UmbralRaptor changed the topic of #kspacademia to: https://gist.github.com/pdn4kd/164b9b85435d87afbec0c3a7e69d3e6d | Dogs are cats. Spiders are cat interferometers. | Космизм сегодня! | Document well, for tomorrow you may get mauled by a ネコバス. | <UmbralRaptor> egg|nomz|egg: generally if your eyes are dewing over, that's not the weather. | <ferram4> I shall beat my problems to death with an engineer. | We can haz pdf
<egg|anbo|egg>
!wpn whitequark
* galois
gives whitequark a distinguished かわいい meow
<egg|anbo|egg>
!wpn -add:adj posh
<galois>
Added adj 'posh'
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* raptop
pokes Jackson with a canada goose
<raptop>
...did he introduce a method that looks like it should work on a whole class of problems, but actually only works on one(1)?
<raptop>
I'm pretty sure that the image charge and the origional charge aren't supposed to be in the same place, but then maybe my assumption for the image-image charge seen from the inside has the wrong position?
<SnoopJeDi>
raptop, what's represented by $z$? This work looks like it's done on-axis, but the PDF asks for all points in space
<raptop>
SnoopJeDi: z is the axis, though I sort of stopped upon getting a contradictory result for the location of the image charge
<raptop>
Uh, the potentials are for all space, but then I just looked at the on-axis cases to try to find the locations of the image charges, and, well
<SnoopJeDi>
Hmm, okay I yanked my copy off the shelf and I'm looking in 4.4, and I see a problem that looks similar to your work in my 4.5 "Boundary-Value Problems with Dielectrics," but his example is a semiinfinite dielectric
<SnoopJeDi>
"Clearly the simplest solution is that for $z < 0$ the potential is equivalent to that of a charge $q''$ at the position $A$ of the actual charge $q$"
<SnoopJeDi>
"Sir, this is a Wendy's."
<raptop>
Yeah, I was working on the assumption that one could do that thing with spheres and dielectrics that one can with spheres and conductors
<SnoopJeDi>
hmm, is that in a previous section? I've forgotten basically all of this to be quite honest with you
<raptop>
Apparently I'll need to end up with a few infinite series of spherical harmonics (well, legendre polynomials) instead?
<raptop>
Somewhere in a previous section, yeah
<SnoopJeDi>
let me see if I can find the conductor one and maybe I'll understand the approach better
<SnoopJeDi>
oh, this is exactly problem 4.5(a) in my text
<raptop>
hrm
<SnoopJeDi>
which I think is the first edition?
<SnoopJeDi>
raptop, does your chapter 4 have an example like this? "The second illustration of electrostatic problems involving dielectrics is that of a dielectric sphere of radius $a$ with dielectric constant $ε$..." on p.p. 133 of mine, accompanying equations that look close to the setup you'd want are eq. 4.54, 4.55 in mine
<SnoopJeDi>
except for the fact that his problem has no charge
<SnoopJeDi>
guess that means it's a nonstarter upon further consideration, but I think you want something closer to that
<SnoopJeDi>
but I will try again to find the conducting spheres thing, I missed it in my scrub through sec 3
<raptop>
let's see... my 4.4 is boundary value problems with dielectrics
<SnoopJeDi>
yea I think it's been shuffled around since my ed.
<raptop>
Examples include the infinite dielectrics with a planar boundary, a dielectric sphere in an infinite field
<raptop>
There's also an illustration of a cavity in an infnite dielectric
<SnoopJeDi>
yea those are the same examples I have. I think the second is closer geometrically to your problem than the first, but your work resembles the first closely
* raptop
should probably grab some food and try an attemp with spherical harmonics then
<SnoopJeDi>
namely section 4.8 at the bottom of pp.73, "We may also generalise the discussion"
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<SnoopJeDi>
and 7.3.2 is following the not-quite-your-problem thing that I was pointing to in Jackson (we used Jackson for his course so the notes rhyme heavily, but he elaborates things much more than does Jackson)
<SnoopJeDi>
so your solution should be of the form 4.138, but you need to apply the discontinuity at the surface of the dielectric
<SnoopJeDi>
err, sorry, not those
<SnoopJeDi>
although the line of reasoning there would let you chase it down by way of Green functions