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schoolبكالوريوس
event_available2026-07-14
السؤال
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3. The Neumann integral for ellipses.
6.1. Find the Neumann integral (no need to solve it) for the case of two concentric
ellipses with semi-major axis A and a and semi-minor axis B and b, respectively.
Assume the two ellipses belong to two parallel planes at a distance 6 from each
other, as shown in Fig. 3.3. [15 points]
6.2. Show that the result found in 6.1 reduces to the Neumann integral for a circle
when A B and a b. Consider then the solution Eq. (4.8) of Eq. (4.7) in page
105 of the notes. Assume A a 1 m and 6e-1/4r with r = 1 mm. Under
these conditions, & can be assumed to be small. Using these numerical values
and following closely all derivations in the notes, compare the approximate
result for M21 when considering the first two terms (i.e., up to 2) in the series
expansions (4.11a) and (4.11b) for F(k) and E(k) in page 106 of the notes
to the approximate result obtained when considering just the first term in the
series expansions (4.12a) and (4.12b) in the notes, i.e., In 4/k, and 1. [10 points]
A
B
이
Figure 3.3
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