitor with $10mF$ 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 $33pF$ and two Folkos with $1,5\mu F$ 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 $30pF$ 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:// | + | Various conventions have been established for designating the capacitance value of a capacitor [[https:// |
| \\ \\ | \\ \\ | ||
| Zeile 987: | Zeile 1287: | ||
| Electrolytic capacitors can explode! | Electrolytic capacitors can explode! | ||
| - | {{youtube> | + | {{youtube> |
| </ | </ | ||
| <callout icon=" | <callout icon=" | ||
| - | - 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_r | + | - 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 $\pm 10\%$. I.e. a calculation accurate to several decimal places is rarely necessary/ | + | - The capacitance value often varies by more than $\pm 10~{ \rm \%}$. I.e. a calculation accurate to several decimal places is rarely necessary/ |
| - 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. | - 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 1007: | Zeile 1307: | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | By the end of this section, you will be able to: |
| - | - be able to recognise | + | - recognize |
| - | - be able to 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, |
| - | - be able to determine the voltage across a single capacitor in a series circuit | + | - determine the voltage across a single capacitor in a series circuit. |
| </ | </ | ||
| Zeile 1026: | Zeile 1326: | ||
| \end{align*} | \end{align*} | ||
| - | Furthermore, | + | Furthermore, |
| \begin{align*} | \begin{align*} | ||
| U_q = U_1 + U_2 + ... + U_n = \sum_{k=1}^n U_k | U_q = U_1 + U_2 + ... + U_n = \sum_{k=1}^n U_k | ||
| Zeile 1037: | Zeile 1337: | ||
| U_q &= &U_1 &+ &U_2 &+ &... &+ &U_n &= \sum_{k=1}^n U_k \\ | U_q &= &U_1 &+ &U_2 &+ &... &+ &U_n &= \sum_{k=1}^n U_k \\ | ||
| U_q &= & | U_q &= & | ||
| - | {{1}\over{C_{ges}}}\cdot \Delta Q &= &&&&&&&& | + | {{1}\over{C_{ |
| \end{align*} | \end{align*} | ||
| Zeile 1043: | Zeile 1343: | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ {{1}\over{C_{ges}}} = \sum_{k=1}^n {{1}\over{C_k}} } | + | \boxed{ {{1}\over{C_{ |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ \Delta Q_k = const.} | + | \boxed{ \Delta Q_k = {\rm const.}} |
| \end{align*} | \end{align*} | ||
| Zeile 1054: | Zeile 1354: | ||
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{U_{ges} \cdot C_{ges} = U_{k} \cdot C_{k} } | + | \boxed{U_{ |
| \end{align*} | \end{align*} | ||
| - | In the simulation below, besides the parallel connected capacitors $C_1$, $C_2$, | + | In the simulation below, besides the parallel connected capacitors $C_1$, $C_2$, |
| * The switch $S$ allows the voltage source to charge the capacitors. | * 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: | * 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: | ||
| Zeile 1088: | Zeile 1388: | ||
| \Delta Q &= & Q_1 &+ & Q_2 &+ &... &+ & Q_n & | \Delta Q &= & Q_1 &+ & Q_2 &+ &... &+ & Q_n & | ||
| \Delta Q &= &C_1 \cdot U &+ &C_2 \cdot U &+ &... &+ &C_n \cdot U & | \Delta Q &= &C_1 \cdot U &+ &C_2 \cdot U &+ &... &+ &C_n \cdot U & | ||
| - | C_{ges} \cdot U &= &&&&&&&& | + | C_{ \rm eq} \cdot U &= &&&&&&&& |
| \end{align*} | \end{align*} | ||
| Zeile 1094: | Zeile 1394: | ||
| < | < | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ C_{ges} = \sum_{k=1}^n C_k } | + | \boxed{ C_{ \rm eq} = \sum_{k=1}^n C_k } |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ U_k = const} | + | \boxed{ U_k = {\rm const.}} |
| \end{align*} | \end{align*} | ||
| </ | </ | ||
| Zeile 1107: | Zeile 1407: | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ {{Q_k}\over{C_k}} = {{\Delta Q}\over{C_{ges}}} } | + | \boxed{ {{Q_k}\over{C_k}} = {{\Delta Q}\over{C_{ |
| \end{align*} | \end{align*} | ||
