Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_and_electronics_1:block09 [2025/10/20 02:20] – angelegt mexleadmin | electrical_engineering_and_electronics_1:block09 [2025/11/01 00:14] (aktuell) – [Block 09 - Force on charges and electric field strength] mexleadmin | ||
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| Zeile 1: | Zeile 1: | ||
| - | ====== Block 09 — Force on charges | + | ====== Block 09 - Force on Charges |
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ===== Learning objectives ===== | ===== Learning objectives ===== | ||
| < | < | ||
| - | | + | By the end of this section, you will be able to: |
| + | | ||
| + | * Explain and apply the **superposition principle** for forces and fields from multiple charges. | ||
| + | * Compute $|\vec{E}|$ for a **point charge** (Coulomb force), identify $\varepsilon$ and check dimensions. | ||
| + | * Determine the force on a charge in an electrostatic field by applying Coulomb' | ||
| + | * The force vector in coordinate representation | ||
| + | * The magnitude of the force vector | ||
| + | * The angle of the force vector | ||
| + | * The direction of the force | ||
| + | * Determine a force vector by superimposing several force vectors using vector calculus. | ||
| </ | </ | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ===== Preparation at Home ===== | ||
| + | |||
| + | And again: | ||
| + | * Please read through the following chapter. | ||
| + | * Also here, there are some clips for more clarification under ' | ||
| + | |||
| + | For checking your understanding please do the following exercise: | ||
| + | * 1.2.3 | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| ===== 90-minute plan ===== | ===== 90-minute plan ===== | ||
| - | - Warm-up (5–10 min): | + | - Warm-up (8–10 min): |
| - | - Recall / Quick quiz ... | + | - Quick recall |
| - | - Core concepts | + | - Dimensions check: show $1~{\rm N/C}=1~{\rm V/m}$. |
| - | - ... | + | - Concept build & demonstrations |
| - | - Practice (10–20 min): ... | + | - Cause–field–effect chain: charges $\Rightarrow \vec{E}(\vec{x}) \Rightarrow \vec{F}=q\, |
| - | - Wrap-up (5 min): ... | + | - Coulomb law $\Rightarrow$ point-charge field magnitude and direction. |
| + | - **Superposition** for two/three charges; vector addition. | ||
| + | - **Field lines**: definition, drawing rules, sources/ | ||
| + | - **Homogeneous vs. inhomogeneous** fields; conductor boundary facts (perpendicular $\vec{E}$, interior field-free). | ||
| + | - Guided simulation (20–25 min) | ||
| + | - Place single and multiple charges; measure $\vec{E}$ at points. | ||
| + | - Practice (10–15 min) | ||
| + | - net field on-axis of two charges; quick peer check. | ||
| + | - Wrap-up (5 min): | ||
| + | - Summary map: charges → $\vec{E}$ → $\vec{F}$; key properties and units. | ||
| ===== Conceptual overview ===== | ===== Conceptual overview ===== | ||
| <callout icon=" | <callout icon=" | ||
| - | - ... | + | - **Fields separate cause and effect**: charges set up a state in space (the field) that exists whether or not a test charge is present. |
| + | - **Coulomb field of a point charge:** $\displaystyle \vec{E}(\vec{r})=\frac{1}{4\pi\varepsilon}\frac{Q}{r^2}\, | ||
| + | - The **electric field** is a **vector field** $\vec{E}(\vec{x})$; | ||
| + | - **Point charge** model: inverse-square law; direction is radial, outward for $Q>0$, inward for $Q<0$. | ||
| + | - **Superposition** holds: for multiple sources, $\vec{E}_{\rm total}=\sum_k \vec{E}_k$ (vector sum at the same point). | ||
| </ | </ | ||
| Zeile 23: | Zeile 57: | ||
| ===== Core content ===== | ===== Core content ===== | ||
| - | ==== 1st sub-chapter | + | ==== Electric Effects |
| - | ... | + | Every day life teaches us that there are various charges and their effects. The image <imgref ImgNr01> depicts a chargeable body that can be charged through charge separation between the sole and the floor. The movement of the foot generates a negative surplus charge in the body, which progressively spreads throughout the body. A current can flow even through the air if a pointed portion of the body (e.g., a finger) is brought into close proximity to a charge reservoir with no extra charges. |
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| - | ==== 2nd sub-chapter ==== | + | First, we shall define certain terms: |
| + | - **{{wp> | ||
| + | - **{{wp> | ||
| + | | ||
