Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
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electrical_engineering_2:the_electrostatic_field [2023/03/23 17:34] ott |
electrical_engineering_2:the_electrostatic_field [2024/03/19 03:19] (aktuell) mexleadmin [Bearbeiten - Panel] |
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| Zeile 1: | Zeile 1: | ||
| - | ====== 1. The Electrostatic Field ====== | + | ====== 1 The Electrostatic Field ====== |
| < | < | ||
| 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, we will differentiate some terms: | + | First, we shall define certain |
| - | - **{{wp> | + | - **{{wp> |
| - | - **{{wp> | + | - **{{wp> |
| - | - **{{wp> | + | - **{{wp> |
| - | In this chapter, only electrostatics are considered. The magnetic fields are therefore | + | Only electrostatics is discussed in this chapter. |
| - | Also, electrodynamics is not considered | + | Furthermore, electrodynamics is not covered |
| ~~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 |
| < | < | ||
| Zeile 65: | Zeile 64: | ||
| </ | </ | ||
| - | 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> | ||
| 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 aware, that in English courses and literature $\vec{E}$ is simply | + | Be aware, that in English courses and literature $\vec{E}, $ is simply |
| </ | </ | ||
| 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 256: | Zeile 255: | ||
| 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 as an **area charge** when the charge is distributed over an area. \\ Examples of this are the floor or a plate of a capacitor. Again, an extended charged object can be considered | + | * It is spoken of as an **area charge** when the charge is distributed over an area. \\ Examples of this are the floor or a plate of a capacitor. Again, an extended charged object can be considered |
| * 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, | ||
| 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 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 the extended conductor) it would be accelerated along field lines. Thus its kinetic energy increases. Because the whole system of plates (for field generation) and charge however does not change its energetic state - thermodynamically the system is closed. From this follows: if the kinetic energy increases, the potential energy must decrease. | + | If this charge 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 345: | Zeile 349: | ||
| \end{align*} | \end{align*} | ||
| - | For a motion perpendicular to the field lines (i.e. from ${ \rm A}$ to ${ \rm C}$) no work is needed - so $W_{ \rm AC}=0$ results - because the formula above is only true for $F_C$ parallel to $s$. The motion perpendicular to the field lines is similar to the movement of weight in the gravitational field at the same height. Or more illustrative: | + | 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*} | ||
| Zeile 352: | Zeile 356: | ||
| The work $W_{ \rm AB}$ 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} | ||
| Zeile 387: | Zeile 391: | ||
| \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. | 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. | No matter how this charge is moved to a point ${ \rm B}$ and back again: as soon as it gets back to point ${ \rm A}$, it has the same energy again. | ||
| Zeile 420: | Zeile 424: | ||
| - Returning to the starting point from any point $A$ after a closed circuit, the __circuit voltage__ along the closed path is 0. \\ A closed path is mathematically expressed as a ring integral: \begin{align} U = \oint \vec{E} \cdot {\rm d} \vec{s} = 0 \end{align} | - 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 $\oint \vec{X} \cdot {\rm d} \vec{s}=0$ 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 as: \begin{align*} \boxed{W_{ \rm AB}= q \cdot U_{ \rm AB}} \end{align*} |
| </ | </ | ||
| Zeile 435: | Zeile 439: | ||
| 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 522: | Zeile 526: | ||
| </ | </ | ||
| - | |||
| - | ==== Tasks ==== | ||
| - | |||
| - | {{page> | ||
| - | {{page> | ||
| - | {{page> | ||
| =====1.4 Conductors in the Electrostatic Field ===== | =====1.4 Conductors in the Electrostatic Field ===== | ||
| Zeile 588: | Zeile 586: | ||
