Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_and_electronics_1:block11 [2025/10/31 15:34] – mexleadmin | electrical_engineering_and_electronics_1:block11 [2025/11/08 14:27] (aktuell) – mexleadmin | ||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| - | ====== Block 11 — Influence and displacement field ====== | + | ===== Block 11 — Influence and Displacement Field ====== |
| ===== Learning objectives ===== | ===== Learning objectives ===== | ||
| < | < | ||
| After this 90-minute block, you can | After this 90-minute block, you can | ||
| - | * ... | + | * explain **electrostatic induction** on conductors and argue why the interior of a conductor is field-free (Faraday cage). |
| + | * distinguish the **electric field strength** $\vec{E}$ from the **electric displacement flux density** $\vec{D}$ and state $ \vec{D} = \varepsilon \vec{E} = \varepsilon_0 \varepsilon_{\rm r}\vec{E}$. | ||
| + | * apply **Gauss’s law** for the displacement field to simple closed surfaces to relate enclosed charge $Q$ and flux $\oint \vec{D}\cdot {\rm d}\vec{A}$. | ||
| + | * determine $E(r)$ for parallel-plate and coaxial geometries starting from $\vec{D}$, then using $\vec{E}=\vec{D}/ | ||
| + | * reason about **surface charge density** $\varrho_A = \Delta Q/\Delta A$ and the normal field at conductor surfaces. | ||
| + | * use typical **relative permittivities** $\varepsilon_{\rm r}$ to estimate field reduction in dielectrics. | ||
| + | * interpret **dielectric strength** $E_0$ (breakdown) and reason about its impact on design limits (safe $E$, spacing, material choice). | ||
| </ | </ | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ===== Preparation at Home ===== | ===== Preparation at Home ===== | ||
| Zeile 14: | Zeile 21: | ||
| For checking your understanding please do the following exercises: | For checking your understanding please do the following exercises: | ||
| - | * ... | + | * 5.4.1 |
| + | * 5.4.4 | ||
| + | * 5.4.5 | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ===== 90-minute plan ===== | ===== 90-minute plan ===== | ||
| - | - Warm-up (x min): | + | - Warm-up (10 min): |
| - | - .... | + | - One-minute recap quiz (Block 10): equipotentials, |
| - | - Core concepts & derivations (x min): | + | - Demo: conductor in external field → Faraday cage effect (refer to the embedded sim in this block). |
| - | - ... | + | - Core concepts & derivations (45 min): |
| - | - Practice (x min): ... | + | - Induction on conductors: charge displacement, |
| - | - Wrap-up (x min): Summary box; common | + | - Polarization of dielectrics; |
| + | - Definitions: | ||
| + | - Worked derivations via $\vec{D}$: | ||
| + | * Parallel plates: $D=Q/A \; | ||
| + | * Coaxial cylinders: $D(r)=Q/ | ||
| + | - Material data: typical $\varepsilon_{\rm r}$; concept of **dielectric strength** $E_0$ and safe design margins. | ||
| + | - Practice (25 min): | ||
| + | - Short board tasks using pillbox surfaces to find $\varrho_A$ on a conductor. | ||
| + | - Mixed-dielectric capacitor slice: split voltages via constant $D$. | ||
| + | - Guided use of the embedded sims to observe field/ | ||
| + | - Wrap-up (10 min): | ||
| + | - Summary box (key formulas, when to start with $D$ vs. $E$). | ||
| + | - Common | ||
| + | - Preview to Block 12 (capacitors from field viewpoint). | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ===== Conceptual overview ===== | ===== Conceptual overview ===== | ||
| <callout icon=" | <callout icon=" | ||
| - | - ... | + | - **Conductors in electrostatics: |
| + | - **Dielectrics (polarization): | ||
| + | - **Displacement field & Gauss’s law:** for any closed surface, the flux of $\vec{D}$ equals the enclosed charge: $Q=\oint \vec{D}\cdot{\rm d}\vec{A}$. \\ Choose the surface to exploit symmetry, get $\vec{D}$ first, then $\vec{E}$ via material law. | ||
| + | - **Permittivity: | ||
| + | - **Design limit:** when $|E|$ exceeds the **dielectric strength** $E_0$, breakdown occurs → current flows. \\ Safe design keeps $|E|\ll E_0$ by material choice and geometry (spacing, shaping to avoid high curvature). | ||