| - | In the simulation below, again besides the parallel connected capacitors $C_1$, $C_2$, | + | In the simulation below, again besides the parallel connected capacitors $C_1$, $C_2$, |
| This derivation is also well explained, for example, in [[https:// | This derivation is also well explained, for example, in [[https:// | ||
| Zeile 1126: | Zeile 1426: | ||
| </ | </ | ||
| - | =====1.9 | + | =====1.9 Configurations |
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson, you should: | + | By the end of this section, you will be able to: |
| - | - be able to recognise a stratification | + | - recognize the different layering |
| - | - know which quantity remains constant | + | - know which quantity remains constant |
| - | - know the constant quantity for a lateral layers as well | + | - be familiar with the equivalent circuits for normal |
| - | - be familiar with the equivalent circuits for perpendicular | + | - calculate the total capacitance of a capacitor with layering |
| - | - be able to calculate the total capacitance of a capacitor with stratification | + | |
| - 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. | ||
| Zeile 1144: | Zeile 1443: | ||
| - | Up to now was assumed only one dielectricum resp. only vacuum within capacitor. Now is looked at more detailed, how multi-layered construction between sheets | + | Up until this point, it was assumed |
| - | Thereby several | + | By doing this, various |
| - | - **perpendicular layering**: There are different dielectrics perpendicular | + | It is possible to tell the following variations apart |
| - | - **lateral layering**: There are different dielectrics parallel | + | |
| - | - **arbitrary configuration**: | + | - **layers are parallel to capacitor plates - dielectrics in series**: \\ The boundary layers |
| + | - **layers are perpendicular to capacitor plates - dielectrics in parallel**: \\ The boundary layers | ||
| + | - **arbitrary configuration**: | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Lateral Configuartion | + | ==== 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 that the boundary layers are parallel to the electrode surfaces. A voltage $U$ is applied to the structure from the outside. \\ | ||
| <WRAP 40em> | <WRAP 40em> | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| - | </ | + | </ |
| - | 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 |
| 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: | 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*} | \begin{align*} | ||
| - | U = \int \limits_{total \, inner \\ volume} \! \! \vec{E} \cdot d \vec{s} = E_1 \cdot d_1 + E_2 \cdot d_2 + E_3 \cdot d_3 | + | U = \int \limits_{\rm path \, inside |
| \tag{1.9.1} | \tag{1.9.1} | ||
| \end{align*} | \end{align*} | ||
| Zeile 1179: | Zeile 1480: | ||
| \begin{align*} | \begin{align*} | ||
| - | Q = \iint_{A} \vec{D} \cdot d \vec{A} = const. | + | Q = \iint_{A} \vec{D} \cdot {\rm d} \vec{A} = {\rm const.} |
| \end{align*} | \end{align*} | ||
| Zeile 1186: | Zeile 1487: | ||
| \begin{align*} | \begin{align*} | ||
| \vec{D_1} \cdot \vec{A} & = & \vec{D_2} \cdot \vec{A} & = & \vec{D_3} \cdot \vec{A} & \quad \quad \quad & | \:\: \vec{D_k} & \parallel \vec{A} \\ | \vec{D_1} \cdot \vec{A} & = & \vec{D_2} \cdot \vec{A} & = & \vec{D_3} \cdot \vec{A} & \quad \quad \quad & | \:\: \vec{D_k} & \parallel \vec{A} \\ | ||
| - | | + | |
| - | | + | |
| - | | + | |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ | + | \boxed{ |
| \tag{1.9.2} | \tag{1.9.2} | ||
| \end{align*} | \end{align*} | ||
| Zeile 1197: | Zeile 1498: | ||
| Using $(1.9.1)$ and $(1.9.2)$ we can also derive the following relationship: | Using $(1.9.1)$ and $(1.9.2)$ we can also derive the following relationship: | ||
| \begin{align*} | \begin{align*} | ||
| - | E_2 = & {{\varepsilon_{r1}}\over{\varepsilon_{r2}}}\cdot E_1 , \quad E_3 = {{\varepsilon_{r1}}\over{\varepsilon_{r3}}}\cdot E_1 \\ | + | E_2 = & {{\varepsilon_{ |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| U = & E_1 \cdot d_1 + & E_2 & \cdot d_2 + & E_3 & \cdot d_3 \\ | U = & E_1 \cdot d_1 + & E_2 & \cdot d_2 + & E_3 & \cdot d_3 \\ | ||
