We´re now partway through our discussion of circuits.
We´ve already introduced resistors.
We´re now going to introduce the other important component of circuits
that we´ll be talking about which is capacitors.
So again, what we´ve done is talk about Ohm´s Law with resistors.
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.
Let´s start with the parallel plate capacitors.
The first capacitors are formally so, were called Leyden jars.
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.
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.
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.
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.
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.
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.
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.
The units of capacitance are coulombs per volt
as you can see just by looking at the equation.
And we call a coulomb per volt a Farad.
And we represent the Farad with a letter F.
As you can see here we have to be once again very careful.
This is a different c from the C for coulombs.
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.
It´s not variable for a problem that can change.
It's just a unit, like a measurement like feet or meters.
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.
So we instead typically measure in much, much smaller units.
For example, a microfarad which is 1/1000000 of a Farad
and sometimes even units smaller than these.
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.
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.
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.
So let´s look at what each of these variables are.
The epsilon that we see, that Greek letter with a little zero over it,
under rather is the permittivity of free space.
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.
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.
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.
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.
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.
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.
It´s the area of the plate divided by the distance between the two plates of your parallel plate capacitor.
We also have additional laws for capacitors just like we have additional laws for the resistors.
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.
And that makes sense if you look at this.
If you imagine these two capacitors which are in parallel.
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.
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.
If the two capacitors instead are in series, what they´re going to do is exactly the opposite.
They´ll act like a smaller capacitor.
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.
With resistors we added resistance in series in a very simple way
like we added capacitors in parallel.
So be sure to keep these distinct in your head
but also make sure you know how to add capacitors and resistance.
Here´s a summary of how adding capacitance works.
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.