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 [2023/03/15 14:10] – mexleadmin | 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 |
| < | < | ||
| < | < | ||
| Zeile 17: | Zeile 16: | ||
| </ | </ | ||
| - | 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 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 coordinate system). | + | - Effects along spatial axes $x$, $y$ and $z$ (Cartesian coordinate system). |
| - 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 $[E]={{[F]}\over{[q]}}=1 ~\rm{{N}\over{As}}=1 ~\rm{{N\cdot m}\over{As \cdot m}} = 1 ~\rm{{V \cdot A \cdot s}\over{As \cdot m}} = 1 ~\rm{{V}\over{m}}$ | + | $E = {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \quad$ with $[E]={{[F]}\over{[q]}}=1 ~{ \rm {N}\over{As}}=1 ~{ \rm {N\cdot m}\over{As \cdot m}} = 1 ~{ \rm {V \cdot A \cdot s}\over{As \cdot m}} = 1 ~{ \rm {V}\over{m}}$ |
| </ | </ | ||
| 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 |
| </ | </ | ||
| Zeile 137: | Zeile 136: | ||
| - | 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>: |
| < | < | ||
| Zeile 148: | Zeile 147: | ||
| </ | </ | ||
| - | For a full picture of the field lines between charges, one has to start with a single charge. The in- and outgoing lines on this charge are drawn in equidistance on the charge. This is also true for the situation with multiple charges. However there, the lines are not necessarily run radially anymore. The test charge is influenced by all the single charges, and therefore the field lines can get bend. | + | For a full picture of the field lines between charges, one has to start with a single charge. The in- and outgoing lines on this charge are drawn in equidistance on the charge. This is also true for the situation with multiple charges. However there, the lines are not necessarily run radially anymore. The test charge is influenced by all the single charges, and therefore the field lines can get bent. |
| < | < | ||
| Zeile 156: | Zeile 155: | ||
| </ | </ | ||
| - | In <imgref ImgNr031> | + | In <imgref ImgNr031> |
| Try the following in the simulation: | Try the following in the simulation: | ||
| * Get accustomed to the simulation. You can... | * Get accustomed to the simulation. You can... | ||
| 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 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 199: | Zeile 198: | ||
| </ | </ | ||
| - | <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 218: | Zeile 217: | ||
| - | ===== 1.2 Electric | + | ===== 1.2 Electric |
| < | < | ||
| Zeile 230: | Zeile 229: | ||
| - determine a force vector by superimposing several force vectors using vector calculus. | - determine a force vector by superimposing several force vectors using vector calculus. | ||
| - state the following quantities for a force vector: | - state the following quantities for a force vector: | ||
| - | - 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 done by giving the correct labeling the subscript of the force. In <imgref ImgNr06> (a) and (b) the convention is shown: A force $\vec{F}_{21}$ acts on charge $Q_2$ and is caused by charge $Q_1$. As a mnemonic you can remember " | + | 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 254: | 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. | ||
| Zeile 270: | Zeile 269: | ||
| <WRAP group>< | <WRAP group>< | ||
| In **homogeneous fields**, magnitude and direction are constant throughout the field range. | 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> | + | This field form is idealized to exist within plate capacitors. e.g., in the plate capacitor (<imgref ImgNr07> |
| < | < | ||
| Zeile 301: | Zeile 300: | ||
| {{youtube> | {{youtube> | ||
| </ | </ | ||
| + | |||
| + | {{page> | ||
| + | {{page> | ||
| + | {{page> | ||
| + | |||
| =====1.3 Work and Potential ===== | =====1.3 Work and Potential ===== | ||
| Zeile 311: | Zeile 315: | ||
| - know how work is defined in the electrostatic field. | - know how work is defined in the electrostatic field. | ||
| - | - describe when work has to be performed and when it does not in the situation of a 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. | ||
| Zeile 322: | 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 " | ||
| Zeile 331: | Zeile 335: | ||
| First, the situation of a charge in a homogeneous electric field shall be considered. As we have seen so far, the magnitude of $E$ is constant and the field lines are parallel. Now a positive charge $q$ is to be brought into this field. | 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. |
| < | < | ||
| Zeile 339: | Zeile 343: | ||
| < | < | ||
