suspected error pg 492, octupole field

Thanks Vasco.

You're right about the quadrupole formula but the vector fields are still empirically wrong.

This field is produced by 1/(zBar^4) which the book claims produces an octupole, but looks to me more like a hexapole:



This one is produced by 1/(zBar^5) and looks like an octupole:



I think I have found an explanation for this relating to dipole orientation.

Equation (19) on page 491 supposes a purely real separation between the two dipoles which are to coalesce into the quadrupole. This is fine but if we try to repeat the same process with 1/(zBar^3) quadrupoles coalescing to an octupole, then again we imply a purely real separation between the quadrupoles. (Otherwise one of the numerators would have picked up some complex component.)

All these purely real separations lead some of the polar lobes to vanish as the quadrupoles coalesce.
An attempt to line up the necessary opposing poles, all on the real axis, leads to a charge configuration something like this:

-+ +- +- -+

As these come together the distinct lobes round the two interior dipoles merge, leaving only six lobes in the limit.
You can see this happening in the center of the following vector field.



The solution is to spread the orientation of the poles of alternating charge evenly around a tiny circle whose radius goes to zero. This ensures that all the dipoles and quadrupoles etc all oppose one another as required.

The function that approximates the effect of these as the radius goes to zero is:

SUM n = 0 to N: (-1)^n/(zBar - epsilon*e^(-i2pi(n/N)))

where N = 8 poles

This sum has surprising symmetry in the numerator. Coefficients of all powers of z vanish except for that of z^(N/2 - 1). The way they vanish is reminiscent of the patterns in a discrete fourier transform, or the sums on page 106.

So for N of 8, this leaves z^3 in the numerator, and with z^8 as the dominant term in the denominator, the final formula is (1/zBar^5).

This method predicts:
(1/zBar^2) for a dipole field
(1/zBar^3) for a quadrupole field
(1/zBar^5) for an octupole field
(1/zBar^9) for an 16-pole field

all verifiable empirically.

The approach in the book implies:
(1/zBar^2) for a dipole field
(1/zBar^3) for a quadrupole field
(1/zBar^4) for an octupole field
(1/zBar^5) for an 16-pole field

What do you think?

Mark

No, not an error then I guess, but a little misleading.

The paragraph ending 'and so on.' on pg 492 does invite the reader to imagine that continuing the sequence, a (2^n)-pole field is given by 1/(zBar^(n+1)). The modifier 'so-called' would have to apologize for an exponentially increasing absence of lobes :-)

Also, while it is true that the 6-lobe field can be created by coalescing 8 poles, it can also be created by coalescing any greater even number. Here is an example with 14 poles, all on the real axis:

-+ +-+-+-+-+- -+

But it seems like it would be more mathematical to name the field after the minimum number of poles needed, which is 6 (if arranged correctly, such as around a circle).

I also think that the fact that a recognizable (2^n)-pole field is given by 1/(zBar^(2^(n-1)+1)) is really surprising and worthy of mention. (An octo-lobed field is given by 1/(zBar^5) !?)

At any rate, I'm splitting hairs, I think the book is excellent, and I'm grateful for your time in looking into my query.

Regards Mark

(I will learn laTeX eventually :-)

So is this forum which demystifies all shortcuts Needham did...