Parallel Plate Capacitors

by Jared Rovny

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    00:01 We're now partway through our discussion of circuits.

    00:04 We've already introduced resistors.

    00:05 We're now going to introduce the other important component of circuits that we'll be talking about which is capacitors.

    00:11 So again, what we've done is talk about Ohm's Law with resistors.

    00:16 Now, we're going to introduce capacitance just before going to more complicated circuits when we put all of these things together. In discussing capacitance, we'll start with the idea of a parallel plate capacitor and what that is and then discuss the capacitor energy and voltage that we can put in a capacitor and then finally I finish up with a discussion of dielectrics and how we can change the capacitance of the capacitor with the material called dielectric.

    00:40 Let's start with the parallel plate capacitors.

    00:43 The first capacitors are formally so, were called Leyden jars.

    00:48 They were just two bits of metal and they were put together like this separated by some material made of glass or some sort of insulator.

    00:55 What they would do with this is attach some sort a voltage source just anything to push current from, and charge from one of these metal objects to the other and then what would happen is you would have two metals just like this, separated by an insulating layer, by glass in this case.

    01:12 Both metals would be conducting and so they would have the charge flow from one to the other because of the applied voltage and then what would happen is because you have negative charge on one of the metal objects and the positive charge on the other metal object, these positive and negative charges would attract each other and they would hold on to each other through the insulator without being able to travel through the insulator and so they'll be sort of stuck in that position so that even if we remove the battery we would still have these charges stored because they would be holding each other in place.

    01:46 So it's almost like a spring system or a latch system where we have electricity on one side and a positive charge on the other side and these would be attracting each other and staying in place for this reason.

    01:57 In general, with the capacitor all it is, is two plates just like the leyden jar, but these two plates or two metals, any two conductors just like this would be storing charge may be plus Q on one of the plates and minus Q on the other plate and then we would apply a voltage to the capacitor to get charge to flow from one to the other creating this a non-equilibrium state where one has a different charge and then the other one.

    02:22 We can ask again how much charge is built up on these two plates if I apply a particular voltage and this defines for us what we call for a given capacitor its capacitance.

    02:33 The capacitance of a capacitor tells us how much charge could we store in that capacitor for a given amount of voltage. Just by rearranging the equation that we have here, we can see that the amount of charge stored on the capacitor is equal to its capacitance times the voltage that we apply to that capacitor.

    02:51 The units of capacitance are coulombs per volt as you can see just by looking at the equation.

    02:58 And we call a coulomb per volt a Farad.

    03:00 And we represent the Farad with a letter F.

    03:03 As you can see here we have to be once again very careful.

    03:06 This is a different c from the C for coulombs.

    03:10 So in the first equation, in the bullet points here, you can see we have the units of capacitance where that c is a variable capacitance which can take on many values. The units of capacitance are equal to coulombs per volt and that c for coulombs is a c for a particular unit.

    03:27 It's not variable for a problem that can change.

    03:29 It's just a unit, like a measurement like feet or meters.

    03:33 So again, these are different c's. One for capacitance and one for coulombs so be careful to keep those distinct. And then secondly, with Farads it turns out that a Farad is quite a big unit in terms of most practical capacitors that we'll be using.

    03:47 So we instead typically measure in much, much smaller units.

    03:50 For example, a microfarad which is 1/1000000 of a Farad and sometimes even units smaller than these.

    03:57 For a parallel plate capacitor, all we have instead of our leyden jar with two metal cups separated by some glass, is instead just two parallel plates.

    04:09 Both in this picture are just circular plates and it turns out for a parallel plate capacitor, we can actually solve for and find what the capacitance is in this capacitor.

    04:19 The capacitance for a parallel plate capacitor which has a distance d between the two plates and area of one of the plates of A and a vacuum between or even something else between is the capacitance being equal to K times epsilon, times A over d.

    04:38 So let's look at what each of these variables are.

    04:40 The epsilon that we see, that Greek letter with a little zero over it, under rather is the permittivity of free space.

    04:48 This constant value and that's all it is, it's just some constant number so again not a variable, and won't be changing is a very small number which you can see written here. It's measured in farads per meter.

    05:01 The relative permittivity which we have put here in K is sometimes called the dielectric constant and has to do with what material we have put between the two plates of our capacitor.

    05:12 In this case we have just a vacuum between the two plates of our capacitor and so K the dielectric constant will just be one because we haven't changed the permittivity of free space.

    05:23 And that's really the way to think about this variable K in this equation is it's a sort of way to edit the permittivity of free space by having it be something than other than just free space.

    05:33 But when you have, as we see here, free space or a vacuum between your capacitor plates, K is just one and you're not editing the permittivity of free space.

    05:41 The A and the d in the equation for the capacitance or the parallel plate capacitor is exactly the A and the d you see here.

    05:47 It's the area of the plate divided by the distance between the two plates of your parallel plate capacitor.

    05:53 We also have additional laws for capacitors just like we have additional laws for the resistors.

    06:00 So if we have capacitors in parallel as you can see here where current can either go in one path or the other path, they simply add because they are just like one larger capacitor.

    06:12 And that makes sense if you look at this.

    06:14 If you imagine these two capacitors which are in parallel.

    06:17 It wouldn't make any difference to the circuit if the capacitors were connected on either side so you have a parallel plate capacitor with simply two bigger plates than you had before.

    06:27 And so for these reason, the total capacitance for a parallel capacitor or two capacitors in parallel in a circuit is simply the sum of the two the capacitances.

    06:37 If the two capacitors instead are in series, what they're going to do is exactly the opposite.

    06:42 They'll act like a smaller capacitor.

    06:44 So putting these together we get the opposite law which is that one over the total capacitance is equal to one over each capacitance added together and notice while these are very important equations to memorize, they're exactly the opposite of what we've talked about with resistors.

    07:00 With resistors we added resistance in series in a very simple way like we added capacitors in parallel.

    07:07 So be sure to keep these distinct in your head but also make sure you know how to add capacitors and resistance.

    07:13 Here's a summary of how adding capacitance works.

    07:19 This is a good reference and you should certainly have everything on this page memorized especially the idea, the concept that these capacitors in parallel act like a larger capacitor, all again capacitor in series are going to add in the opposite way and act like a smaller capacitor.

    About the Lecture

    The lecture Parallel Plate Capacitors by Jared Rovny is from the course Circuit Elements.

    Included Quiz Questions

    1. Its capacitance increases.
    2. Its capacitance decreases.
    3. Its capacitance remains unchanged.
    4. It initially decreases then increases to its initial value.
    5. It initially increases then decreases.
    1. The amount of charge per applied voltage
    2. The amount of charge per flowing current
    3. The amount of current per applied voltage
    4. The amount of voltage per applied charge
    5. The amount of voltage per applied current
    1. It doubles.
    2. It gets multiplied by 4.
    3. It reduces to one-half.
    4. It reduces to one-quarter.
    5. It does not depend on the area.
    1. 5 μF
    2. 10 μF
    3. 15 μF
    4. 100 μF
    5. 1 μF
    1. 20 μF
    2. 10 μF
    3. 15 μF
    4. 100 μF
    5. 5 μF

    Author of lecture Parallel Plate Capacitors

     Jared Rovny

    Jared Rovny

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