| - | 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 \rm{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: $[U] = 1~\rm{V}$ | + | - 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 372: | 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 381: | 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 404: | 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 413: | 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 426: | Zeile 435: | ||
| </ | </ | ||
| - | In the previous subchapter the term voltage got a more general meaning. | + | In the previous subchapter, the term voltage got a more general meaning. |
| This shall be now applied to investigate the electric field a bit more. | This shall be now applied to investigate the electric field a bit more. | ||
| Once a charge $q$ moves perpendicular to the field lines, it experiences neither energy gain nor loss. | 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 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 | + | 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 437: | Zeile 446: | ||
| In <imgref ImgNr98>, | In <imgref ImgNr98>, | ||
| - | * The equipotential surfaces are drawn with a fixed step size, e.g. $1~\rm{V}$, $2~\rm{V}$, $3~\rm{V}$, ... . | + | * The equipotential surfaces are drawn with a fixed step size, e.g. $1~{ \rm V}$, $2~{ \rm V}$, $3~{ \rm V}$, ... . |
| * Since the electric field is higher near charges, equipotential surfaces are also closer together there. | * 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 $90°$ | ||
| Zeile 452: | 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 467: | Zeile 476: | ||
| 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> |
| < | < | ||
| Zeile 476: | Zeile 485: | ||
| \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 506: | Zeile 515: | ||
| <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 539: | Zeile 542: | ||
| </ | </ | ||
| - | 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. |
| - | At the following, charges at an electric conductor are investigated. | + | In the following, charges at an electric conductor are investigated. |
| These charges are only free to move within the conductor. | These charges are only free to move within the conductor. | ||
| - | At first an ideal conductor without resistance is considered. | + | 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 ==== | ||
| Zeile 548: | Zeile 551: | ||
| 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 additional charges create an electric field. Thus, a resultant force acts on each charge. | ||
| - | The cause of this force are the electric fields of the surrounding electric charges. So the charges repel and move apart. \\ | + | The causes |
| < | < | ||
| Zeile 567: | Zeile 570: | ||
| <panel type=" | <panel type=" | ||
| - | Point discharge is a well-known phenomenon, which can be seen as {{wp> | + | Point discharge is a well-known phenomenon, which can be seen as {{wp> |
| < | < | ||
| Zeile 577: | Zeile 580: | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| {{drawio> | {{drawio> | ||
| </ | </ | ||
| - | In the <imgref ImgNr194> | + | In the <imgref ImgNr194> |
| - | In order to cope with this complex shape and the wanted charge density, the following path shall be taken: | + | To cope with this complex shape and the 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 $U_{AB} = \int_{C}^{G} \vec{E} \cdot d \vec{s}$ 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, $C$ is a point at distance $r_0$ from the charge and $G$ is the ground potential at infinity. | + | - It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm CG} = \int_{ |
| - Compare the field and the potentials of the different spherical conductors in <imgref ImgNr194>, | - 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? | - 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? | - What can be conducted for the field of the three situations in (b) and (d), when the total charge on the surface is considered to be always the same? | ||
| - For spherical conductors the surface charge density is constant. Given that this charge density leads to the overall charge $q$, how is $\varrho_A$ depending on the radius $r$ of a sphere? | - For spherical conductors the surface charge density is constant. Given that this charge density leads to the overall charge $q$, how 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, for charges on the surface of the curvature to the left and to the right it represents | + | - 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 |
