electrical_engineering_and_electronics_1:block12 [MEXLE Wiki]

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====== Block 12 - Capacitors and Capacitance ======

===== Learning objectives =====
<callout>
After this 90-minute block, you can 
  - define a **capacitor** and **capacitance** $C$ and use it for an ideal plate capacitor, including unit checks $[C]={\rm F}$. 
  - relate fields and material. 
  - compute $C$ for key geometries (parallel plates, coaxial, spherical) and explain how $A$, $d$, $\varepsilon_{\rm r}$ scale $C$. 
</callout>

===== Preparation at Home =====

Well, again 
  * read through the present chapter and write down anything you did not understand.
  * Also here, there are some clips for more clarification under 'Embedded resources' (check the text above/below, sometimes only part of the clip is interesting). 

For checking your understanding please do the following exercises:
  * ...

===== 90-minute plan =====
  - Warm-up (8 min):
    - Quick recall quiz: $C=\dfrac{Q}{U}$, $\vec{D}=\varepsilon\vec{E}$, units of $E$ (${\rm V/m}$), $D$ (${\rm C/m^2}$). 
    - Estimate: how $C$ changes when $A$ doubles or $d$ halves (plate model). 
  - Core concepts & derivations (60 min):
    - From fields to $C$ (plate capacitor): $U=\int\vec{E}\cdot{\rm d}\vec{s}=E\,d$, $Q=\oint\vec{D}\cdot{\rm d}\vec{A}=D\,A$, $\Rightarrow C=\varepsilon_0\varepsilon_{\rm r}\dfrac{A}{d}$. Dimensional check: $[\varepsilon_0\varepsilon_{\rm r}A/d]={\rm F}$. 
    - Other geometries: coaxial and spherical capacitor formulas; where fields are highest (edge intuition kept qualitative). 
  - Practice (20 min):
    - Mini-calcs: (i) $C$ of given $A,d,\varepsilon_{\rm r}$; (ii) coaxial $C$ per length; (iii) allowable $U$ from $E_0$ and $d$; (iv) energy at given $U$. 
    - Discuss the provided “glass plate in capacitor” task.
  - Wrap-up (2 min): Summary box + pitfalls checklist; connect to next block (capacitor circuits).

===== Conceptual overview =====
<callout icon="fa fa-lightbulb-o" color="blue">
  - A **capacitor** is two conductors separated by a dielectric. It stores **charge** and **energy** in the electric field; no conduction current flows through the ideal dielectric. :contentReference[oaicite:15]{index=15}
  - **Capacitance** measures how much charge per volt: $C=\dfrac{Q}{U}$. For parallel plates, $C=\varepsilon_0\varepsilon_{\rm r}\dfrac{A}{d}$ → increase $A$ or $\varepsilon_{\rm r}$, decrease $d$ to raise $C$. :contentReference[oaicite:16]{index=16}
  - **Other geometries:** Closed forms exist for coaxial and spherical capacitors; useful as building blocks and for cables/sensors. :contentReference[oaicite:18]{index=18}
</callout>

===== Core content =====


==== Capacitor ====

  * A capacitor can "store" charges. The total charge of a two-plate capacitor is in general 0. 
  * From the mechanical point of view a capacitor has two electrodes (= conductive areas), which are separated by a dielectric (= non-conductor).
  * In a capacitor an electric field can be established without charge carriers moving internally.
  * The characteristic of the capacitor is the capacitance $C$.
  * In addition to the capacitance, every capacitor also has resistance and inductance. However, both of these are usually very small and are neglected for an ideal capacitor.
  * Examples of capacitor are
    * the electrical component "capacitor",
    * an open switch,
    * a wire related to ground,
    * a human being
$\rightarrow$ Thus, for any arrangement of two conductors separated by an insulating material, a capacitance can be specified.