| - | ... | + | Only electrostatics is discussed in this chapter. For the time being, magnetic fields are thus excluded. |
| + | Furthermore, | ||
| - | ==== n'th sub-chapter | + | ==== Fields |
| - | ... | + | The concept of a field will now be briefly discussed in more detail. |
| + | - The introduction of the field distinguishes the cause from the effect. | ||
| + | - The field in space is caused by the charge $Q$. | ||
| + | - As a result of the field, the charge $q$ in space feels a force. | ||
| + | - This distinction is brought up again in this chapter. \\ It is also fairly obvious in electrodynamics at high frequencies: | ||
| + | -There are different-dimensional fields, just like physical quantities: | ||
| + | - In a **scalar field**, each point in space is assigned a single number. \\ For example, | ||
| + | - a temperature field $T(\vec{x})$ on a weather map or in an object | ||
| + | - a pressure field $p(\vec{x})$ | ||
| + | - Each point in space in a **vector field** is assigned several numbers in the form of a vector. This reflects the action as it occurs along the spatial coordinates. \\ As an example. | ||
| + | - gravitational field $\vec{g}(\vec{x})$ pointing to the object' | ||
| + | - electric field $\vec{E}(\vec{x})$ | ||
| + | - magnetic field $\vec{H}(\vec{x})$ | ||
| + | - A tensor field is one in which each point in space is associated with a two- or more-dimensional physical quantity - that is, a tensor. Tensor fields are useful in mechanics (for example, the stress tensor), but they are not required in electrical engineering. | ||
| + | Vector fields are defined as follows: | ||
| + | - Effects along spatial axes $x$, $y$ and $z$ (Cartesian coordinate system). | ||
| + | - Effect in magnitude and direction vector (polar coordinate system) | ||
| - | ===== Common pitfalls ===== | + | <panel type=" |
| - | * ... | + | |
| - | ... | + | |
| - | + | ||
| - | ===== Exercises ===== | + | |
| - | ==== Quick checks ==== | + | Place a negative charge $Q$ in the middle of the simulation and turn off the electric field. The latter is accomplished by using the hook on the right. The situation is now close to reality because a charge appears to have no effect at first glance. |
| - | # | + | A sample charge $q$ is placed near the existing charge $Q$ for impact analysis (in the simulation, the sample charge is called " |
| - | # | + | |
| - | Here is a simple exercise | + | < |
| + | </ | ||
| + | {{url> | ||
| + | Take a charge ($+1~{ \rm nC}$) and position it. \\ Measure the field across a sample charge (a sensor). | ||
| - | # | + | </ |
| - | Here is the solution of the Exercise 1 | + | ~~PAGEBREAK~~ ~~CLEARFIX~~ |
| + | <callout icon=" | ||
| + | - Fields describe a physical state of space. | ||
| + | - Here, a physical quantity | ||
| + | - The electrostatic field is described by a vector field. | ||
| + | </ | ||
| - | # | + | ==== The electric Field ==== |
| - | # | + | |
| + | We had already considered the charge as the central quantity of electricity in [[block02]] and recognized it as a multiple of the elementary charge. | ||
| + | Now, we want to determine the electric field of charges. For this, a measurement of its magnitude and direction is now required. The **Coulomb force** between two charges $Q_1$ and $Q_2$ is: | ||
| - | # | + | \begin{align*} |
| - | # | + | F_C = {{{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1 \cdot Q_2} \over {r^2}}} |
| + | \end{align*} | ||
| - | Here is another simple exercise ... | + | The force on a (fictitious) sample charge $q$ is now considered to obtain a measure of the magnitude of the electric field. |
| - | # | + | \begin{align*} |
| + | F_C &= {{{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1 \cdot q} \over {r^2}}} \\ | ||
| + | &= \underbrace{{{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}}}_\text{=independent of q} \cdot q \\ | ||
| + | \end{align*} | ||
| - | Here is the solution | + | As a result, the left part is a measure of the magnitude |
| - | # | + | <WRAP centeralign> |
| - | # | + | $E = {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \quad$ with $[E]={{[F]}\over{[q]}}=1 ~{ \rm {N}\over{As}}=1 ~{ \rm {N\cdot m}\over{As \cdot m}} = 1 ~{ \rm {V \cdot A \cdot s}\over{As \cdot m}} = 1 ~{ \rm {V}\over{m}}$ |
| + | </ | ||
| - | ==== Longer exercises ==== | + | The result is therefore |
| + | \begin{align*} | ||
| + | \boxed{F_C | ||
| + | \end{align*} | ||