| - Set up this equality formula based on the formula for the potential from question 1. | - 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 $r_2$. | - 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 ist the relationship between the bending of the surface and the charge density? | + | - What is the relationship between the bending of the surface and the charge density? |
| </ | </ | ||
| Zeile 612: | Zeile 615: | ||
| {{url> | {{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 parallel component would cause a charge shift in the metal. | * The field lines leave the surface again at right angles. Again, a parallel component would cause a charge shift in the metal. | ||
| - | This effect of charge displacement in conductive objects by an electrostatic field is called | + | 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). | Induced charges can be separated (<imgref ImgNr11> right). | ||
| - | If we look at the separated induced charges without the external field, their field is again just as strong in magnitude as the external field only in opposite direction. | + | If we look at the separated induced charges without the external field, their field is again just as strong in magnitude as the external field only in the opposite direction. |
| <callout icon=" | <callout icon=" | ||
| Zeile 629: | 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/ | 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. | + | 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. | As the potential between two points is independent of the path between them, a path along the surface is also possible without energy expenditure. | ||
| Zeile 638: | 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 649: | Zeile 651: | ||
| 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. | + | 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. | 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 665: | Zeile 667: | ||
| {{page> | {{page> | ||
| + | <wrap anchor # | ||
| <panel type=" | <panel type=" | ||
| - | Given is the two-dimesional | + | Given is the two-dimensional |
| - | In the picture there are 4 positions marked with numbers. \\ \\ | + | In the picture, there are 4 positions marked with numbers. \\ \\ |
| Order the numbered positions by increasing charge density! | Order the numbered positions by increasing charge density! | ||
| Zeile 679: | Zeile 682: | ||
| - | --> Answer | + | # |
| $\varrho_2 < \varrho_3 < \varrho_1 < \varrho_4$ | $\varrho_2 < \varrho_3 < \varrho_1 < \varrho_4$ | ||
| Zeile 686: | Zeile 689: | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | <-- | + | |
| + | # | ||
| + | |||
| </ | </ | ||
| - | =====1.5 The Electric Displacement Field and Gauss' | + | =====1.5 The Electric Displacement Field and Gauss' |
| < | < | ||
| Zeile 713: | Zeile 720: | ||
| * ... we investigated the __effect__ of the electric field onto a (probe) charge, which can be calculated by $\vec{F}= \vec{E}\cdot q$. | * ... we investigated the __effect__ of the electric field onto a (probe) charge, which can be calculated by $\vec{F}= \vec{E}\cdot q$. | ||
| * ... the field $\vec{E}$ is principally a property of the space and the charges inside of it. | * ... the field $\vec{E}$ is principally a property of the space and the charges inside of it. | ||
| - | * ... we also only had a look on "empty space" containing charges and/or ideally conducting components | + | * ... we also only had a look at "empty space" containing charges and/or ideally conducting components |
| - | The in 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 " | + | 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 " | ||
| - | In order to investigate this situation, we want to consider two conductive plates (X) and (Y) with the area $\Delta A$ in the electrostatic field $\vec{E}$ in vacuum a little more exactly. For this purpose, the plates shall first be brought into the field separately. | + | To investigate this situation, we want to consider two conductive plates (X) and (Y) with the area $\Delta A$ in the electrostatic field $\vec{E}$ in a vacuum a little more exactly. For this purpose, the plates shall first be brought into the field separately. |
| < | < | ||
| Zeile 725: | Zeile 733: | ||
| < | < | ||
| - | 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 to the negative outer plate (1) positive charges get induced on (X). Equally, near to positive outer plate (2) negative charges get induced on (Y). Graphically speaking, for each field line ending on the pair of plates, a single charge must move from one plate to the other. The direction of the movement is in similar to the direction of $\vec{E}$. This ability to separate charges (i.e. to generate electrostatic induction) is another property of space. This property is independent of any matter inside | + | 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 | + | 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 |
