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

[ForgetfulPhysicist]

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?

 

[Meir Achuz]

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.

 

[ForgetfulPhysicist]

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?"

 

[ForgetfulPhysicist]

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.

 

[vanhees71]

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.

 

[Meir Achuz]

"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.