| </ | </ | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ===== Core content ===== | ===== Core content ===== | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ==== Electric Field inside of a conductor ==== | ==== Electric Field inside of a conductor ==== | ||
| Zeile 54: | Zeile 84: | ||
| </ | </ | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ==== Electrostatic Induction ==== | ==== Electrostatic Induction ==== | ||
| Zeile 122: | Zeile 152: | ||
| - The internal measurable electric field is compensated | - The internal measurable electric field is compensated | ||
| - | To have an uncompensated field in the following the **electric displacement flux density $\vec{D}$** is introduced. | + | To have an uncompensated field in the following the **electric displacement flux density $\vec{D}$** is introduced. |
| The electric displacement flux density is only focusing on the __cause__ of the electric fields. | The electric displacement flux density is only focusing on the __cause__ of the electric fields. | ||
| - | As we have seen, its effect can differ since the space can also " | + | As we have seen, its effect can differ since the space can also " |
| The electric displacement flux density is only related to the causing charges $Q$. Thie relationship is shown in the following. | The electric displacement flux density is only related to the causing charges $Q$. Thie relationship is shown in the following. | ||
| Zeile 146: | Zeile 176: | ||
| </ | </ | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ==== Dielectric Constant (Permittivity) ==== | ||
| + | |||
| + | Dielectric materials reduce the electric field inside them. How much die field is reduced is given by a material dependent constant the **dielectric constant** or **permittivity** $\varepsilon_r$. It is unitless and a ratio related to the unhindered field in vacuum. | ||
| + | |||
| + | \begin{align*} | ||
| + | {{D}\over{E}} = \varepsilon = \varepsilon_0 \cdot \varepsilon_{ \rm r} \\ | ||
| + | \boxed{D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E} | ||
| + | \end{align*} | ||
| + | |||
| + | Some values of the relative permittivity $\varepsilon_{ \rm r}$ for dielectrics are given in <tabref tab01> | ||
| + | |||
| + | |||
| + | <WRAP 30em> | ||
| + | |||
| + | < | ||
| + | ^ material | ||
| + | | air | $\rm 1.0006$ | ||
| + | | paper | $\rm 2$ | | ||
| + | | PE, PP | $\rm 2.3$ | | ||
| + | | PS | $\rm 2.5$ | | ||
| + | | hard paper | $\rm 5$ | | ||
| + | | glass | $\rm 6...8$ | ||
| + | | water ($20~°{ \rm C}$) | $\rm 80$ | | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ==== Typical Geometries ==== | ||
| + | |||
| + | The " | ||
| + | This shall be shown with the two most common geometries (which are the only one necessary for this course). | ||
| + | |||
| + | <WRAP group> | ||
| + | <WRAP half column> | ||
| + | === Field of a parallel Plates === | ||
| + | \\ | ||
| + | * Nearly all of the field is between the plate (see <imgref ImgNr294> | ||
| + | * All of the $D$-field of the charges is between the plates, and therefore through the area $A$ of the plates (see <imgref ImgNr294> | ||
| + | * Given $D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E$, the eletric field $E$ is: $$ E = {{Q}\over{ \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot A}} $$\\ | ||
| + | |||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | | ||
| + | |||
| + | {{drawio> | ||
| + | </ | ||
| + | <WRAP half column> | ||
| + | === Field of a coaxial cylindrical Plates === | ||
| + | \\ | ||
| + | * All of the field is between the plate (see <imgref ImgNr295> | ||
| + | * All of the $D$-field of the charges on the inner plate penetrates through any cylintrical area $A(l,r) = 2 \pi \cdot l \cdot r$. \\ → The $D$-Field is given as: $$ D = {{Q}\over{A(l, | ||
| + | * Again given $D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E$, the eletric field $E$ is: $$ E = {{Q}\over{ \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot 2 \pi \cdot l \cdot r}} $$\\ | ||
| + | |||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | | ||