| \begin{align*} | \begin{align*} | ||
| Zeile 733: | Zeile 741: | ||
| \end{align*} | \end{align*} | ||
| - | ON the other hand one could also only focus onto the induced charges on the surfaces: In the shown arrangement (homogeneous field, all surfaces parallel to each other), the surface charge density $\varrho_A = {{\Delta Q}\over{\Delta A}}$ thus electrostatic induction is proportional to the external field $E$. It holds: | + | On the other hand one could also only focus on the induced charges on the surfaces: In the shown arrangement (homogeneous field, all surfaces parallel to each other), the surface charge density $\varrho_A = {{\Delta Q}\over{\Delta A}}$ thus electrostatic induction is proportional to the external field $E$. It holds: |
| \begin{align*} | \begin{align*} | ||
| Zeile 748: | Zeile 756: | ||
| * Similar to the electric field $\vec{E}$ also the flux density is a field. | * Similar to the electric field $\vec{E}$ also the flux density is a field. | ||
| - | * It can be interpreted as an vector field. pointing in the same direction as the electric field $\vec{E}$. | + | * It can be interpreted as a vector field. pointing in the same direction as the electric field $\vec{E}$. |
| - | * The electric displacement field has the unit " | + | * 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 electric field $\vec{E}$ was defined. This resulted from the Coulomb force, i.e. the __action on a sample charge__. The electric displacement field, on the other hand, is not described by an action, but __caused by charges__. | + | Why is now a second field introduced? This shall become clearer in the following, but first, it shall be considered again how the electric field $\vec{E}$ was defined. This resulted from the Coulomb force, i.e. the __action on a sample charge__. The electric displacement field, on the other hand, is not described by an action, but __caused by charges__. |
| 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, $\varepsilon$ is $\varepsilon_0$, | + | 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 was considered and only a surface perpendicular to the field lines. Thus only equipotential surfaces (e.g. a metal foil) were investigated. | Up to now, only a homogeneous field was considered and only a surface perpendicular to the field lines. Thus only equipotential surfaces (e.g. a metal foil) were investigated. | ||
| - | In that case it was found that the charge is equal to the electric displacement field on the surface: $\Delta Q = D\cdot \Delta A$. | + | In that case, it was found that the charge is equal to the electric displacement field on the surface: $\Delta Q = D\cdot \Delta A$. |
| This formula is now to be extended to arbitrary surfaces and inhomogeneous fields. | This formula is now to be extended to arbitrary surfaces and inhomogeneous fields. | ||
| - | As with the potential and other physical problems, the problem is to be broken down into smaller sub-problems, | + | As with the potential and other physical problems, the problem is to be broken down into smaller sub-problems, |
| * 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: Inhomogenity → Solution: infinitesimal | + | === 1. Problem: Inhomogenity → Solution: infinitesimal |
| - | 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*} | ||
| Zeile 785: | Zeile 793: | ||
| === 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) area vector and $\vec{D}$-field: | + | Now assume an arbitrary surface. Thus the $\vec{D}$-field no longer penetrates through the surface at right angles. But for the electrostatic induction, only the rectangular part was relevant. So only this part has to be considered. This results from consideration of the cosine of the angle between (right-angled) area vector and $\vec{D}$-field: |
| \begin{align*} | \begin{align*} | ||
| - | dQ = D\cdot | + | {\rm d}Q = D\cdot |
| \end{align*} | \end{align*} | ||
| Zeile 806: | Zeile 814: | ||
| === 3. Summing up === | === 3. Summing up === | ||
| - | Since so far only infinitesimally small surface pieces were considered must now be integrated again to a total surface. If a closed enveloping surface around a body is chosen, the result is: | + | Since so far only infinitesimally small surface pieces 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. | ||
| Zeile 828: | 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 834: | 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 |
| </ | </ | ||
| Zeile 849: | Zeile 860: | ||
| <panel type=" | <panel type=" | ||
