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Bipolar Doc Question REVIEWED

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A brief working note dated 11.27.15 by Phil, in the support files for the bowl-in-toroidals electrostatics project. It asks how upper and lower constant-u curves (0<u1<π and π<u2<2π) are related when they lie on the same circle. Using the circle center yc = a/tan u and radius R = a/|sin u|, it concludes u2 = u1 + π. The file name suggests the note was reviewed.

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Bipolar Doc Questions PhL 11.27.15 Question 1: Look at the lines of constant u. On the top imagine there is some 0 < u1 < π value u1 and on the bottom there is some π < u2 < 2π value. How are u1 and u2 related if they both lie on the same circle? Answer: What do we know? Both constant-u curves lie on the same circle, which is this circle x2 + (y- yc)2 = R2 yc = a /tanu R = a/|sinu| . (2.6) The equation R = a/|sinu| gives one clue. It says this |sin(u1)| = |sin(u2)| For the upper half where u1 lies in (0,π) we know that sin(u1) is positive. For the lower half, where u2 lies in (π,2π), we have sin(u2) is negative. This our equation is sin(u1) = - sin(u2) But we can write sin(u1) = - sin(u1±π) so this suggests that u2 = u1 ± π. Since u1 is in the lower range, it must be the plus sign. So u2 = u1 + π. u2 - u1 = π // answer to the question. __________________________________________________________________________