| + | |||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ==== Dielectric strength of dielectrics ==== | ==== Dielectric strength of dielectrics ==== | ||
| Zeile 172: | Zeile 266: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ===== Common pitfalls ===== | ===== Common pitfalls ===== | ||
| - | * ... | + | * Mixing up **cause** and **effect**: using $\oint \vec{E}\cdot{\rm d}\vec{A}$ to count charge. Use **$\vec{D}$** for Gauss’s law with charge; convert to $\vec{E}$ only via $\vec{E}=\vec{D}/ |
| + | * Forgetting that the **interior of a conductor is field-free** in electrostatics and that $E$ is **normal** to an ideal conducting surface (no tangential $E$ on the surface). | ||
| + | * Assuming induced charges fill the **volume** of a conductor. They reside on the **surface**; | ||
| + | * Ignoring that **$D$ is continuous** in the normal direction across simple dielectric interfaces when no free surface charge is present; consequently, | ||
| + | * Treating $\varepsilon_{\rm r}$ as a constant in all contexts. Real materials can be frequency/ | ||
| + | * Checking breakdown with voltage only. The limit is on **field** $E$; always relate geometry (e.g., plate spacing, curvature) to $E$ and compare to **$E_0$** with units (e.g., $\,{\rm kV/mm}$). | ||
| ===== Exercises ===== | ===== Exercises ===== | ||
| Zeile 229: | Zeile 328: | ||
| # | # | ||
| - | |||
| - | </ | ||
| - | |||
| - | |||
| - | <panel type=" | ||
| - | |||
| - | A plate capacitor with a distance of $d = 2 ~{ \rm cm}$ between the plates and with air as dielectric ($\varepsilon_{ \rm r}=1$) gets charged up to $U = 5~{ \rm kV}$. | ||
| - | In between the plates, a thin metal foil with the area $A = 45~{ \rm cm^2}$ is introduced parallel to the plates. | ||
| - | |||
| - | Calculate the amount of the displaced charges in the thin metal foil. | ||
| - | |||
| - | <button size=" | ||
| - | * What is the strength of the electric field $E$ in the capacitor? | ||
| - | * Calculate the displacement flux density $D$ | ||
| - | * How can the charge $Q$ be derived from $D$? | ||
| - | </ | ||
| - | |||
| - | <button size=" | ||
| - | $Q = 10 ~{ \rm nC}$ | ||
| - | </ | ||
| - | |||
| - | </ | ||
| - | |||
| - | |||
| - | <panel type=" | ||
| - | |||
| - | An ideal plate capacitor with a distance of $d_0 = 7 ~{ \rm mm}$ between the plates gets charged up to $U_0 = 190~{ \rm V}$ by an external source. | ||
| - | The source gets disconnected. After this, the distance between the plates gets enlarged to $d_1 = 7 ~{ \rm cm}$. | ||
| - | |||
| - | - What happens to the electric field and the voltage? | ||
| - | - How does the situation change (electric field/ | ||
| - | |||
| - | <button size=" | ||
| - | * Consider the displacement flux through a surface around a plate | ||
| - | </ | ||
| - | |||
| - | <button size=" | ||
| - | - $U_1 = 1.9~{ \rm kV}$, $E_1 = 27~{ \rm kV/ | ||
| - | - $U_1 = 190~{ \rm V}$, $E_1 = 2.7~{ \rm kV/ | ||
| - | </ | ||
| - | |||
| - | </ | ||
| - | |||
| - | |||
| - | <panel type=" | ||
| - | |||
| - | An ideal plate capacitor with a distance of $d_0 = 6 ~{ \rm mm}$ between the plates and with air as dielectric ($\varepsilon_0=1$) is charged to a voltage of $U_0 = 5~{ \rm kV}$. | ||
| - | The source remains connected to the capacitor. In the air gap between the plates, a glass plate with $d_{ \rm g} = 4 ~{ \rm mm}$ and $\varepsilon_{ \rm r} = 8$ is introduced parallel to the capacitor plates. | ||
| - | |||
| - | 1. Calculate the partial voltages on the glas $U_{ \rm g}$ and on the air gap $U_{ \rm a}$. | ||
| - | |||
| - | # | ||
| - | * Build a formula for the sum of the voltages first | ||
| - | * How is the voltage related to the electric field of a capacitor? | ||
| - | # | ||
| - | |||
| - | # | ||
| - | |||
| - | The sum of the voltages across the glass and the air gap gives the total voltage $U_0$, and each individual voltage is given by the $E$-field in the individual material by $E = {{U}\over{d}}$: | ||
| - | \begin{align*} | ||
| - | U_0 &= U_{\rm g} + U_{\rm a} \\ | ||
| - | &= E_{\rm g} \cdot d_{\rm g} + E_{\rm a} \cdot d_{\rm a} | ||
| - | \end{align*} | ||
| - | |||