| - | A plate capacitor with a distance of $d = 2 ~\rm{cm}$ between the plates and with air as dielectric ($\varepsilon_r=1$) gets charged up to $U = 5~\rm{kV}$. | + | A plate capacitor with a distance of $d = 2 ~{ \rm cm}$ between the plates and with air as dielectric ($\varepsilon_{ \rm r}=1$) gets charged up to $U = 5~{ \rm kV}$. |
| - | In between the plates a thin metal foil with the area $A = 45~\rm{cm^2}$ is introduced parallel to the plates. | + | In between the plates a thin metal foil with the area $A = 45~{ \rm cm^2}$ is introduced parallel to the plates. |
| Calculate the amount of the displaced charges in the thin metal foil. | Calculate the amount of the displaced charges in the thin metal foil. | ||
| Zeile 861: | Zeile 872: | ||
| <button size=" | <button size=" | ||
| - | $Q = 10 ~\rm{nC}$ | + | $Q = 10 ~{ \rm nC}$ |
| </ | </ | ||
| Zeile 869: | Zeile 880: | ||
| <panel type=" | <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. | + | 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 $d_1 = 7 ~\rm{cm}$. | + | 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? | - What happens to the electric field and the voltage? | ||
| - | - How does the situation change (electric field / voltage), when the source is not disconnected? | + | - How does the situation change (electric field/ |
| <button size=" | <button size=" | ||
| - | * consider | + | * Consider |
| </ | </ | ||
| <button size=" | <button size=" | ||
| - | - $U_1 = 1.9~\rm{kV}$, $E_1 = 27~\rm{kV/m}$ | + | - $U_1 = 1.9~{ \rm kV}$, $E_1 = 27~{ \rm kV/m}$ |
| - | - $U_1 = 190~\rm{V}$, $E_1 = 2.7~\rm{kV/ | + | - $U_1 = 190~{ \rm V}$, $E_1 = 2.7~{ \rm kV/ |
| </ | </ | ||
| Zeile 889: | Zeile 900: | ||
| <panel type=" | <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 $U_0 = 5~\rm{kV}$. | + | An ideal plate capacitor with a distance of $d_0 = 6 ~{ \rm mm}$ between the plates and with air as dielectric ($\varepsilon_0=1$) is charged to a voltage of $U_0 = 5~{ \rm kV}$. |
| - | The source remains connected to the capacitor. In the air gap between the plates, a glas plate with $d_g = 2 ~\rm{mm}$ and $\varepsilon_r | + | The source remains connected to the capacitor. In the air gap between the plates, a glass plate with $d_{ \rm g} = 4 ~{ \rm mm}$ and $\varepsilon_{ \rm r} = 8$ is introduced parallel to the capacitor plates. |
| - | - Calculate the partial voltages on the glas $U_g$ and on the air gap $U_a$. | + | 1. Calculate the partial voltages on the glas $U_{ \rm g}$ and on the air gap $U_{ \rm a}$. |
| - | - What would be the maximum allowed thickness of a glas plate, when the electric field in the air-gap shall not exceed $E_{max}=12~\rm{kV/cm}$? | + | |
| - | <button size=" | + | # |
| * build a formula for the sum of the voltages first | * build a formula for the sum of the voltages first | ||
| * How is the voltage related to the electric field of a capacitor? | * How is the voltage related to the electric field of a capacitor? | ||
| - | </ | + | # |
| - | <button size=" | + | # |
| - | - $U_a = 4~\rm{kV}$, $U_g = 1 ~\rm{kV}$ | + | |
| - | | + | 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} &= 4 ~{\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 g} \cdot d_{\rm g} + E_{\rm a} \cdot d_{\rm a} \\ | ||
| + | &= {{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*} | ||
| + | # | ||
| + | |||
| + | |||
| + | # | ||
| + | $d_{ \rm g} = 2.10~{ \rm mm}$ | ||
| + | # | ||
| </ | </ | ||
| Zeile 910: | Zeile 991: | ||
| <panel type=" | <panel type=" | ||
| - | Two two concentric spherical conducting plates | + | Two concentric spherical conducting plates |
| - | The radius of the inner sphere is $r_i = 3~\rm{mm}$, the inner radius from the outer sphere is $r_o = 9~\rm{mm}$. | + | The radius of the inner sphere is $r_{ \rm i} = 3~{ \rm mm}$, and the inner radius from the outer sphere is $r_{ \rm o} = 9~{ \rm mm}$. |
| - | - What is the capacity of this capacitor, given that air is used as dielectric? | + | - What is the capacity of this capacitor, given that air is used as a dielectric? |
| - | - What would be the limit value of the capacity, when the inner radius of the outer sphere is going to infinity ($r_o \rightarrow \infty$)? | + | - 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=" | <button size=" | ||
| * What is the displacement flux density of the inner sphere? | * What is the displacement flux density of the inner sphere? | ||
| Zeile 922: | Zeile 1003: | ||
| <button size=" | <button size=" | ||
| - | - $C = 0.5~pF$ | + | - $C = 0.5~{ \rm pF}$ |
| - | - $C_{\infty} = 0.33~\rm{pF}$ | + | - $C_{\infty} = 0.33~{ \rm pF}$ |
| </ | </ | ||
| Zeile 929: | Zeile 1010: | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