==== Capacitance $C$ ====

The capacitance $C$ can be derived as follows:
  - A plate capacitor has a nearly homogenious field. \\ Therefore, it is given for the voltage: \\ $$U = \int \vec{E} {\rm d} \vec{s} = E \cdot l$$ \\ and hence \\ $$E= {{U}\over{l}}$$ or with $D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E $ $$D= \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{U}\over{l}}$$
  - Furthermore, the charge $Q$ can be given as $${Q= \rlap{\Large \rlap{\int_A} \int} \, \LARGE \circ} \; \vec{D} \cdot {\rm d} \vec{A} $$ The idealizion for the plate capacitor leads to: $$Q=D \cdot A$$.
  - 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_{ \rm r} $).
  - So it is reasonable to determine a proportionality factor ${{Q}\over{U}}$.

The capacitance $C$ of an idealized plate capacitor is defined as

\begin{align*}
\boxed{C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{A}\over{l}} =  {{Q}\over{U}}}
\end{align*}

Some of the main results here are:
  * The capacity can be increased by increasing the dielectric constant $\varepsilon_{ \rm r} $, given the the same geometry.
  * As near together the plates are as higher the capacity will be. 
  * As larger the area as higher the capacity will be.

This relationship can be examined in more detail in the following simulation:

--> Capacitor lab#

If the simulation is not displayed optimally, [[https://phet.colorado.edu/sims/cheerpj/capacitor-lab/latest/capacitor-lab.html?simulation=capacitor-lab&locale=de|this link]] can be used.

{{url>https://phet.colorado.edu/sims/cheerpj/capacitor-lab/latest/capacitor-lab.html?simulation=capacitor-lab&locale=de 900,800 noborder}}

<--

The <imgref ImgNr171> shows the topology of the electric field inside a plate capacitor.

<WRAP>
<imgcaption ImgNr171 | Topological situation inside a plate capacitor>
</imgcaption> <WRAP>
{{url>https://www.falstad.com/vector2de/vector2de.html?f=ChargedPlateDipole&fc=Floor%3A%20field%20magnitude&fl=Overlay%3A%20equipotentials&d=vectors&m=Mouse%20%3D%20Adjust%20Angle&st=20&vd=32&a1=63&a2=16&rx=77&ry=5&rz=107&zm=1.165 600,400 noborder}}
</WRAP></WRAP>

==== Symbols ====

  * The symbol of a general capacitor is given be two parallel lines nearby each other. \\
  * Since **electrolytic capacitors** can only withstand voltage in one direction, the **polarisation** is often shown by a curved electrode (US) or a unfilled one (EU). \\ Be aware that electrolytic capacitors can explode, once used in the wrong direction.

{{drawio>electrical_engineering_and_electronics_1:CapSymbols01.svg}}

==== Designs and types of capacitors ====

To calculate the capacitance of different designs, the definition equations of $\vec{D}$ and $\vec{E}$ are used. This can be viewed in detail, e.g., in [[https://www.youtube.com/watch?v=kAXg1xMkR_4&ab_channel=PatrickKaploo|this video]]. \\ Based on the geometry, different equations result (see also <imgref ImgNr17>).

<WRAP>
<imgcaption ImgNr17 | geometry of capacitors>
</imgcaption>
{{drawio>GeometryCapacitors.svg}}
</WRAP>

^Shape of the Capacitor^  Parameter                                                                            ^  Equation for the Capacity  ^
|plate capacitor       | area $A$ of plate \\ distance $l$ between plates                                      | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{A}\over{l}} \end{align*}|
|cylinder capacitor    |radius of outer conductor $R_{ \rm o}$ \\ radius of inner conductor $R_{ \rm i}$ \\ length $l$       | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot 2\pi {{l}\over{{\rm ln} \left({{R_{ \rm o}}\over{R_{ \rm i}}}\right)}} \end{align*}|
|spherical capacitor   |radius of outer spherical conductor $R_{ \rm o}$ \\ radius of inner spherical conductor $R_{ \rm i}$ | \begin{align*}C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot 4 \pi {{R_{ \rm i} \cdot R_{ \rm o}}\over{R_{ \rm o} - R_{ \rm i}}} \end{align*} |

~~PAGEBREAK~~ ~~CLEARFIX~~

<WRAP>
<imgcaption ImgNr116 | Structural shapes of Capacitors>
</imgcaption>
{{drawio>DesignsCapacitors.svg}}
</WRAP>

