Jackson Sec 2.6 on "general solution" of charge near sphere

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Discussion Overview

The discussion revolves around Section 2.6 of Jackson's book on Electromagnetism, specifically addressing the nature of a sphere in the context of a nearby point charge. Participants seek clarification on whether the sphere is a conducting sphere and what type of interaction exists between the sphere and the point charge.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions whether the solution at the end of Section 2.6 pertains to a conducting sphere, noting that a conducting sphere would not have a potential varying with azimuth and polar angle.
  • Another participant asserts that the sphere in Section 2.6 is not a conducting sphere but relates to the Dirichlet Green's function, which describes the potential due to a point charge outside a conducting sphere.
  • A follow-up inquiry seeks to clarify the nature of the sphere if it is not conducting, specifically asking if it is a dielectric sphere or a non-interacting sphere.
  • One participant expresses frustration, suggesting that the scenario presented is a non-interacting sphere near a point charge, which they view as lacking practical relevance.
  • Another participant discusses the Dirichlet condition and the role of Green's functions in determining the electrostatic potential for a unit charge near a grounded conducting surface.
  • A later reply indicates that the type of sphere is irrelevant to the discussion, suggesting that participants should refer to Jackson's section on Green's functions for clarity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the sphere or the relevance of the example presented in Jackson's text. Multiple competing views remain regarding the interaction between the point charge and the sphere.

Contextual Notes

There are unresolved assumptions regarding the nature of the sphere and its interaction with the point charge, as well as the implications of the Dirichlet Green's function in this context.

ForgetfulPhysicist
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Hi , I'd like a little bit of clarification about Section 2.6 from Jackson's classic book on E & M.

Section 2.6 starts out with the problem of a "conducting sphere" near a point charge, but then it confusingly veers away to a problem where potential is prescribed to vary with azimuth and polar angle. So my question is: can somebody verify that the solution at the end of Section 2.6 is NOT for a conducting sphere? After all, a conducting sphere would NOT have potential varying in the azimuth etc...

Further, if it's NOT a conducting sphere then what is the interaction between the "nearby point charge" and the sphere? Is it a dielectric sphere? Is it a completely non-interacting sphere?
 
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The sphere in 2.6 is not a conducting sphere, but the Dirichlet Green's function is the potential given by a point charge outside a conducting sphere.
 
Please continue on to answer my follow on questions: "if it's NOT a conducting sphere then what is the interaction between the "nearby point charge" and the sphere? Is it a dielectric sphere? Is it a completely non-interacting sphere?"
 
It seems the answer is "it's a completely non-interacting sphere near a point charge"... which seems to be a very useless, rarely occurring, seldom-real-world-application, mathterbation example for Jackson to spend our time on.
 
For the Dirchlet condition, i.e., ##G(\vec{x},\vec{x}')=0## for ##\vec{x}' \in S## (where ##S## is the surface under consideration, i.e., in this case the sphere), the Green's function is formally the electrostatic potential ##\phi(\vec{x}')## for a unit charge located in ##\vec{x}## at presence of a conducting "grounded" surface ##S##. For a sphere it can be determined using the method of image charges.
 
"Is it a dielectric sphere? Is it a completely non-interacting sphere?"
It doesn't matter what kind of sphere it is. Read Jackson's section on Green's functions.
 

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