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184_notes:examples:week7_cylindrical_capacitor [2017/10/06 17:10] – [Finding the Capacitance of a Cylindrical Capacitor] tallpaul | 184_notes:examples:week7_cylindrical_capacitor [2018/06/19 15:37] – curdemma | ||
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+ | [[184_notes: | ||
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=====Finding the Capacitance of a Cylindrical Capacitor===== | =====Finding the Capacitance of a Cylindrical Capacitor===== | ||
Find the capacitance of a cylindrical capacitor. The structure of the capacitor is a cylindrical shell inside another cylindrical shell. The two shells become oppositely charged when the capacitor is connected to a power source. The length of the cylinders is $L$, and their radii are $a$ and $b$, with $a<b$. | Find the capacitance of a cylindrical capacitor. The structure of the capacitor is a cylindrical shell inside another cylindrical shell. The two shells become oppositely charged when the capacitor is connected to a power source. The length of the cylinders is $L$, and their radii are $a$ and $b$, with $a<b$. | ||
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===Approximations & Assumptions=== | ===Approximations & Assumptions=== | ||
* The cylinders are much longer than they are far from one another, i.e., $L >> a, b$. | * The cylinders are much longer than they are far from one another, i.e., $L >> a, b$. | ||
+ | * Cylinders are uniformly charged. | ||
===Representations=== | ===Representations=== | ||
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* We represent the situation below. | * We represent the situation below. | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
====Solution==== | ====Solution==== | ||
- | Let's start with node $A$. Incoming current | + | In order to find capacitance, |
- | $$I_{A\rightarrow B} = I_{out}-I_2 = I_{in}-I_2 = I_1-I_2 = 5 \text{ A}$$ | + | |
+ | In order to arrive at potential difference, | ||
+ | |||
+ | We will end up integrating in the radial direction (a convenient choice, as this is how the electric field is directed!) from $a$ to $b$, which will give us the potential difference between the two cylinders. | ||
+ | |||
+ | In order to find the electric field between the cylinders, we will use Gauss' Law. Below, we show a Gaussian surface that is cylindrical and fits inside the capacitor, with a radius | ||
+ | |||
+ | [{{ 184_notes: | ||
+ | |||
+ | We have done a [[184_notes: | ||
+ | |||
+ | The integral is over the entire Gaussian surface, upon which $\vec{E}\bullet | ||
- | We do a similar analysis for node $B$. Incoming current is $I_{A\rightarrow B}$, and outgoing current | + | We write $E(s)$ because the electric field magnitude depends only on the distance from the central vertical axis. Note, this notation is read as " |
- | $$I_{B\rightarrow D} = I_{out}-I_3 = I_{in}-I_3 | + | |
- | For node $C$, incoming current is $I_2$ and $I_3$. There is no outgoing current defined yet! $I_{C\rightarrow D}$ must be outgoing | + | At this point, we can apply Gauss' Law. We equate flux to the charge enclosed divided by $\epsilon_0$: |
- | $$I_{C\rightarrow D} = I_{out} = I_{in} = I_2+I_3 = 7 \text{ A}$$ | + | \begin{align*} |
+ | \frac{Q_{enclosed}}{\epsilon_0} &= \int\vec{E}\bullet \text{d}\vec{A} \\ | ||
+ | &= \int E(s)\text{d}A \\ | ||
+ | &= E(s) \int \text{d}A \\ | ||
+ | & | ||
+ | & | ||
+ | \end{align*} | ||
- | Lastly, | + | The last thing we need is $Q_{enclosed}$. This is simply the fraction of $Q$ that the Gaussian surface encloses. Since the height of the Gaussian cylinder |
- | $$I_{D\rightarrow battery} = I_{out} = I_{in} = I_{B\rightarrow D}+I_{B\rightarrow D} = 8 \text{ A}$$ | + | $$E(s)=\frac{Q_{enclosed}}{\epsilon_0}\frac{1}{2\pi sh} = \frac{Q}{2\pi\epsilon_0 L s}$$ |
- | Notice that $I_{D\rightarrow battery}=I_1$. | + | This is the magnitude |
+ | \begin{align*} | ||
+ | |\Delta V| &= \int_{r_i}^{r_f} \vec{E}\bullet \text{d}\vec{r} \\ | ||
+ | & | ||
+ | & | ||
+ | & | ||
+ | & | ||
+ | \end{align*} | ||
- | {{ 184_notes: | + | Finally, we can find the capacitance based on the charge of one cylinder and the potential we just found: |
+ | $$C=\frac{Q}{\Delta V} = \frac{2\pi\epsilon_0 L}{\log\left(\frac{b}{a}\right)}$$ | ||
+ | Notice that the capacitance no longer depends on the charge Q, but only on the shape/size of our capacitor ($a$, $b$, and $L$ ) |