In <imgref ImgNr116> different designs of capacitors can be seen:
  - **{{wp>variable_capacitor|rotary variable capacitor}}** (also variable capacitor or trim capacitor).
    - 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 of $A$.
    - The size of the area is increased by the number of plates. Nevertheless, only small capacities are possible because of the necessary distance.
    - Air is usually used as the dielectric; occasionally, small plastic or ceramic plates are used to increase the dielectric constant.
  - **{{wp>Ceramic_capacitor#Multi-layer_ceramic_capacitors_(MLCC)|multilayer capacitor}}**
    - 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.
    - The multilayer ceramic capacitor is also referred to as KerKo or MLCC.
    - The variant shown in (2) is an SMD variant (surface mound device).
  - Disk capacitor
    - 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).
  - **{{wp>Electrolytic capacitor}}**, in German also referred to as  //Elko// for //__El__ektrolyt__ko__ndensator//
    - 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.
    - 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.
    - The electrolytic capacitor is built up wrapped, and often has a cross-shaped predetermined breaking point at the top for gas leakage.
  - **{{wp>film capacitor}}**, in German also referred to as  //Folko//, for //__Fol__ien__ko__ndensator//.
    - 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.
    - In the event of an electrical breakdown, the foil enables "self-healing": the metal coating evaporates locally around the breakdown. Thus the short-circuit is canceled again.
    - With some manufacturers, this type is referred to as  MKS (__M__metallized foil__c__capacitor, Polye__s__ter).
  - **{{wp>Supercapacitor}}** (engl. Super-Caps)
    - 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.
    - Supercapacitors can achieve very large capacitance values (up to the Kilofarad range), but only have a low maximum voltage

~~PAGEBREAK~~ ~~CLEARFIX~~

<WRAP 30em>
<imgcaption ImgNr17 | types of capacitors>
{{elektrotechnik_1:kondensatorensmd.jpg}}{{elektrotechnik_1:kondensatorentht.jpg}}
</imgcaption>
</WRAP>

In <imgref ImgNr17> are shown different capacitors:
  - The above two SMD capacitors
    - On the left a $100~{ \rm µF}$ electrolytic capacitor
    - On the right a $100~{ \rm nF}$ MLCC in the commonly used {{wp>Surface-mount_technology#Packages}} 0603 ($1.6~{ \rm mm}$ x $0.8~{ \rm mm}$)
  - 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 $+$
    - 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
Various conventions have been established for designating the capacitance value of a capacitor [[https://www.eit.edu.au/resources/different-types-of-capacitors/|various conventions]].

<callout icon="fa fa-exclamation" color="red" title="Note:">
  - 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.
  - 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/possible.

  - 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.

</callout>

~~PAGEBREAK~~ ~~CLEARFIX~~

===== Common pitfalls =====

  * **Mixing up geometry symbols.** Use $d$ (or $l$) strictly as **plate spacing** and $A$ as **active area** in $C=\varepsilon_0\varepsilon_{\rm r}\dfrac{A}{d}$. Check units to catch mistakes. 
  * **Forgetting the field relations.** $U=\int \vec{E}\cdot{\rm d}\vec{s}$ and $Q=\oint \vec{D}\cdot{\rm d}\vec{A}$; without them, layered-dielectric problems are guessed instead of solved. 
  * **Assuming conduction through the dielectric.** The apparent “current through a capacitor” is displacement-related; no charge carriers traverse the ideal dielectric.
  * **Real-part issues.** Ignoring polarity of electrolytics and tolerance spreads ($\pm 10~\%$ and more) causes design errors; pick suitable component types. 

===== Exercises =====


<panel type="info" title="Task 5.5.1 induced Charges"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>

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="xs" type="link" collapse="Loesung_5_5_1_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_5_5_1_Tipps" collapsed="true">
  * 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 $D$?
</collapse>

<button size="xs" type="link" collapse="Loesung_5_5_1_Endergebnis">{{icon>eye}} Result</button><collapse id="Loesung_5_5_1_Endergebnis" collapsed="true">
$Q = 10 ~{ \rm nC}$
</collapse>

</WRAP></WRAP></panel>


<panel type="info" title="Task 5.5.2 Manipulating a Capacitor I"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>

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/voltage), when the source is not disconnected?

