Partial Dielectrics

by Patrick Ford
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Hey, guys, let's do an example. What is the new capacitance of the two capacitors that air partially filled with di electrics 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 die electric that fills the top half off this capacitor. What this is going to be like is this is going to be like to capacitors in parallel. Sorry. In Siris. Okay, imagine if this was plus Q Then we'd have a minus que Here on the inner surface of this die electric Ah, plus Q on the outer surface of the die electric 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 di electric and the lower capacitor does not will call those C one and C two. So see, one is gonna be Kappa. Absolutely not. That area is a but the distances d over to. So this is to Kappa Absolutely not a over D. Once I rearrange it, C two is not gonna have any dye electric. So no, Kappa, it's absolutely not. The area is still a and the distance is still deal over to. So this is to absolutely not a over T. Now, since these Aaron Siri's, then I would say that the equivalent capacitance. So the total capacitance of this physical capacitor is one. Oversee equals one over to Kappa Absolute. Not over d plus one over to absolutely not a over d. Okay, the least common denominator is to Kappa absolute. Not over d. So I need a Kappa over a Kappa. So this is gonna be one plus Kappa over to Kappa. Absolute. Not 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 one plus Kappa. Absolutely not a over D. Now, this coefficient right here looks kind of like an effective die electric constant, right? Like if you put half of a die electric through, then this looks almost like its own die electric constant. Right? That's just a number times Epsilon. I still not over d Sorry. It's just a number of times, the capacitance. So it looks like it's effective dialect. Your constant. Okay, Now let's do part B. This time we put a dye electric halfway through on the left side. Now it's the same. Here is gonna be the potential difference or the voltage. That's just gonna be whatever it is across the plates, whether or not the dye electric is there. Okay, so these look like they're in parallel, right? They both have the same voltage. So let's find I'll call this one C one. I'll call this one C two. Let's find those capacitance is to see one is gonna be Kappa. Absolutely not. What's the area, though? Well, it has half of the capacitor, so I was half of the area, So this is a over to, but the distance is the same. So this is what we'll call it Kappa over two. Excellent. Not a over D. C. Two is just gonna be absolutely not. No Kappa. Right, Because it's in a vacuum. The area once again is half. It has half the capacitor. So this is a over to de This is just one half. Absolutely not. Over D Now, these air in parallel. Okay, so I could just add them. Okay, Okay, Once I've added them, we get this one hat in this case over two. Plus this one over to They have the same denominator. So I could just say it's K plus 1/2 and notice. This also looks like a normal capacitor, but with some sort of weird die electric constant. But either way, this is the capacitance for part B. All right, guys, Thanks for watching.