For the capacitor network shown in Figure 18.49 , the potential difference across ab is 36 V. Fi

For the capacitor network shown in Figure 18.49 , the...

Key Concepts

Charge Distribution and Voltage Division

Charge distribution in a capacitor network and the division of voltage across each capacitor depend on the configuration of the network. In series circuits, all capacitors share the same charge, but the voltage across each capacitor is inversely proportional to its capacitance. In parallel circuits, each capacitor experiences the same voltage, and the stored charge is directly proportional to its capacitance. These principles help in determining the individual parameters of each capacitor in a network.

Equivalent Capacitance

Equivalent capacitance is the concept of replacing a network of capacitors with a single capacitor that has the same effect on the overall circuit. For capacitors in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals of each capacitor’s capacitance, while for capacitors in parallel, the equivalent capacitance is the sum of the individual capacitances. This simplification is essential when calculating charge distribution and energy storage in complex circuits.

Energy Stored in Capacitors

The energy stored in a capacitor is determined by the formula U = 1/2 CV^2, where C is the capacitance and V is the voltage across the capacitor. This equation is essential for calculating both the total energy stored in a network and the energy stored in each individual capacitor. Understanding this relationship is key to analyzing the performance and efficiency of circuits that incorporate capacitors.

Capacitance

Capacitance is a measure of a capacitor's ability to store electric charge per unit voltage. It is defined as the ratio of the electric charge on each conductor to the potential difference between them. This property is fundamental in analyzing circuits with capacitors, helping to determine how much charge is stored and how the voltage is distributed across different components.

Series and Parallel Capacitor Combinations

In circuits, capacitors can be connected in series, in parallel, or in complex combinations. In a series configuration, the same charge flows through each capacitor, and the overall voltage is divided among them inversely based on their capacitances. In contrast, in a parallel configuration, the voltage across each capacitor is the same, while the total stored charge is the sum of the charges on each capacitor. Understanding these rules is crucial for simplifying and analyzing capacitor networks.

Transcript

00:01 Okay, so in this problem we're given two capacitors in series with one another like this.

00:08 And they have 150 and 120 nanofarad capacitances.

00:16 So 150 nanofarad and then 120 nanofarad.

00:23 And what else are we given? and then the total difference is 36 volts from a to b.

00:31 So here's a, here's b, v delta v is 120 volts.

00:41 And the question is, then, what's the total charge stored in the network? so to do that, let's find the equivalent capacitance.

00:53 So a, we want to get q, close question mark.

00:57 So c equivalent, if they're in series, it adds inverse.

01:01 So we're going to do 1 over 150 plus 1 over 120.

01:09 Both of those are nanofarads, but it's a little painstaking to write.

01:13 So i'm not going to write, at least small things.

01:17 So i'm going to plug that into a calculator to see what i get.

01:28 So that's 66 .7 nanofarads.

01:38 And then we can figure out the charts.

01:40 It's going to be put on the system because q is equal to cv.

01:45 C is q or v, so q is cv.

01:48 And so we just want to take this 66 .7 and multiply it by 120 volts to get 8 ,000 times 10 to the minus 9.

02:00 So let's see, times 10 to the minus 9, 8 times 10 to the minus 6.

02:08 So that's going to be 8 microculomes.

02:16 And for part b, we want to get the truce.

02:20 Charge on each capacitor.

02:22 So each capacitor is going to be, it's going to get this eight microculem because the charge will come out from the battery onto here.

02:32 And that's going to put eight microculems here, polarize this to be minus eight, polarized this little metal piece connecting the plates to make this plus eight.

02:43 And then this is going to be minus eight.

02:47 So we could, yeah, i can kind of imagine.

02:50 And so therefore it's going to be eight microculems on each because they're in parallel.

03:02 And c asks the total energy stored and the network.

03:12 So what we want to do is do, i think you can actually let's like test this.

03:20 So you could do the charge on the energy on each and add them.

03:25 You could also use the equivalent capacitance.

03:28 So i'm just going to test this idea out.

03:31 So let's do this simpler one first.

03:33 I think it's going to be you should use the equivalent capacitance and then the voltage.

03:40 So let's use u as one -half cv squared.

03:47 And then i'll use c equivalent.

03:51 So plugging that into a calculator, let's see.

03:55 I get, oh, so let's see, one half, 6 .7, 10 of the minus 9, and then b squared...

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