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For the capacitor network shown in Figure 18.49 , the potential difference across ab is 36 V. Find (a) the total charge stored in this network, (b) the charge on each capacitor, (c) the total energy stored in the network, (d) the energy stored in each capacitor, and (e) the potential difference across each capacitor.

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Physics 101 Mechanics

Physics 102 Electricity and Magnetism

Chapter 18

Electric Potential and Capacitanc

Kinetic Energy

Potential Energy

Energy Conservation

Electric Charge and Electric Field

Gauss's Law

Electric Potential

Capacitance and Dielectrics

University of Michigan - Ann Arbor

University of Sheffield

McMaster University

Lectures

13:02

In physics, potential energy is the energy possessed by a body or a system due to its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units (SI) is the joule (J). One joule is the energy expended (or work done) in applying a force of one newton through a distance of one metre (1 newton metre). The term potential energy was introduced by the 19th century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concepts of potentiality. Potential energy is associated with forces that act on a body in a way that the work done by these forces on the body depends only on the initial and final positions of the body, and not on the specific path between them. These forces, that are called potential forces, can be represented at every point in space by vectors expressed as gradients of a scalar function called potential. Potential energy is the energy of an object. It is the energy by virtue of a position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force that works against the force field of the potential. This work is stored in the field, which is said to be stored as potential energy.

18:38

In physics, electric flux is a measure of the quantity of electric charge passing through a surface. It is used in the study of electromagnetic radiation. The SI unit of electric flux is the weber (symbol: Wb). The electric flux through a surface is calculated by dividing the electric charge passing through the surface by the area of the surface, and multiplying by the permittivity of free space (the permittivity of vacuum is used in the case of a vacuum). The electric flux through a closed surface is zero, by Gauss's law.

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For the capacitor network …

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07:36

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05:30

05:56

05:44

03:29

03:31

01:56

04:54

For the capacitor net- wor…

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24.30 For the capacitor ne…

05:28

05:38

Okay, so this problem were given to capacitors in serious with one another like this. And, um, they have 1 51 120 Nano fair. Odd capacitance is so on 50 you know, fair ad and then 1 20 Nano fair. Odd. And what else are we given? And then the total difference is 36 folds from A to B. So here is a Here's b felt to be is ah, 120 volts. And the question is, then what's the total charge stored in the network? So to do that, let's find the equivalent capacitance. So a we want to get que question, mark. So see, equivalent, if they're in serious and adds inverse, So we're gonna do one over 1 50 plus one over 1 20 both of those air Naret nano fair odds when it's little painstaking too, right? So I'm not gonna write I least small things. So I'm gonna put got into a calculator to see what I got. So that 66.7, you know, Franz, and then we can figure out the charge is gonna be put on the system because, Q, this sequel to see VC is cure be so Q s c B. And so we just want to take this 66.7 and multiply it by 120 folds to get 8000 times 10 to the minus nine. So, see, thanks. 10 to the minus nine, eight times 10 to the minus six. So that's gonna be eat micro cool ums. And for a part B, we want to get the charge in each camp, ask capacitors. So, um, each capacitor is gonna be it's going to get this eat my group, whom, because the charge will come out from the battery onto here. And that's gonna put eight micro Coombs here polarizes to be minus eight, polarized this little metal piece connecting the plates to make this plus eight. And then this is gonna be minus eight. I'm sort of You could Yeah, I kind of imagine um And so So therefore, it's gonna be eight micro cool ums on each concerns because they're in parallel and see asks, um, the total energy storage and the network. So, um, we want to do is due. I think you can actually looks like test this so you could do the charge on the energy on each and add them. But I think you could also use the equivalent capacitance. So I'm just gonna test this idea out. Um, so let's do this simpler one for so I think it's gonna be you just used the equivalent capacitance and then the voltage. So let's use, um, you is 1/2 c v squared and then all you see equivalent. So plugging that into a calculator, Um, see, I got it. Oh, so let's see. 6.77 minus nine and the square. So it's 1 20 squared, so it's 4.8 times 10 to the minus four jewels. And then I can also imagine just adding the the um adding the, um sorry, adding the energy of each capacitor. That's we don't really know the voltage on each, but we do know the queue. So that's used the one that involves Q and C. So it's gonna be 1/2 Q squared oversee, so it's gonna be 1/2 and then they each have the same cues. I'll just use that Q squared over. See, um, plus one over two C one plus one over 1/2 Q squared over see too. Oh, then you can kind of see that That's equal to 1/2 Q squared over, see equivalent. And then, um I mean, for fun. We can just plug this in so we do 1/2 que square and oversee equivalent. So 0.5 times it's squared and then see equivalent 66.7. Don't worry about exponents, and we do in fact get the same answer. So there's two ways to approach yet I think this way is a little easier, But are there any of their questions in this one? Oh, yes, there are more questions. So that was D Oh, I just found the energy total. It's that was see, And we need to find the energy in each one. Um, so I kind of got a head start on that, and then so I'll go ahead and do that. So, um, you one is gonna be 1/2. Um, he was squared over, See? So, um, we said there is eight micro cool ums. And then see the 1st 1 is, um, 150 nano fare ads. So let me go ahead and plug that into a calculator. So 1/2 times um, e micro fad. Micro cruel. Um I mean, okay, go on. Half eat micro hula MB squared C is equal to 1 50 you know, for odds you like a split screen or something. Really have room for that. Um, okay, so then I got 2.13 times 10 of the minus for so just forget this. 2.13 times, 10 to the minus for and then the remaining energy is gonna be Ah, the total minus. You won. So the total energy is, um, 4.8. Just used to sick Briggs times 10 to the minus four. Minus 2.1. Times 10 to the minus. For So that's gonna be 2.7 times 10 to the minus four jewels. So that's D and then the answer T Let's see a potential difference across each capacitor. Okay, so be QC. Over. Visa V is equal to a Q over. See? So, um, so V one is cute, Juan. Oversee one. So that's just eight micro Coolum divided by one 50 I think. Yeah. 1 50 Nana. Cool of nano. Fair odds. And then I'll just go ahead and write out the equation. for what it is for the other one. So then that's the same eat micro whom divided by 1 20 nano fair odds. And, of course, my crow's 10 of the minus six Nano's 10 to the minus nine. So eight divided by 1 50 I'm standing in my six, a friend of the minus nine problems 53 Waltz And then this one is 66.7, right, just to 67 boats.

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