| - | The displacement field $D$ must be continuous across the different materials since it is only based on the charge $Q$ on the plates. | ||
| - | \begin{align*} | ||
| - | D_{\rm g} &= D_{\rm a} \\ | ||
| - | \varepsilon_0 \varepsilon_{\rm r, g} \cdot E_{\rm g} &= \varepsilon_0 | ||
| - | \end{align*} | ||
| - | |||
| - | Therefore, we can put $E_\rm a= \varepsilon_{\rm r, g} \cdot E_\rm g $ into the formula of the total voltage and rearrange to get $E_\rm g$: | ||
| - | \begin{align*} | ||
| - | U_0 &= E_{\rm g} \cdot d_{\rm g} + \varepsilon_{\rm r, g} \cdot E_{\rm g} \cdot d_{\rm a} \\ | ||
| - | &= E_{\rm g} \cdot ( d_{\rm g} + \varepsilon_{\rm r, g} \cdot d_{\rm a}) \\ | ||
| - | |||
| - | \rightarrow E_{\rm g} &= {{U_0}\over{d_{\rm g} + \varepsilon_{\rm r, g} \cdot d_{\rm a}}} | ||
| - | \end{align*} | ||
| - | |||
| - | Since we know that the distance of the air gap is $d_{\rm a} = d_0 - d_{\rm a}$ we can calculate: | ||
| - | \begin{align*} | ||
| - | E_{\rm g} &= {{5' | ||
| - | & | ||
| - | \end{align*} | ||
| - | |||
| - | By this, the individual voltages can be calculated: | ||
| - | \begin{align*} | ||
| - | U_{ \rm g} &= E_{\rm g} \cdot d_\rm g &&= 250 ~\rm{{kV}\over{m}} \cdot 0.004~\rm m &= 1 ~{\rm kV}\\ | ||
| - | U_{ \rm a} &= U_0 - U_{ \rm g} &&= 5 ~{\rm kV} - 1 ~{\rm kV} & | ||
| - | |||
| - | \end{align*} | ||
| - | # | ||
| - | |||
| - | |||
| - | # | ||
| - | $U_{ \rm a} = 4~{ \rm kV}$, $U_{ \rm g} = 1 ~{ \rm kV}$ | ||
| - | # | ||
| - | |||
| - | |||
| - | 2. What would be the maximum allowed thickness of a glass plate, when the electric field in the air-gap shall not exceed $E_{ \rm max}=12~{ \rm kV/cm}$? | ||
| - | |||
| - | # | ||
| - | Again, we can start with the sum of the voltages across the glass and the air gap, such as the formula we got from the displacement field: $D = \varepsilon_0 \varepsilon_{\rm r, g} \cdot E_{\rm g} = \varepsilon_0 | ||
| - | Now we shall eliminate $E_\rm g$, since $E_\rm a$ is given in the question. | ||
| - | \begin{align*} | ||
| - | U_0 & | ||
| - | &= {{E_\rm a}\over{\varepsilon_{\rm r, | ||
| - | \end{align*} | ||
| - | |||
| - | The distance $d_\rm a$ for the air is given by the overall distance $d_0$ and the distance for glass $d_\rm g$: | ||
| - | \begin{align*} | ||
| - | d_{\rm a} = d_0 - d_{\rm g} | ||
| - | \end{align*} | ||
| - | |||
| - | This results in: | ||
| - | \begin{align*} | ||
| - | U_0 &= {{E_{\rm a}}\over{\varepsilon_{\rm r,g}}} \cdot d_{\rm g} + E_{\rm a} \cdot (d_0 - d_{\rm g}) \\ | ||
| - | {{U_0}\over{E_{\rm a} }} &= {{1}\over{\varepsilon_{\rm r,g}}} \cdot d_{\rm g} + d_0 - d_{\rm g} \\ | ||
| - | & | ||
| - | d_{\rm g} &= { { {{U_0}\over{E_{\rm a} }} - d_0 } \over { {{1}\over{\varepsilon_{\rm r,g}}} - 1 } } & | ||
| - | \end{align*} | ||
| - | |||
| - | With the given values: | ||
| - | \begin{align*} | ||
| - | d_{\rm g} &= { { 0.006 {~\rm m} - {{5 {~\rm kV} }\over{ 12 {~\rm kV/cm}}} } \over { 1 - {{1}\over{8}} } } &= { {{8}\over{7}} } \left( { 0.006 - {{5 }\over{ 1200}} } \right) | ||
| - | \end{align*} | ||
| - | # | ||
| - | |||
| - | |||
| - | # | ||
| - | $d_{ \rm g} = 2.10~{ \rm mm}$ | ||
| - | # | ||
| - | |||
| - | </ | ||
| - | |||
| - | |||
| - | <panel type=" | ||
| - | |||
| - | Two concentric spherical conducting plates set up a spherical capacitor. | ||
| - | The radius of the inner sphere is $r_{ \rm i} = 3~{ \rm mm}$, and the inner radius from the outer sphere is $r_{ \rm o} = 9~{ \rm mm}$. | ||
| - | |||
| - | - What is the capacity of this capacitor, given that air is used as a dielectric? | ||
| - | - What would be the limit value of the capacity when the inner radius of the outer sphere goes to infinity ($r_{ \rm o} \rightarrow \infty$)? | ||
| - | <button size=" | ||
| - | * What is the displacement flux density of the inner sphere? | ||
| - | * Out of this derive the strength of the electric field $E$ | ||
| - | * What ist the general relationship between $U$ and $\vec{E}$? Derive from this the voltage between the spheres. | ||
| - | </ | ||
| - | |||
| - | <button size=" | ||
| - | - $C = 0.5~{ \rm pF}$ | ||
| - | - $C_{\infty} = 0.33~{ \rm pF}$ | ||
| - | </ | ||
| </ | </ | ||