| 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 $U_{ \rm CG} = \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ with $\vec{E} ={{1} \over {4\pi\cdot\varepsilon}} \cdot {{q} \over {r^2}} \cdot \vec{e}_r $, where $\vec{e}_r$ is the unit vector pointing radially away, ${ \rm C}$ is a point at distance $r_0$ from the charge and ${ \rm G}$ is the ground potential at infinity. | - It is good to first calculate the potential field of a point charge. \\ For this calculate $U_{ \rm CG} = \int_{ \rm C}^{ \rm G} \vec{E} \cdot {\rm d} \vec{s}$ with $\vec{E} ={{1} \over {4\pi\cdot\varepsilon}} \cdot {{q} \over {r^2}} \cdot \vec{e}_r $, where $\vec{e}_r$ is the unit vector pointing radially away, ${ \rm C}$ is a point at distance $r_0$ from the charge and ${ \rm G}$ is the ground potential at infinity. | ||
| - Compare the field and the potentials of the different spherical conductors in <imgref ImgNr194>, | - Compare the field and the potentials of the different spherical conductors in <imgref ImgNr194>, | ||
| Zeile 594: | Zeile 592: | ||
| - 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, charges on the surface of the curvature to the left and to the right represent the same situation as in (a). For the next step, it is important that by this, the potentials of the left sphere with $q_1$ and $r_1$ and the right sphere with $q_2$ and $r_2$ are the same. | + | - Now, the situation in (c) shall be considered. Here, all components are conducting, i.e. the potentials on the surface are similar. Both spheres shall be considered to be as far away from each other, that they show an undisturbed field nearby their surfaces. In this case, charges on the surface of the curvature to the left and the right represent the same situation as in (a). For the next step, it is important that by this, the potentials of the left sphere with $q_1$ and $r_1$ and the right sphere with $q_2$ and $r_2$ are the same. |
| - Set up this equality formula based on the formula for the potential from question 1. | - 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 $r_2$. | ||
| Zeile 623: | Zeile 621: | ||
| * The field lines leave the surface again at right angles. Again, a parallel component would cause a charge shift in the metal. | * 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 656: | Zeile 654: | ||
| 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? | ||
| Zeile 669: | Zeile 667: | ||
| {{page> | {{page> | ||
| + | <wrap anchor # | ||
| <panel type=" | <panel type=" | ||
| Zeile 683: | Zeile 682: | ||
| - | --> Answer | + | # |
| $\varrho_2 < \varrho_3 < \varrho_1 < \varrho_4$ | $\varrho_2 < \varrho_3 < \varrho_1 < \varrho_4$ | ||
| Zeile 690: | Zeile 689: | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | <-- | + | |
| + | # | ||
| + | |||
| </ | </ | ||
| - | =====1.5 The Electric Displacement Field and Gauss' | + | =====1.5 The Electric Displacement Field and Gauss' |
| < | < | ||
| Zeile 722: | Zeile 725: | ||
| The effect can differ since the space can also " | 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 a vacuum a little more exactly. For this purpose, the plates shall first be brought into the field separately. | + | To investigate this situation, we want to consider two conductive plates (X) and (Y) with the area $\Delta A$ in the electrostatic field $\vec{E}$ in a vacuum a little more exactly. For this purpose, the plates shall first be brought into the field separately. |
| < | < | ||
| Zeile 730: | 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 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 at <imgref ImgNr12> b) and c), it is evident, that for larger plates (X) and (Y) more charges get displaced. So, in order to get a constant value by dividing displacement flux by the corresponding area. This leads to the **electric displacement field $D$** (sometimes also displacement flux density), which is defined as: | + | This movement is represented with the **displacement flux $\Psi$**. The displacement flux is given by the amount of moved charge $\Psi = n \cdot e = Q$, with the unit $[\Psi]= [Q] = 1~{ \rm C}$. When looking at <imgref ImgNr12> b) and c), it is evident, that for larger plates (X) and (Y) more charges get displaced. So, to get a constant value by dividing displacement flux by the corresponding area. This leads to the **electric displacement field $D$** (sometimes also displacement flux density), which is defined as: |
| \begin{align*} | \begin{align*} | ||
| Zeile 756: | Zeile 759: | ||
| * 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. | ||
| Zeile 775: | Zeile 778: | ||
| === 1. Problem: Inhomogenity → Solution: infinitesimal Area === | === 1. Problem: Inhomogenity → Solution: infinitesimal Area === | ||
