This preview shows pages 1–3. Sign up to view the full content.
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 + (ba) x/h
(Do you see that? It's the formula for a straight line with intercept a and slope (ba)/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 + (ba)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/(ba)) / (a + (ba) 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,.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2) 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":
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 ROGERS
 Resistance

Click to edit the document details