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September 14

Electric field between a round pipe and a larger rectangular pipe co-axial with the round pipe

Electric field strength is defined in terms of volts per metre and is useful for estimating the force on charged particles. For example, if we have two parallel plates, x metres apart, plate area large compared to spacing, and one at a potential y volts with respect to the other, there is an electric field between them of strength y/x V/m.

If we have two long round pipes, one having outside radius r₁ and the other inside radius r₂, the r₁ pipe inside the r₂ pipe, and the r₁ pipe at a potential y volts wrt the r₂ pipe, the electric field between them is more concentrated near the surface of the inner pipe. Most textbooks on electrostatics give a formula for the field strength immediately outside the inner pipe: y / [r₁ Ln(r₂/r₁)]

I need a formula for the case where the outer pipe is of rectangular cross section, inside dimensions m x n metres. This isn’t covered in the books I checked.

Intuitively, providing the diameter of the round inner pipe is small compared to the rectangular pipe dimensions, the field strength very close to the round inner pipe will be evenly distributed around it.

Where can such a formula be found, or how can such a formula be derived? 58.167.248.33 (talk) 03:23, 14 September 2015 (UTC)[reply]

This is 2D electrostatic problem, which can be formulated in terms of complex analysis. To solve this it in this specific case you need to find a holomorphic function that maps a rectangle into a circle. Ruslik_Zero 20:14, 14 September 2015 (UTC)[reply]

I had a look at the two wiki articles you linked, but I am none the wiser. Seems to be a branch of math I haven't done. 121.215.38.158 (talk) 00:24, 15 September 2015 (UTC)[reply]

You can read this. Ruslik_Zero 20:21, 15 September 2015 (UTC)[reply]

I'm still none the wiser. The website seems to be about finding the field strength at any single point (x,y) and assumes a level of math I don't have. Square or rectangular section pipes are not covered by the worked examples. How you get from that to the field strength close to the round inner pipe inside a rectangular section pipe is still a mystery to me. 120.145.26.25 (talk) 03:42, 16 September 2015 (UTC)[reply]

Central Binomial Coefficient Divisible by n³

Are there any central binomial coefficients $2n \choose n$ divisible by $n^{3}$ ? I wasn't able to find any for $1<n\leq 10^{5}$ . (OEIS does contain a sequence of CBC's divisible by $n^{2}$ , for instance). I believe this might be related to Wolstenholme's theorem. — 79.113.233.141 (talk) 08:53, 14 September 2015 (UTC)[reply]

Wolstenholme's theorem would imply that

{2p \choose p}\equiv 2{\pmod {p^{3}}}

for p a prime >3. So it rules out n a prime but it's probably not much help otherwise. You're not really interested in the remainder as long as it's not 0, so in a sense Wolstenholme's theorem does too much and you can eliminate n a prime much more easily. In fact it's not too hard to eliminate prime powers as an application of Kummer's theorem. Without going into a lot of detail, I believe this can be extended to say that if p is the largest prime dividing n then p<(2n)^1/4, which would eliminate a large percentage of possible n's. I would suggest extending the search to at least 10¹⁰ before drawing any conclusions; I can't find any argument that would prove there are no such n and it seems likely that they exist even if they are very rare. --RDBury (talk) 17:20, 14 September 2015 (UTC)[reply]

Try 154836. --JBL (talk) 17:27, 14 September 2015 (UTC)[reply]

Just to verify; 154836 = 2²⋅3²⋅11⋅17⋅23 and 2⋅154836 choose 154836 is 2⁹⋅3⁷⋅11³⋅17³⋅23³⋅other primes. --RDBury (talk) 21:07, 14 September 2015 (UTC)[reply]

JBL, did you also check all other numbers in between 100,000 and 154,836 ? — 79.118.161.220 (talk) 02:18, 15 September 2015 (UTC)[reply]

Yes; assuming Mathematica can be trusted (not a certain) this is the unique solution below 500,000.--JBL (talk) 03:27, 15 September 2015 (UTC)[reply]

Thank you for helping me eliminate the first 500,000 numbers from the list of possible suspects. :-) — 79.118.161.220 (talk) 04:06, 15 September 2015 (UTC)[reply]

The only other example below 10⁶ is 985,320. — 79.113.245.131 (talk) 10:47, 16 September 2015 (UTC)[reply]

Nice. So I think we can conclude that there is likely a sparse but infinite set of solutions. --JBL (talk) 14:40, 16 September 2015 (UTC)[reply]

Resolved

– Thanks to RDBury's suggestion, the search-method is now very time-efficient.

September 14

Electric field between a round pipe and a larger rectangular pipe co-axial with the round pipe

Central Binomial Coefficient Divisible by n3

Central Binomial Coefficient Divisible by n³