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
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,.