For the system of four capacitors shown in Figure P26.19, find (a) the total energy stored in th

step 1 So we have a configuration of capacitors that looks like this. We have four of them in a circuit

For the system of four capacitors shown in Figure P26.19,...

Key Concepts

Energy Storage in Capacitors

This concept relies on the well-known formula E = 1/2 * C * V^2, which quantifies the amount of energy stored in a capacitor given its capacitance (C) and the voltage (V) across it. It is a foundational idea in electromagnetism and circuit theory that is used to analyze energy storage on both a single capacitor level and within more complex networks.

Equivalent Capacitance in Networks

When capacitors are interconnected in series or parallel arrangements, their overall effect can be simplified by calculating an equivalent capacitance. This approach is essential in circuit analysis as it allows a complex network of capacitors to be represented as a single capacitor, making it easier to determine the overall energy storage and study the system’s behavior under applied voltages.

Series and Parallel Combinations

The organization of capacitors in series or parallel drastically affects both the voltage distribution and the charge on each capacitor. Recognizing how series circuits cause voltage drops to divide among capacitors while parallel circuits maintain the same voltage across each component is critical for accurate energy calculations in circuits, as these configurations shape the effective capacitance and energy distribution.

Energy Distribution in Capacitor Networks

Even though the overall energy stored in the network can be calculated using the equivalent capacitance, the energy on each individual capacitor is determined by its specific voltage and the local charge distribution. Comparing the sum of the individually computed energies to the energy computed from the equivalent capacitance can reveal differences that are important for understanding energy partitioning and the nuances of energy conservation within a network.

Transcript

00:01 So we have a configuration of capacitors that looks like this.

00:05 We have four of them in a circuit.

00:13 And this one is 3, this one is 6, they're all on microferids, this one is 2, and this is 4, so those are the capacitance.

00:30 So i can combine the top branch and the bottom branch in series, and then add the 2 up to get the total equivalent capacitance.

00:38 So the top branch we have when capacitors are combined in series, we have three times six, and then divide by three plus six.

00:57 And that gives us two microferds.

01:03 And then for the bottom branch, we have two times four divided by two plus four, which is 1 .33 microferds.

01:18 So now we have something looks like this with one capacitor on the top.

01:27 This is 2, and this is 1 .33.

01:42 And to find the equivalent capacitance now, we add the 2 up, so 2 plus 1 .33, which is 3 .33 microferids.

01:58 We also have a battery that's 90 volts.

02:03 And to find the total energy in the circuit, take 1 .5 cv squared.

02:17 And this is the equivalent capacitance.

02:21 So that's going to be 1 .5 times 3 .33, which is rounded, times 90 squared, which comes out to 1 .35 times 10 to the minus 2, 13 .5 millajoules.

02:38 This is joules.

02:44 Now, to find the energy in each individual capacitor, well, the important thing i know here is that the voltage on the top and bottom branch of this circuit is the same.

02:57 So then i don't have to worry about voltages, and i can just calculate the energy in terms of charge and capacitance.

03:05 I also know that the charge through each of these pairs of capacitors on the top and bottom is going to be the same through the individual one in each pair.

03:15 It doesn't matter because we have the same current going through them.

03:20 So i can use this bottom circuit to say that the charge...

03:28 Let me label these, actually.

03:30 This will be c1, c2, c3, and c4.

03:38 So the charge on 1 is going to be the same as the charge on 2, and it's just this.

03:46 So that's going to be, we have the capacitance times the voltage.

03:56 And like i said, the voltage is going to be the same through either branch.

03:59 So the voltage is always 90...