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184_notes:examples:week7_charging_capacitor [2017/10/06 00:16] – [Looking at a Capacitor as it's Charging] tallpaul | 184_notes:examples:week7_charging_capacitor [2017/10/09 15:37] – dmcpadden | ||
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=====Looking at a Capacitor as it's Charging===== | =====Looking at a Capacitor as it's Charging===== | ||
- | Suppose you have a parallel plate capacitor that is disconnected from any power source and is discharged. At time $t=0$, the capacitor is connected to a battery. | + | Suppose you have a parallel plate capacitor that is disconnected from any power source and is discharged. At time $t=0$, the capacitor is connected to a battery. |
===Facts=== | ===Facts=== | ||
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===Lacking=== | ===Lacking=== | ||
- | * Graphs of $I$, $Q$, $\Delta V$$. | + | * Graphs of $I$, $Q$, $\Delta V$. |
===Approximations & Assumptions=== | ===Approximations & Assumptions=== | ||
- | * The power source is connected correctly with respect to the capacitor and there are no other circuit elements (except for the wire). | + | * There are no other circuit elements (except for the wire). |
* The wire itself has a small resistance, just so we do not have infinite current at $t=0$. | * The wire itself has a small resistance, just so we do not have infinite current at $t=0$. | ||
* Practically speaking, the capacitor becomes "fully charged" | * Practically speaking, the capacitor becomes "fully charged" | ||
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====Solution==== | ====Solution==== | ||
- | Let's start with node $A$. Incoming current | + | Before the capacitor |
- | $$I_{A\rightarrow B} = I_{out}-I_2 = I_{in}-I_2 = I_1-I_2 = 5 \text{ A}$$ | + | |
- | We do a similar analysis for node $B$. Incoming current is $I_{A\rightarrow B}$, and outgoing current is $I_3$. Since $I_{A\rightarrow B}=5 \text{ A}$ and $I_3=4 \text{ A}$, we need $I_{B\rightarrow D}$ to be outgoing to balance. To satisfy the Node Rule, we set | + | {{ 184_notes: |
- | $$I_{B\rightarrow D} = I_{out}-I_3 = I_{in}-I_3 = I_{A\rightarrow B}-I_3 = 1 \text{ A}$$ | + | |
- | For node $C$, incoming | + | Slightly after time $t=0$, charges start to move (setting up a current) and the capacitor begins to charge. Due to the high electric field at the very beginning, the charge flows very quickly at first. Since current |
- | $$I_{C\rightarrow D} = I_{out} = I_{in} = I_2+I_3 = 7 \text{ A}$$ | + | |
- | Lastly, we look at node $D$. Incoming current is $I_{B\rightarrow D}$ and $I_{C\rightarrow D}$. Since there is no outgoing current defined yet, $I_{D\rightarrow battery}$ must be outgoing to balance. To satisfy the Node Rule, we set | + | {{ 184_notes: |
- | $$I_{D\rightarrow battery} = I_{out} = I_{in} = I_{B\rightarrow D}+I_{B\rightarrow D} = 8 \text{ A}$$ | + | |
- | Notice that $I_{D\rightarrow battery}=I_1$. This will always be the case for currents going in and out of the battery (approximating | + | During this time, the current is smaller than at the beginning, but it is also decreasing at a slower rate, since the plates are not charging |
- | {{ 184_notes:6_nodes_with_arrows.png?300 |Circuit with Nodes}} | + | {{ 184_notes:7_cap_time_3.png?300 |Capacitor |
+ | |||
+ | Based on what we have argued so far, the graph of the current in the circuit should look something like this: | ||
+ | |||
+ | {{ 184_notes: | ||
+ | |||
+ | Notice that we also used the charge of the capacitor in our argument for why the current drops off like it does. We basically argue that the charge graph will look like a flipped current graph: rapidly accumulating charge at first, and gradually slowing down in accumulation until equilibrium (" | ||
+ | |||
+ | {{ 184_notes: | ||
+ | |||
+ | Lastly, we wish to produce a graph of the potential difference between the plates. We can pull from a [[184_projects: | ||
+ | $$E_{plate} = \frac{Q_{plate}}{2\epsilon_0 A_{plate}}$$ | ||
+ | Notice that we do not necessarily need to remember this equation -- the electric field of everything we have looked at so far has been directly proportional to the charge (not $Q^2$ or $Q^5$ or anything strange). | ||
+ | |||
+ | We also remember that potential difference across a constant electric field is just the electric field times the distance: | ||
+ | $$\Delta V = E\cdot \Delta d$$ | ||
+ | Since the distance between the plates is unchanging and the electric field simply scales with $Q$, we can expect the voltage graph to have the same shape as the charge graph: | ||
+ | |||
+ | {{ 184_notes: | ||
+ | |||
+ | That's it! We can even do one more check to be sure we are happy with our result. Notice the the capacitance of a capacitor can be represented as the charge on a plate (or something else, depending on the shape of the capacitor) divided by the potential difference between the plates (or whatever): $$C = \frac{Q}{\Delta V}$$ | ||
+ | |||
+ | We know that capacitance is a constant, since it is an intrinsic property of the capacitor' |