<button size="xs" type="link" collapse="Loesung_5_5_2_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_5_5_2_Tipps" collapsed="true">
  * Consider the displacement flux through a surface around a plate
</collapse>

<button size="xs" type="link" collapse="Loesung_5_5_2_Endergebnis">{{icon>eye}} Result</button><collapse id="Loesung_5_5_2_Endergebnis" collapsed="true">
  - $U_1 = 1.9~{ \rm kV}$, $E_1 = 27~{ \rm kV/m}$ 
  - $U_1 = 190~{ \rm V}$, $E_1 = 2.7~{ \rm kV/m}$ 
</collapse>

</WRAP></WRAP></panel>


<panel type="info" title="Task 5.5.3 Manipulating a Capacitor II"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>

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 $U_{ \rm a}$.

#@HiddenBegin_HTML~1531T,Tipps~@#
  * Build a formula for the sum of the voltages first 
  * How is the voltage related to the electric field of a capacitor?
#@HiddenEnd_HTML~1531T,Tipps~@#

#@HiddenBegin_HTML~1531P,Path~@#

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  \cdot E_{\rm a} 
\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}  &= {{5'000 ~\rm V}\over{0.004 ~{\rm m} + 8 \cdot 0.002 ~{\rm m}}} \\
         &= 250 ~\rm{{kV}\over{m}}
\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*}
#@HiddenEnd_HTML~1531P,Path~@#


#@HiddenBegin_HTML~1531R,Results~@#
$U_{ \rm a} = 4~{ \rm kV}$, $U_{ \rm g} = 1 ~{ \rm kV}$ 
#@HiddenEnd_HTML~1531R,Results~@#


2. What would be the maximum allowed thickness of a glass plate, when the electric field in the air-gap shall not exceed $E_{ \rm max}=12~{ \rm kV/cm}$?

#@HiddenBegin_HTML~1532P,Path~@#
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  \cdot E_\rm a $. \\
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,g}}}   \cdot d_{\rm g} + E_{\rm a} \cdot d_{\rm a} \\
\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} \cdot ({{1}\over{\varepsilon_{\rm r,g}}} - 1) + d_0 \\
d_{\rm g}                &= { { {{U_0}\over{E_{\rm a} }} - d_0 } \over { {{1}\over{\varepsilon_{\rm r,g}}} - 1 } }     &= { { d_0 - {{U_0}\over{E_{\rm a} }} } \over { 1 - {{1}\over{\varepsilon_{\rm r,g}}} } }                   
\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)  {~\rm m}                   
\end{align*}
#@HiddenEnd_HTML~1532P,Path~@#


#@HiddenBegin_HTML~1532R,Results~@#
$d_{ \rm g} = 2.10~{ \rm mm}$
#@HiddenEnd_HTML~1532R,Results~@#

</WRAP></WRAP></panel>


<panel type="info" title="Task 5.5.4 Spherical capacitor"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>

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="xs" type="link" collapse="Loesung_1_5_4_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_1_5_4_Tipps" collapsed="true">
  * 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.
</collapse>

<button size="xs" type="link" collapse="Loesung_1_5_4_Endergebnis">{{icon>eye}} Result</button><collapse id="Loesung_1_5_4_Endergebnis" collapsed="true">
  - $C = 0.5~{ \rm pF}$ 
  - $C_{\infty} = 0.33~{ \rm pF}$
</collapse>

</WRAP></WRAP></panel>


<panel type="info" title="Task 5.5.5 Applying Gauss's law: Electric Field of a line charge"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>

{{youtube>NyRjHj2uy6k}}

</WRAP></WRAP></panel>


===== Embedded resources =====

<WRAP column half>
The background behind the dielectric constant $\varepsilon_{ \rm r} $ and the field is explained in the following video \\
{{youtube>rkntp3_cZl4}}
</WRAP>

<WRAP column half>
Electrolytic capacitors can explode! \\
{{youtube>sW0a9d_vWoc}}
</WRAP>
~~PAGEBREAK~~ ~~CLEARFIX~~