| Zeile 958: | 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 970: | 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 987: | 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 999: | 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 1014: | Zeile 1095: | ||
| < | < | ||
| - | ^ material | + | ^ material |
| - | | air | 1.0006 | + | | air | $\rm 1.0006$ | |
| - | | paper | 2 | | + | | paper | $\rm 2$ | |
| - | | hard paper | 5 | | + | | PE, PP |
| - | | glass | 6...8 | | + | | PS |
| - | | PE, PP | + | | hard paper | $\rm 5$ | |
| - | | PS | + | | glass | $\rm 6...8$ |
| - | | water ($20~°\rm{C}$) | 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 1033: | 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 | + | ^ Material |
| - | | air | 0.1...0.3 | + | | air |
| - | | SF6 gas | 8 | | + | | SF6 gas |
| - | | vacuum | + | | insulating oils | $\rm 5...30$ | |
| - | | insulating oils | 5...30 | + | | vacuum |
| - | | quartz | + | | quartz |
| - | | PP, PE | 50 | | + | | PP, PE | $\rm 50$ | |
| - | | PS | 100 | | + | | PS | $\rm 100$ | |
| - | | distilled water | 70 | | + | | distilled water | $\rm 70$ | |
| </ | </ | ||
| </ | </ | ||
| Zeile 1072: | Zeile 1153: | ||
| - 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, | ||
| - | - 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 1083: | 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 1092: | 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 1101: | 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: | Some of the main results here are: | ||
| - | * The capacity can be increased by increasing the dielectric constant $\varepsilon_r | + | * 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 near together the plates are as higher the capacity will be. | ||
| * As larger the area as higher the capacity will be. | * As larger the area as higher the capacity will be. | ||
| - | The beckground | + | The background |
| {{youtube> | {{youtube> | ||
| Zeile 1115: | Zeile 1196: | ||
| 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 1123: | Zeile 1204: | ||
| <-- | <-- | ||
| - | The <imgref ImgNr171> | + | The <imgref ImgNr171> |
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| {{url> | {{url> | ||
| Zeile 1142: | Zeile 1223: | ||
| ^Shape of the Capacitor^ | ^Shape of the Capacitor^ | ||
| - | |plate capacitor | + | |plate capacitor |
| - | |cylinder capacitor | + | |cylinder capacitor |
| - | |spherical capacitor | + | |spherical capacitor |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 1157: | 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 1163: | 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, metalized layer. | - A material similar to a "chip bag" is used as an insulator: a plastic film with a thin, metalized layer. | ||
| - The construction shows a high pulse load capacitance and low internal ohmic losses. | - 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 1194: | 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 $100~\rm{µF}$ electrolytic capacitor | + | - On the left a $100~{ \rm µF}$ electrolytic capacitor |
| - | - On the right a $100~\rm{nF}$ 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 $10~\rm{mF}$ in blue, the positive terminal is marked with $+$ | + | - a big electrolytic capacitor with $10~{ \rm mF}$ in blue, the positive terminal is marked with $+$ |
| - | - in the second row is a Kerko with $33~\rm{pF}$ and two Folkos with $1,5~\rm{µF}$ each | + | - in the second row is a Kerko with $33~{ \rm pF}$ and two Folkos with $1,5~{ \rm µF}$ each |
| - | - in the bottom row you can see a trim capacitor with about $30~\rm{pF}$ and a tantalum electrolytic capacitor and another electrolytic capacitor | + | - in the bottom row you can see a trim capacitor with about $30~{ \rm pF}$ and a tantalum electrolytic capacitor and another electrolytic capacitor |
| Various conventions have been established for designating the capacitance value of a capacitor [[https:// | Various conventions have been established for designating the capacitance value of a capacitor [[https:// | ||
| Zeile 1210: | Zeile 1291: | ||
| <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 $\pm 10~\rm{\%}$. 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 1230: | Zeile 1311: | ||
| By the end of this section, you will be able to: | By the end of this section, you will be able to: | ||
| - | - recognise | + | - recognize |
| - 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, | ||
