I'm not sure how to use limiting principle to estimate the N fraction when r -> 0 and small area dA stays fixed

$ \lim N_{r\to\infty} = \lim\delta A/(4\pi r^2)_{r\to\infty}$ = 1/2 N ? How?

  • $\begingroup$ Since $\delta A$ is fixed, then it can't vary with $r$, hence the $\lim_{r->\infty} \delta A/4\pi r^2=(\delta A/4 \pi )\lim_{r->\infty} \frac {1}{r^2}=0$. And the mystery variable $N$ must have a value of $N=0$. $\endgroup$ – Cinaed Simson Apr 5 at 3:09
  • $\begingroup$ Answer is 1/2 number of events N. how do you interpret the final explanation for flashcard 16 under "mathematics and statistics"? -> great.cwru.edu $\endgroup$ – Quesop Apr 5 at 3:25
  • $\begingroup$ To send me email, you need @cinaed in the comment. Also, I don't do flash cards - that's your homework. $\endgroup$ – Cinaed Simson Apr 5 at 18:38
  • $\begingroup$ Since $4\pi r^2$ is the surface of sphere, and $\delta A$ is a small area of the sphere, you probably want the $lim_{r \rightarrow 0} \delta A/4\pi r^2$. Think of the surface as a balloon and $\delta A$ is small fixed patch on the ballon. Let the air out of the balloon until the small patch prevents the balloon from from shrinking further. The patch should now shield roughly half of the balloon. So you probably don't want to take $r \rightarrow 0$. Maybe $r \rightarrow \delta r$ such that $lim_{r \rightarrow \delta r} \delta A/4\pi r^2 =1/2$. It's hard to be precise. $\endgroup$ – Cinaed Simson Apr 5 at 18:43
  • $\begingroup$ Sorry, I did get an email - I was just looking for different one. $\endgroup$ – Cinaed Simson Apr 5 at 19:15

Your problem is a mismatch between the expression of which you are finding the limit and the answer you are expecting. They are answers to different questions.

Your expression 𝛿𝐴/(4𝜋𝑟2) is the fraction of a sphere at radius r covered by dA if the area were spread onto the surface of the sphere. This will actually diverge as r goes to 0 because dA will be many times the area of the sphere for sufficiently small r.

Your expected answer, 1/2, is the limit as r goes to 0 of the fraction of events passing through a FLAT dA which is tangent to the sphere of radius r.

I suggest taking the limit of a new expression derived from the geometry I describe in my sentence about your expected answer.

  • $\begingroup$ so dA = r^2 dsolidangle = r^2 theta^2 phi for small theta and phi angles. so working backwards theta^2 *phi must be 2*pi. I don't see how setting theta^2 *phi to 2pi would make dA flat? please clarify? $\endgroup$ – Quesop Apr 5 at 2:08
  • $\begingroup$ No, flat is not the only possibility. There are infinitely many expressions which have a limit of 1/2 as r goes to 0. I don’t know what your goal is with this limit, so I just chose the first physically reasonable possibility that occurred to me which would yield your expected result of 1/2. Maybe the quantity you were trying to find was something else entirely. $\endgroup$ – Duncan Harris Apr 5 at 2:19
  • $\begingroup$ The argument I am reading is "There are dA/(4*pir^2) events with dA much smaller than 4*pir^2 (total surface area). As r goes to 0 (dA fixed) we get half the total events since dA now gets hit by emissions into half the spherical volume" My question is about the limit in second sentence. I think it is a non sequitur. $\endgroup$ – Quesop Apr 5 at 2:31
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    $\begingroup$ You can’t assume both that dA is constant and that it is much smaller than r^2 as r goes to zero. They are contradictory assumptions so you can only derive nonsense. $\endgroup$ – Duncan Harris Apr 5 at 2:34
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    $\begingroup$ I find the last question on that flash card silly if a quantitative answer was expected. How do you put the finite area dA an infinitesimal distance from the origin? There isn’t some obviously best choice of how to do so, and your choice dramatically affects the answer. I think that if I had been answering the question on a test I would have written the same expression as you, and then said that it was meaningless for spheres smaller than dA. $\endgroup$ – Duncan Harris Apr 5 at 3:17

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