capa 9 - Phys 1120, CAPA #9 solutions 1) The trick here is...

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Phys 1120, CAPA #9 solutions 1) The trick here is (as the problem hint suggested) to think of this as a stack of little pancakes, each of thickness "dx". (See the picture) They are all in a row, in series, so the total resistance is the sum of the little resistances, R = Integral(dR). So what is the resistance "dR" of a little pancake? Well, a pancake is simply like a tiny cylinder, and we know R of that, it's just R = rho * L / Area, or in this case dR = rho * dx / Area. They GAVE us rho, and we just need to stare at the picture and think about the area of the little pancakes. That would be "pi r^2", but r DEPENDS on x! So now it's a geometry puzzle, how does r vary as x moves from 0 (at the left) to h (at the right)? r=a when x=0, and r=b when x=h, and it varies linearly, so the formula must be r = a + (b-a) x/h (Do you see that? It's the formula for a straight line with intercept a and slope (b-a)/h. If you didn't come up with it on your own, *think about it* till you see how you could have gotten it yourself!) So we're all set up, R = rho*integral(from 0 to h) of (dx / [pi * (a + (b-a)x/h)^2] (Do you see that? it's just adding up rho dx / pi r^2. ..) The integral may look nasty, but it's not so bad, basically like integral(1/x^2) = 1/x, except here we have integral (1/(c + d x)^2) = (1/d) (1/[c+dx]) Do you see why? Once again, don't take my word for it, work out that integral! How do you normally do that - go back to your Calc 1 book, or just think about the rules of integration, but we do expect you do be able to do integrals of this difficulty level . .. If you can't, talk to someone for a little help or review, like one of the professors or your TA, or. ..) So I get R = (rho/pi) * (-h/(b-a)) / (a + (b-a) x/h), all evaluated from 0 to h. This gives me a brief mess, and then I factored and simplified, and got (rho/pi) * h/(ab). Wow, very simple! It almost makes me think there must be a neat trick that I missed, but anyway, that's my result. At this point, after doing algebra and integrals like this, I *really* want to stop and look at the result -does it make sense? It says R = rho h / (pi a b). If h gets longer, it gets bigger - definitely reasonable! Units are right: rho * (distance)/(distance^2), that's good. As the faces get bigger, (either a OR b), the resistance goes down. Also makes sense! So nothing about it seems crazy to me. In fact, given that R = rho *Length/Area, if you had to make a GUESS for a formula, this seems extremely logical. Length is just. .. length (h). And area is, well, some sort of "average"area - it's not pi a^2, it's not pi b^2, it's inbetween, pi a b! Kind of cool,.
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2) Doesn't look at first like it should be hard, but this one does involve a little algebra. Let me define currents I1, I2, and I3 all to be going DOWN the page. Remember, that's arbitrary, doesn't matter what I pick! (Do you see this? If I guess wrong, they'll just come out negative, no big deal). They are the 3 unknowns. Loop 1, clockwise through the "left half":
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capa 9 - Phys 1120, CAPA #9 solutions 1) The trick here is...

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