| Zeile 1245: | 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 1256: | 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 1262: | 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 1273: | 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 1307: | 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 1313: | 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 1326: | 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 1353: | Zeile 1434: | ||
| By the end of this section, you will be able to: | By the end of this section, you will be able to: | ||
| - | - recognise | + | - recognize |
| - | - 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 | - calculate the total capacitance of a capacitor with layering | ||
| - know the law of refraction at interfaces for the field lines in the electrostatic field. | - know the law of refraction at interfaces for the field lines in the electrostatic field. | ||
| Zeile 1363: | 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**: | ||
| < | < | ||
| Zeile 1377: | Zeile 1459: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Lateral Configuration | + | ==== Dielectrics in Series |
| First, the situation is considered that the boundary layers are parallel to the electrode surfaces. A voltage $U$ is applied to the structure from the outside. \\ | First, the situation is considered 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 1398: | 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 1405: | 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 1416: | 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*} | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| {{url> | {{url> | ||
| </ | </ | ||
| - | The situation can also be transferred to a coaxial structure of a cylindrical capacitor or concentric structure of spherical capacitors. | + | The situation can also be transferred to a coaxial structure of a cylindrical capacitor or the concentric structure of spherical capacitors. |
| <callout icon=" | <callout icon=" | ||
| - | Lateral configuration 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 1462: | 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 |
| Zeile 1484: | Zeile 1566: | ||
| </ | </ | ||
| - | This situation can also be transferred to a coaxial structure of a cylindrical capacitor or concentric structure of spherical capacitors. | + | This situation can also be transferred to a coaxial structure of a cylindrical capacitor or the concentric structure of spherical capacitors. |
| <callout icon=" | <callout icon=" | ||
| - | In the case of perpendicular configuration, | + | |
| - | - A perpendicular | + | Conclusions: |
| - | - The electric field in the capacitor | + | - The layering |
| - | - Considering | + | - The electric field for dielectrics |
| - | - The parallel components | + | - We also found some results for the $E$ and $D$ fields |
| - | - The parallel components | + | - The tangential component |
| + | - The tangential component | ||
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 1501: | Zeile 1585: | ||
| However, some hints can be derived from the previous types of layering: | 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 parallel | + | * The tangential |
| * 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 parallel | + | * The tangential |
| <WRAP 30em> | <WRAP 30em> | ||
| Zeile 1516: | Zeile 1600: | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | Since $\vec{D} = \varepsilon_{0} \varepsilon_{r} \cdot \vec{E}$ the direction of the fields must be the same. \\ | + | 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 { { { \rm{tan} \alpha_1 } \over { \rm{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. |
| - | + | ||
| - | {{youtube> | + | |
| - | + | ||
| - | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | |
| Zeile 1543: | Zeile 1623: | ||
| </ | </ | ||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| - | <panel type=" | + | <panel type=" |
| < | < | ||
| Zeile 1552: | Zeile 1649: | ||
| </ | </ | ||
| - | Two parallel capacitor plates face each other with a distance $d_K = 10~\rm{mm}$. A voltage of $U = 3' | + | 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 ($\varepsilon_{r, | + | Parallel to the capacitor plates there is a glass plate ($\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 $E_{0,G} =12 ~\rm{kV/cm}$ must not exceed. | + | - What is the maximum thickness of the glass pane if the electric field $E_{ \rm 0, G} =12 ~{ \rm kV/cm}$ must not exceed? |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 1576: | Zeile 1673: | ||
| ====== Further links ====== | ====== Further links ====== | ||
| - | * [[https:// | + | * [[https:// |
| ====== additional Links ====== | ====== additional Links ====== | ||
| Zeile 1587: | Zeile 1684: | ||
| - | A really | + | A great introduction |
| examples: | examples: | ||