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Capacitance of Cylindrical Capacitor

Patrick Ford
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Hey, guys, let's do an example. What is the capacitance per unit? Length off. Two concentric, infinitely long cylindrical shells, one of radius A and one of radius. Be with a less than be Consider the charge on each cylinder to be plus or minus. Q. So what they're talking about is we have some cylindrical shell that's infinitely long with Radius A and some other concentric cylindrical shell that's also infinitely long of radius be and they have plus and minus Q. And this will form a capacitor. So what we want to do is find the capacitance. Now the capacitance is gonna be the charge per unit. Voltage. So what we have to do is find the voltage between these two cylinders between this distance right here and then divide the charge by that. Okay, that potential difference. That voltage is gonna be negative, integral of e dot dx. Okay, in whatever direction, once again, we're working in the radial direction. Now, between these two, the electric field is on Lee going to depend upon the inner cylinder. That's what gasses lost says. And that electric field is going to be K sorry to K lambda over our or hat So our integral is gonna look like from a to B two k lambda over our our hat dotted into D are are happy because we're doing the radial direction. So this whole thing is gonna look like the negative Integral from A to B of two K lambda over r D R. Okay, And once again, this is also a very easy integral one. Over R is just the log. So this becomes negative. Two k Lambda Ellen of our from A to B. This whole thing is gonna be negative. Two k lambda Ln of B minus, Ellen of A. We can combine these two by saying that's be divided by a So it's negative two K Lambda Elena be divided by a Okay, let me give myself some breathing room here. Another trick that we could do to simplify this because everyone's gonna do it is we can bring this negative inside the log, and that's just gonna reciprocate the division. So this is going to be two k Lambda Ellen, a divided by beef. But we're not done. We still need to find the capacitance. The capacitance is gonna be Q over V. Let me give myself just a little bit more space. It's Q over V, which is going to be Q over to. Okay, Lambda Ellen A over B. Now, what is Lambda Lambda is the charge per unit Length. Right. So this is gonna be Q over to K Q over L Ln a over b So those queues cancel the l comes into the numerator, and this is going to be L two k. Ellen A over B. So what I need to do is I need to divide this l over. I'm looking for the capacitance per unit length. The reason is is because this is an infinitely long cylinder, which means that Ellis infinity, which means that the capacitance is also infinity. But the capacitance per unit length is not infinity. If you remember, that K is 1/4 Pi epsilon. Not then 1/2 K becomes two pi epsilon. Not, And this is over, Ellen. A over B and that is the capacitance per unit. Length off infinitely long concentric cylindrical shells. All right, guys, Thanks for watching