Intro To Dielectrics

Hey guys. Let's do an example. what is the new capacitance of the two capacitors that are partially filled with dielectrics shown in the following figure, okay? We have an A and B situation going on here. So, let's just address them each distinctly, okay? For Part A, we have a dielectric that fills the top half of this capacitor, what this is going to be like is, this is going to be like two capacitors in parallel, sorry, in series, okay? Imagine, if this was plus Q then we'd have a minus Q here on the inner surface of this dielectric, a plus Q on the outer surface of the dielectric, and a minus Q on this plate. So, this looks like a single capacitor and this looks like a single capacitor, where the upper capacitor has a dielectric and the lower capacitor does not, we'll call those C1 and C2. So, C1 is going to be Kappa, Epsilon naught, the area is a but the distance is d over 2, so this is 2 Kappa, Epsilon naught, A over d, once I rearrange it, C2 is not going to have any dielectric. So, no Kappa it's epsilon naught, the area is still A and the distance is still d over 2, so this is 2 epsilon naught A over d. Now, since these are in series, then I would say the equivalent capacitance, so the total capacitance of this physical capacitor is 1 over C equals 1 over 2 Kappa epsilon naught naught, A over d plus 1 over 2 epsilon naught A over d, okay? The least common denominator is 2 Kappa epsilon naught, A over d. So, I need a Kappa over a Kappa, so this is going to be 1 plus Kappa over 2 Kappa epsilon naught A over d, okay? So, the whole thing is going to be, let me give myself just a little bit of room here, C equals to Kappa over 1 plus Kappa Epsilon naught, A over d. Now, this coefficient right here looks kind of like an effective dielectric constant, right? Like, if you put half of a dielectric through then this looks almost like its own dielectric constant, right? That's just a number times epsilon, epsilon naught, A over V, sorry, it's just a number times the capacitance, so it looks like it's effective dielectric constant, okay?

Now, let's do Part B this time, we put a dielectric halfway through on the left side. Now, what's the same here is going to be the potential difference or the voltage, that's just going to be whatever it is across the plates whether or not the dielectric is there, okay? So, these look like they are in parallel, right? They both have the same voltage. So, let's find, I'll call this one C1 I'll call this 1 C2, let's find those capacitances, C1 is going to be Kappa Epsilon naught, what's the area though? Well, it has half of the capacitor so it has half of the area, so this is a over 2 but the distance is the same, so this is, we'll call it Kappa over 2, epsilon naught, A over d, c2 is just going to be epsilon naught, no Kappa, right? Because it's in a vacuum, the area once again is 1/2, it has half the capacitor, so this is A over 2 d, this is just 1/2 epsilon naught A over d. Now, these are in parallel so I can just add them, okay? Once I've added them we get this, one half, this k over 2 plus this 1 over 2, they have the same denominator. So, I can just say it's k plus 1 over 2 and notice this also looks like a normal capacitor but with some sort of weird dielectric constant, but either way, this is the capacitance for Part B. Alright guys, thanks for watching.