| - | # | + | The unit of $E$ is $\rm 1 {{N}\over{As}} = 1 {{V}\over{m}} $ |
| - | # | + | |
| - | Here is a longer exercise ... | + | <callout icon=" |
| - | # | + | - 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. | ||
| + | </ | ||
| - | Here is the solution of the Exercise 1 | + | <callout icon=" |
| - | # | + | At a measuring point $P$, a charge $Q$ produces an electric field $\vec{E}(Q)$. This electric field is given by |
| - | # | + | - the magnitude $|\vec{E}|=\Bigl| {{1} \over {4\pi\cdot\varepsilon}} \cdot {{Q_1} \over {r^2}} \Bigl| $ and |
| + | - the direction of the force $\vec{F_C}$ experienced by a sample charge on the measurement point $P$. This direction is indicated by the unit vector $\vec{e_{ \rm r}}={{\vec{F_C}}\over{|F_C|}}$ in that direction. | ||
| + | Be aware that in English courses and literature $\vec{E} $ is simply referred to as the electric field, and the electric field strength is the magnitude $|\vec{E}|$. In German notation, the // | ||
| + | </ | ||
| - | Here are the Exercises given by {{tagtopic>...}} | + | The direction of the electric field is switchable in <imgref ImgNr02> via the " |
| - | ===== Embedded resources ===== | + | ==== Direction of the Coulomb force and Superposition |
| - | <WRAP column half> | + | In the case of the force, only the direction has been considered so far, e.g., direction towards the sample charge. For future explanations, |
| - | Here are the youtube resource 1 | + | |
| - | {{youtube>...}} | + | 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: $\vec{F_3}= \vec{F_{31}}+\vec{F_{32}}$. \\ | ||
| + | <imgref ImgNr06> Fig. (d) shows that for a charged surface, the force on a charge on top of this surface is always perpendicular to the surface itself. | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| </ | </ | ||
| + | |||
| + | ==== Energy required to Displace a Charge in the electric Field ==== | ||
| + | |||
| + | Now we want to see, whether we can derive the required energy to displace a charge in the electric field. \\ | ||
| + | |||
| + | Since we know the force on a charge in an electrical field $\vec{E}$ (= Coulomb-Force $\vec{F}_C = q \cdot \vec{E} $), we can borrow some relationships from mechanics for the energy $\Delta W$: | ||
| + | |||
| + | \begin{align*} | ||
| + | \Delta W = \int \vec{F} d\vec{r} = q \int \vec{E} d\vec{r} | ||
| + | \end{align*} | ||
| + | |||
| + | Looks familiar? Maybe not on the first sight. But we already had defined the fraction of the energy difference per charge ${{\Delta W}\over{q}}$ as voltage $U$! \\ | ||
| + | Therefore: | ||
| + | |||
| + | \begin{align*} | ||
| + | \boxed{U = \int \vec{E} d\vec{r} } | ||
| + | \end{align*} | ||
| + | |||
| + | We will apply this relationship in multiple of the upcoming blocks. | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | |||
| + | ===== Common pitfalls ===== | ||
| + | * Treating **force** and **field** as the same thing; forgetting $\vec{F}=q\, | ||
| + | * Mixing units (${\rm N}$, ${\rm C}$, ${\rm V}$, ${\rm m}$): not recognizing $1~{\rm N/C}=1~{\rm V/m}$. | ||
| + | * Drawing **field lines** as closed loops or allowing them to **intersect** (source field: start at $+$, end at $-$; no crossings). | ||
| + | * Ignoring **vector addition** in superposition (adding magnitudes instead of vectors). | ||
| + | * Assuming field exists **only** when a test charge is present; the field is a property of space due to sources. | ||
| + | * Using point-charge formulas too near extended objects; not identifying **homogeneous vs. inhomogeneous** regions. | ||
| + | * Forgetting conductor boundary facts: lines must be **perpendicular** to ideal conducting surfaces; interior **$|\vec{E}|=0$** in electrostatics. | ||
| + | | ||
| + | ===== Exercises ===== | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | {{youtube> | ||
| + | |||
| + | </ | ||
| + | |||
| + | |||
| + | {{page> | ||
| + | {{page> | ||
| + | {{page> | ||
| + | |||
| + | <panel type=" | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | ===== Embedded resources ===== | ||
| + | |||
| + | < | ||
| + | The online book ' | ||
| + | * Chapter [[https:// | ||
| + | * Chapter [[https:// | ||
| + | * Chapter [[https:// | ||
| + | * Chapter [[https:// | ||
| + | </ | ||
| <WRAP column half> | <WRAP column half> | ||
| - | Here are the youtube resource 2 | + | Intro into electric field |
| - | {{youtube> | + | {{youtube> |
| </ | </ | ||
| - | ... | + | |