| - | First, we shall still assume an observation surface perpendicular to the field lines, but an inhomogeneous field. In the inhomogeneous field, the magnitude of $D$ is no longer constant. | + | 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. |
| $Q = D\cdot A$ | $Q = D\cdot A$ | ||
| Zeile 790: | 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*} | ||
| Zeile 811: | 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} | ||
| Zeile 821: | Zeile 824: | ||
| The " | The " | ||
| This can be compared with a bordered swamp area with water sources and sinks: | This can be compared with a bordered swamp area with water sources and sinks: | ||
| - | * The sources in the marsh correspond to the positive charges, and the sinks to the negative charges. The formed water corresponds to the $D$-field. | + | * The sources in the marsh correspond to the positive charges and the sinks to the negative charges. The formed water corresponds to the $D$-field. |
| * The sum of all sources and sinks equals in this case just the water stepping over the edge. | * The sum of all sources and sinks equals in this case just the water stepping over the edge. | ||
| Zeile 836: | 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 847: | Zeile 850: | ||
| <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 877: | Zeile 880: | ||
| <panel type=" | <panel type=" | ||
| - | An ideal plate capacitor with a distance of $d_0 = 7 ~{ \rm mm}$ between the plates | + | An ideal plate capacitor with a distance of $d_0 = 7 ~{ \rm mm}$ between the plates |
| 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 $d_1 = 7 ~{ \rm cm}$. | ||
| Zeile 898: | Zeile 901: | ||
| An ideal plate capacitor with a distance of $d_0 = 6 ~{ \rm mm}$ between the plates and with air as dielectric ($\varepsilon_0=1$) is charged to a voltage of $U_0 = 5~{ \rm kV}$. | An ideal plate capacitor with a distance of $d_0 = 6 ~{ \rm mm}$ between the plates and with air as dielectric ($\varepsilon_0=1$) is charged to a voltage of $U_0 = 5~{ \rm kV}$. | ||
| - | The source remains connected to the capacitor. In the air gap between the plates, a glass plate with $d_{ \rm g} = 2 ~{ \rm mm}$ and $\varepsilon_{ \rm r} = 8$ is introduced parallel to the capacitor plates. | + | The source remains connected to the capacitor. In the air gap between the plates, a glass plate with $d_{ \rm g} = 4 ~{ \rm mm}$ and $\varepsilon_{ \rm r} = 8$ is introduced parallel to the capacitor plates. |
| - Calculate the partial voltages on the glas $U_{ \rm g}$ and on the air gap $U_{ \rm a}$. | - Calculate the partial voltages on the glas $U_{ \rm g}$ and on the air gap $U_{ \rm a}$. | ||
| Zeile 1007: | Zeile 1010: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| <callout icon=" | <callout icon=" | ||
| - | - The material constant $\varepsilon_{ \rm r}$ 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 $\varepsilon_{ \rm r}$ 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 $\varepsilon_{ \rm r}$ depends on the possibility | + | Suppose |
| </ | </ | ||
| Zeile 1044: | Zeile 1047: | ||
| * 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, | ||
| Zeile 1091: | Zeile 1094: | ||
| * 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 1123: | Zeile 1126: | ||
| 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 1151: | Zeile 1154: | ||
| ^Shape of the Capacitor^ | ^Shape of the Capacitor^ | ||
| |plate capacitor | |plate capacitor | ||
| - | |cylinder capacitor | + | |cylinder capacitor |
| |spherical capacitor | |spherical capacitor | ||
| Zeile 1165: | Zeile 1168: | ||
| - **{{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 1171: | Zeile 1174: | ||
| - 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 a 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. aluminum, tantalum or niobium. | - Different metals can be used as the oxidized electrode, e.g. aluminum, tantalum or niobium. | ||
| Zeile 1182: | Zeile 1185: | ||
| - 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 wrapped and often has a cross-shaped predetermined breaking point at the top for gas leakage. | - The electrolytic capacitor is built up wrapped and often has a cross-shaped predetermined breaking point at the top for gas leakage. | ||
| - | - **{{wp> | + | - **{{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, | + | - With some manufacturers, |
| - **{{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. | ||
| Zeile 1218: | Zeile 1221: | ||