| - | U = & E_1 \cdot d_1 + & {{\varepsilon_{r1}}\over{\varepsilon_{r2}}}\cdot E_1 & \cdot d_2 + & {{\varepsilon_{r1}}\over{\varepsilon_{r3}}}\cdot E_1 & \cdot d_3 \\ | + | U = & E_1 \cdot d_1 + & {{\varepsilon_{ |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | U = & E_1 \cdot (d_1 + {{\varepsilon_{r1}}\over{\varepsilon_{r2}}} \cdot d_2 + {{\varepsilon_{r1}}\over{\varepsilon_{r3}}}\cdot d_3 ) \\ | + | U = & E_1 \cdot (d_1 + {{\varepsilon_{ |
| - | E_1 = & {{U}\over{ d_1 + \large{{\varepsilon_{r1}}\over{\varepsilon_{r2}}} \cdot d_2 + \large{{\varepsilon_{r1}}\over{\varepsilon_{r3}}}\cdot d_3 }} | + | E_1 = & {{U}\over{ d_1 + \large{{\varepsilon_{ |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ E_1 = {{U}\over{ \sum_{k=1}^n \large{{\varepsilon_{r1}}\over{\varepsilon_{rk}}} \cdot d_k}} } \quad \text{and} \; E_k = {{\varepsilon_{r1}}\over{\varepsilon_{rk}}}\cdot E_1 | + | \boxed{ E_1 = {{U}\over{ \sum_{k=1}^n \large{{\varepsilon_{ |
| \end{align*} | \end{align*} | ||
| - | The situation can also be transferred to a coaxial structure of a cylindrical capacitor or concentric structure of spherical capacitors. | + | < |
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | The situation can also be transferred to a coaxial structure of a cylindrical capacitor or the concentric structure of spherical capacitors. | ||
| <callout icon=" | <callout icon=" | ||
| - | Cross-stratification results in: | + | Conclusions: |
| - | - A perpendicular | + | - The layering |
| - | - The flux density is constant | + | - The flux density |
| - | - Considering | + | - We also found some results for the $E$ and $D$ fields __along the field line__. These parts of the fields |
| - | - The normal component of the electric field $E_n$ changes abruptly at the interface. | + | - The normal component of the electric field $E_{ \rm n}$ changes abruptly at the interface. |
| - | - The normal component of the flux density $D_n$ is continuous at the interface: $D_{n1} = D_{n2}$ | + | - The normal component of the flux density $D_{ \rm n}$ is continuous at the interface: $D_{ \rm n1} = D_{ \rm n2}$ |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Perpendicular Configuartion | + | ==== Dielectrics in Parallel |
| - | Now the boundary layers should be perpendicular to the electrode | + | Now the boundary layers should be perpendicular to the equipotential |
| <WRAP 40em> | <WRAP 40em> | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 1237: | Zeile 1544: | ||
| \begin{align*} | \begin{align*} | ||
| - | U = \int \limits_{total \, inner \\ volume} \! \! \vec{E} \cdot d \vec{s} = E_1 \cdot d = E_2 \cdot d = E_3 \cdot d | + | U = \int \limits_{\rm path \, inside |
| \end{align*} | \end{align*} | ||
| Since $d$ is the same for all dielectrics, | Since $d$ is the same for all dielectrics, | ||
| - | with the electric flux density $D_k = \varepsilon_{rk} \varepsilon_{0} \cdot E_k$ results: | + | with the electric flux density $D_k = \varepsilon_{{ \rm r}k} \varepsilon_{0} \cdot E_k$ results: |
| \begin{align*} | \begin{align*} | ||
| - | { { D_1 } \over { \varepsilon_{r1} } } = { { D_2 } \over { \varepsilon_{r2} } } = { { D_3 } \over { \varepsilon_{r3} } } = { { D_k } \over { \varepsilon_{rk} } } | + | { { D_1 } \over { \varepsilon_{ |
| \end{align*} | \end{align*} | ||
| 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. \\ | 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. \\ | 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 | + | Therefore, as more charges accumulate |
| - | This situation can also be transferred to a coaxial structure of a cylindrical capacitor or concentric structure of spherical capacitors. | + | |
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | This situation can also be transferred to a coaxial structure of a cylindrical capacitor or the concentric structure of spherical capacitors. | ||
