 ## Physics

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26. Capacitors & Dielectrics

# Capacitance Using Calculus

1
example

## Capacitance of Spherical Capacitor 3m
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Hey, guys, let's do an example. What is the capacitance of two concentric spherical shells? One of radius A and one of radius. Be with a less than be consider the charge on each sphere to be plus or minus que All right, remember that the capacitance mathematically is going to be the charge divided by the potential difference. Okay, for this arbitrary arrangement, what we need to do is find the potential difference between these two plates right here. In order to do that, we're gonna use our calculus equation that it's just the electric field dotted into our direction that we're looking at in this case, we're looking at the our direction, the radial direction. Okay, what is the electric fields gonna be between these two spherical shells? Well, if I look at an arbitrary point between them, it's on Lee going to be due to the inner sphere. That's what God's law tells us that is going to be que que over r squared our hat. So are integral is gonna look like the integral from a to B of cake over r squared our hat dotted into d. R r half just in the radial direction That's what we're integrating. All right, So this whole thing looks like the negative integral of a to be cake. Your r squared D r. Okay, that's a really easy integral right. It's one over r squared, which is negative. One over R. So this is gonna be positive. Que one over our from a to B. This is gonna be cake one over B minus one, over a. All right. I'm gonna give myself a little bit of room here. Now, you can leave this exactly like this. We're not done yet, But you can leave this answer like this. I'm just gonna write it in a different way because most books included in a different way Que que I'm gonna find the least common denominator, which is a B. So this is gonna be a over a B minus B over a B. That's gonna be cake times a minus, bi over a b. Okay. And I wrote it like that because this is how most books we're going to write this. Now. What we need to do is we need to find the capacitance, which is the charge per unit. Voltage. So this is Q over que que a minus bi over a B. You see that? Those accused cancel This is one over K A B over a minus B. And if you remember, that K is 1/4 pi. Absolutely not. Just remember that relationship. This is four pi s l A not a B over a minus bi. That is the capacitance off these concentric spherical shells. Alright, guys, that's it. Thanks for watching.
2
example

## Capacitance of Cylindrical Capacitor 4m
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