Calculate the equivalent capacitance between points a and b...

Calculate the equivalent capacitance between points $a$ and $b$ in Figure $\mathrm{P} 26.73$ . Notice that this system is not a simple series or parallel combination. Suggestion: Assume a potential difference $\Delta V$ between points $a$ and $b .$ Write expressions for $\Delta V_{a b}$ in terms of the charges and capacitances for the various possible pathways from $a$ to $b$ and require conservation of charge for those capacitor plates that are connected to each other.

Key Concepts

Conservation of Charge

The principle of charge conservation dictates that the total charge in isolated parts of a circuit remains constant. When analyzing networks with interconnected capacitors, charge conservation is applied to the nodes where capacitor plates are connected, ensuring that the sum of charge entering a node equals the sum exiting.

Equivalent Capacitance

This concept involves replacing a complex network of capacitors with a single capacitor that has the same effect on the overall circuit. It is the capacitance value that would store the same amount of charge for a given potential difference across its terminals as the entire network does.

Capacitor Network Analysis

In networks that are not simply series or parallel, the analysis requires breaking down the circuit into different pathways and identifying relationships between voltages and charges. This systematic approach often involves writing equations for the potential differences along multiple paths and imposing constraints from charge conservation.

Potential Difference and Voltage Drops

This concept involves expressing the voltage difference between two points as the sum of individual voltage drops across components. In capacitor networks, setting up these expressions is fundamental to ensuring that the net voltage between the points remains consistent regardless of the chosen path.

Capacitor Relationship (Q = CV)

This fundamental relationship states that the charge stored on a capacitor is directly proportional to its capacitance and the voltage across it. It serves as a basis for formulating the equations needed to relate the charges on various capacitors in the network.

Transcript

00:02 We have a bunch of capacitors set up something like this, with one here, one in the middle, like this.

00:21 So this is between points a and b.

00:26 And now i'm going to label these c1, c2, and c3.

00:29 So this is c1, c2, c3, and then these two are the same as c1 and c2.

00:37 So as the problem suggests, i'm going to write the voltage from a to b well let's see i can take a couple different paths i can take this path i can take this path or i can take this path so for the middle path that would be i can do it as a sum of voltages for each capacitor so the voltage is charge over capacitance so q2 over c2 plus q3 over c3 plus q1 over c1 and then i can do the same for the bottom path so that's q1 over c1 plus another q1 over c1 i can do the same for the top path which is q2 over c2 plus another q2 over c2 so all these three expressions have to be equivalent because they all sum to the total voltage so if i look at the first and second equation, i can deduce that q1 over c1 because i have this and this, so then this q1 over c1 has to equal q2 over c2 plus q3 over c3, that combination.

02:16 And then looking at the first and the last equation, we have q2 over c2 and q2 over c2, so this one has to be equal to these two.

02:27 So q2 over c2 equals q3 over c3 plus q1 over c1.

02:36 So now i'm going to solve for q3 over c3 in both cases.

02:41 So for the first one, this is going to be q1 over c1 minus q2 over c2.

02:51 And for the second one, we get q2 over c2 minus q1 over c1.

03:01 But this is just the opposite of the first one.

03:04 So this equals minus q3 over c3.

03:08 So if q3 over c3 equals minus q3 over c3, that means q3 over c3, the only way that can be true is if it's zero.

03:19 And that's the voltage across c3, which is this middle capacitor...