| <callout icon=" | <callout icon=" | ||
| - | In the case of longitudinal stratification, | + | |
| - | - A perpendicular | + | Conclusions: |
| - | - The electric field in the capacitor | + | - The layering |
| - | - Considering | + | - The electric field for dielectrics |
| - | - The tangential | + | - We also found some results for the $E$ and $D$ fields |
| - | - The tangential | + | - The tangential |
| + | - The tangential | ||
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Arbitrary | + | ==== Arbitrary |
| With arbitrary configuration, | With arbitrary configuration, | ||
| - | However, some hints can be derived from the previous types of stratification: | + | However, some hints can be derived from the previous types of layering: |
| * Electric field $\vec{E}$: | * Electric field $\vec{E}$: | ||
| - | * The normal component $E_{n}$ is discontinuous at the interface: $\varepsilon_{r1} \cdot E_{n1} = \varepsilon_{r2} \cdot E_{n2}$ | + | * The normal component $E_{ \rm n}$ is discontinuous at the interface: $\varepsilon_{ |
| - | * The tangential component $E_{t}$ is continuous at the interface: $ E_{t1} = E_{t2}$ | + | * The tangential component $E_{ \rm t}$ is continuous at the interface: $ E_{ \rm t1} = E_{ \rm t2}$ |
| * Electric displacement flux density $\vec{D}$: | * Electric displacement flux density $\vec{D}$: | ||
| - | * The normal component $D_{n}$ is continuous at the interface: $ D_{n1} = D_{n2}$ | + | * The normal component $D_{ \rm n}$ is continuous at the interface: $ D_{ \rm n1} = D_{ \rm n2}$ |
| - | * The tangent | + | * The tangential |
| <WRAP 30em> | <WRAP 30em> | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Since $\vec{D} = \varepsilon_{0} \varepsilon_{r} \cdot \vec{E}$ the direction of the fields must be the same. \\ | + | < |
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | Since $\vec{D} = \varepsilon_{0} \varepsilon_{ | ||
| Using the fields, we can now derive the change in the angle: | Using the fields, we can now derive the change in the angle: | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed { { { tan \alpha_1 } \over { tan \alpha_2 | + | \boxed { { { \tan \alpha_1 } \over { \tan \alpha_2 |
| \end{align*} | \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. | + | 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. |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ====Tasks==== | ||
| - | Different dielectrics in the capacitor | + | <panel type=" |
| - | {{youtube> | + | {{youtube> |
| - | {{youtube>0ZxbPGKA2Po}} | + | </ |
| - | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | <panel type=" |
| - | ====Tasks==== | + | {{youtube> |
| - | <panel type=" | + | </WRAP>< |
| - | {{youtube> | + | <panel type=" |
| + | |||
| + | {{youtube> | ||
| </ | </ | ||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| - | <panel type=" | + | <panel type=" |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Two parallel capacitor plates face each other with a distance $d_K = 10mm$. A voltage of $U = 3'000V$ is applied to the capacitor. Parallel to the capacitor plates there is a glass plate ($\varepsilon_{r, | + | 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_{ | ||
| - | - Calculate the partial voltages $U_G$ in the glass and $U_L$ in the air gap. | + | - 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 $E_{0,G} =12 kV/cm$ 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~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 1334: | Zeile 1666: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 1341: | Zeile 1673: | ||
| ====== Further links ====== | ====== Further links ====== | ||
| - | * [[https:// | + | * [[https:// |
| ====== additional Links ====== | ====== additional Links ====== | ||
| Zeile 1352: | Zeile 1684: | ||
| - | A really | + | A great introduction |
| examples: | examples: | ||