| <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_{ \rm r} \cdot {{A}\over{l}} $ . The capacitance value is given. | - The calculation of the capacitance is usually __not__ via $C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{A}\over{l}} $ . The capacitance value is given. | ||
| Zeile 1324: | Zeile 1327: | ||
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ U_k = const} | + | \boxed{ U_k = {\rm const.}} |
| \end{align*} | \end{align*} | ||
| </ | </ | ||
| Zeile 1361: | Zeile 1364: | ||
| By the end of this section, you will be able to: | By the end of this section, you will be able to: | ||
| - | - recognize the different layering of dielectrics and distinguish between a perpendicular and a lateral layering | + | - recognize the different layering of dielectrics and distinguish between a normal (perpendicular) and a tangential (lateral) layering |
| - | - know which quantity remains constant | + | - know which quantity remains constant |
| - | - know the constant quantity for 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 1371: | Zeile 1373: | ||
| - | Up to now was assumed only one dielectric | + | 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 1385: | Zeile 1389: | ||
| ~~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: | ||
| Zeile 1439: | Zeile 1443: | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| {{url> | {{url> | ||
| Zeile 1447: | Zeile 1451: | ||
| <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 $E_{ \rm n}$ 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 $D_{ \rm n}$ is continuous at the interface: $D_{ \rm n1} = D_{ \rm n2}$ | - The normal component of the flux density $D_{ \rm n}$ is continuous at the interface: $D_{ \rm n1} = D_{ \rm n2}$ | ||
| Zeile 1457: | Zeile 1461: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Perpendicular Configuration | + | ==== 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 1483: | Zeile 1487: | ||
| 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 1495: | Zeile 1499: | ||
| <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 1510: | Zeile 1516: | ||
| * Electric field $\vec{E}$: | * Electric field $\vec{E}$: | ||
| * The normal component $E_{ \rm n}$ is discontinuous at the interface: $\varepsilon_{ \rm r1} \cdot E_{ \rm n1} = \varepsilon_{ \rm r2} \cdot E_{ \rm n2}$ | * The normal component $E_{ \rm n}$ is discontinuous at the interface: $\varepsilon_{ \rm r1} \cdot E_{ \rm n1} = \varepsilon_{ \rm r2} \cdot E_{ \rm n2}$ | ||
| - | * The parallel | + | * The tangential |
| * Electric displacement flux density $\vec{D}$: | * Electric displacement flux density $\vec{D}$: | ||
| * The normal component $D_{ \rm n}$ is continuous at the interface: $ D_{ \rm n1} = D_{ \rm 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 1527: | Zeile 1533: | ||
| </ | </ | ||
| - | Since $\vec{D} = \varepsilon_{0} \varepsilon_{ \rm r} \cdot \vec{E}$ the direction of the fields must be the same. \\ | + | Since $\vec{D} = \varepsilon_{0} \varepsilon_{ \rm r} \cdot \vec{E} |
| Using the fields, we can now derive the change in the angle: | Using the fields, we can now derive the change in the angle: | ||
| Zeile 1534: | Zeile 1540: | ||
| \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 1551: | Zeile 1553: | ||
| </ | </ | ||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| - | <panel type=" | + | <panel type=" |
| < | < | ||
| Zeile 1561: | Zeile 1580: | ||
| Two parallel capacitor plates face each other with a distance $d_{ \rm K} = 10~{ \rm mm}$. A voltage of $U = 3' | Two parallel capacitor plates face each other with a distance $d_{ \rm K} = 10~{ \rm mm}$. A voltage of $U = 3' | ||
| - | Parallel to the capacitor plates there is a glass plate ($\varepsilon_{ \rm r,G}=8$) with a thickness $d_{ \rm G} = 3~{ \rm mm}$ in the capacitor. | + | Parallel to the capacitor plates there is a glass plate ($\varepsilon_{ \rm r, G}=8$) with a thickness $d_{ \rm G} = 3~{ \rm mm}$ in the capacitor. |
| - Calculate the partial voltages $U_{ \rm G}$ in the glass and $U_{ \rm A}$ in the air gap. | - Calculate the partial voltages $U_{ \rm G}$ in the glass and $U_{ \rm A}$ in the air gap. | ||
| - | - What is the maximum thickness of the glass pane if the electric field $E_{ \rm 0,G} =12 ~{ \rm kV/cm}$ must not exceed? | + | - What is the maximum thickness of the glass pane if the electric field $E_{ \rm 0, G} =12 ~{ \rm kV/cm}$ must not exceed? |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 1595: | Zeile 1614: | ||
| - | A really | + | A great introduction to electric and magnetic fields (but a bit too deep for this course) can be found in the [[https:// |
| examples: | examples: | ||