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Post by mondo on Jul 14, 2023 18:37:33 GMT
On page 491, below the second equation for $D(z)$ author defines a "dipole moment" but he doesn't say explicitly what is it. I assume it is a complex number $d = ke^{-i\phi}$ base on the description, above equation for $D(z)$ and a formulae (1) p.456 for a net force acting on a dublet. Am I right here?
Few more questions:
1. "Thus the Polya vector field of $d/z^2$ is a diploe whoes axis points in the direction of $d$ and whose strength is $|d|$." I don't get what is the axis of this polya vector, and how does it point into the direction of $d$?
2. On page 493, author asks for a prove that $z^{1}$ corresponds to a quadrupole at an infinity. I used (19) as suggested to get $d[\frac{1}{\overline{z} - \epsilon} - \frac{1}{\overline{z} - \epsilon}] = d[\frac{2\epsilon}{\overline{z}^2 - \epsilon^2}]$ but it doesn't look like a quadrupole at infinity.
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Post by mondo on Jul 16, 2023 21:01:20 GMT
For question #1, The axis of a Polya vector field points in the direction of $d$ because $d = ke^{i\phi}$ and hence $D(z) = \frac{ke^{-i\phi}}{\overline{z}^2} = ke^{i\phi}$ this shows a common direction?
Also, In the above equation I said $D(z) = ke^{i\phi}$ but in the book, in the last paragraph of subchapter 9 on page 493 author says $D(z)= -ke^{i\phi}$, why $-k$? If it is based on equation (18) then we can see as $\epsilon -> \infty$ $D(z) = \frac{ke^{-i\phi}}{\overline{z}}$ so I don't get why $-k$ there. Author also says the proportion $\frac{S}{\epsilon} =k\pi $ but shouldn't he rather say $S\epsilon = k\pi$ as he did on the previous page, multiple times i.e $k = 4d\epsilon$ or under the equation (18) "so that $2\epsilon S$ remains constant"
For the question #2 I think it is trivial and in fact already solved by the author in the preceding example - equation (19) shows just this, a limiting equation for a source and sink of $z^{1}$.
Thank you.
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Post by mondo on Jul 16, 2023 22:22:48 GMT
As for the ration $S/\epsilon$ - I overlooked that on page 493 author talks about different limit, this time $\epsilon -> \infty$ so yes, it makes sense to define the ratio $\frac{S}{\epsilon}$ as a constant.
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Post by Admin on Jul 17, 2023 6:05:44 GMT
Mondo
Sorry I haven't replied to your questions. I'm tied up at the moment doing other things. I will get back to you as soon as I can.
Vasco
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Post by mondo on Jul 17, 2023 7:08:47 GMT
Vasco,
That's fine. Let me sum up the current status of this thread:
1. I have been able to figure out why on page 493 there is -k. It is because we take the limit when $\epsilon -> \infty$ and the negative term in the denominator, next to $\epsilon$ dominates here.
2. For the question why the axis of a Polya vector field points in the direction of $d$, I think it is because $d = ke^{i\phi}$ and hence $D(z) = \frac{ke^{-i\phi}}{\overline{z}^2} = ke^{i\phi}$ this shows a common direction.
3. I am a bit confused why the dipole strengths is |d| and not |d/z|. After all the Polya vector field is $d/z^2$.
Thank you.
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