Professor Anderson

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The other type of capacitor circuit that we need to deal with is of course when we add them in series. So let's go back to our circuit. And now we're going to take capacitor C1 and we're gonna add capacitor C2 in series. One after the other along the wire. Let's think about where are all the charges. On the top plate of C1, there is going to be positive Q. If there is positive Q on that top plate, then there must be minus Q on the bottom plate. The whole capacitor has to stay neutral. If there is minus Q on the bottom plate, then there has to be positive Q on the second plate of capacitor, on the top plate of Capacitor 2. And therefore there is minus Q on the bottom plate of C2. There is a voltage drop across Capacitor 1 which is V1. There is a voltage drop across Capacitor 2 which is V2. But they all have the same amount of charge on them. So, Kirchhoff's laws told us the following. If we start at some point in a circuit and we go around the circuit, the sum of the voltage increases has to be equal to the sum of the voltage drops. So Kirchhoff's law says the following. V going up minus V1 going down minus V2 going down has to be equal to zero. This is Kirchhoff's circuit law, okay? Or his loop rule. If I move the V1 and the V2 over to the other side, then this just becomes V equals V1 plus V2. All right, but these things have the same charge on each capacitor. And, there is a relationship between charge and voltage. The same charge on C1 and C2. And we know what that is, right? Q equals CV. So if Q equals CV, then we must have V equals Q over C. Therefore, I can write V1 equals Q over C1. V2 equals Q over C2. And now I can take those two things and stick them right into this equation. So what does this equation become? It becomes Q over C1 plus Q over C2. But we also know what V is in the case of the equivalent circuit. In the equivalent circuit, we have one capacitor. It was in series and so it's going to have a value C sub s. So I could say V is equal to Q over C sub s. So this whole line right here equals Q over C sub s. And now you see exactly what happens. Right? If I look at this equation right here, I have Q over C sub 1 plus Q over C sub 2 equals Q or C sub s. I can divide the right and left of that equation by Q and I get 1 over C sub s equals 1 over C1 plus 1 over C2. This is the rule for series capacitors. When you add capacitors in series, you add their inverses. Whereas in parallel, we added their direct capacitances. So in this case, capacitance is in fact going to decrease. When I add two capacitors in series, the resultant capacitance is going to be less than both of them. The whole capacitance decreases. When I added them in parallel, the capacitance increased. All right so let's see if we can summarize the rule for resistors and capacitors.

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The other type of capacitor circuit that we need to deal with is of course when we add them in series. So let's go back to our circuit. And now we're going to take capacitor C1 and we're gonna add capacitor C2 in series. One after the other along the wire. Let's think about where are all the charges. On the top plate of C1, there is going to be positive Q. If there is positive Q on that top plate, then there must be minus Q on the bottom plate. The whole capacitor has to stay neutral. If there is minus Q on the bottom plate, then there has to be positive Q on the second plate of capacitor, on the top plate of Capacitor 2. And therefore there is minus Q on the bottom plate of C2. There is a voltage drop across Capacitor 1 which is V1. There is a voltage drop across Capacitor 2 which is V2. But they all have the same amount of charge on them. So, Kirchhoff's laws told us the following. If we start at some point in a circuit and we go around the circuit, the sum of the voltage increases has to be equal to the sum of the voltage drops. So Kirchhoff's law says the following. V going up minus V1 going down minus V2 going down has to be equal to zero. This is Kirchhoff's circuit law, okay? Or his loop rule. If I move the V1 and the V2 over to the other side, then this just becomes V equals V1 plus V2. All right, but these things have the same charge on each capacitor. And, there is a relationship between charge and voltage. The same charge on C1 and C2. And we know what that is, right? Q equals CV. So if Q equals CV, then we must have V equals Q over C. Therefore, I can write V1 equals Q over C1. V2 equals Q over C2. And now I can take those two things and stick them right into this equation. So what does this equation become? It becomes Q over C1 plus Q over C2. But we also know what V is in the case of the equivalent circuit. In the equivalent circuit, we have one capacitor. It was in series and so it's going to have a value C sub s. So I could say V is equal to Q over C sub s. So this whole line right here equals Q over C sub s. And now you see exactly what happens. Right? If I look at this equation right here, I have Q over C sub 1 plus Q over C sub 2 equals Q or C sub s. I can divide the right and left of that equation by Q and I get 1 over C sub s equals 1 over C1 plus 1 over C2. This is the rule for series capacitors. When you add capacitors in series, you add their inverses. Whereas in parallel, we added their direct capacitances. So in this case, capacitance is in fact going to decrease. When I add two capacitors in series, the resultant capacitance is going to be less than both of them. The whole capacitance decreases. When I added them in parallel, the capacitance increased. All right so let's see if we can summarize the rule for resistors and capacitors.