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

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• ForgetfulPhysicist
In summary: The conversation is about the potential of a conducting sphere near a point charge and how the solution at the end of Section 2.6 is not for a conducting sphere. The potential is given by a point charge outside the sphere and the Green's function is determined using the method of image charges. The type of sphere (conducting or dielectric) does not affect the solution.
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?

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.

## 1. What is the general solution for the charge near a sphere?

The general solution for the charge near a sphere is given by the famous Laplace's equation, which states that the potential at any point in space is equal to the average of the potential at all points on the surface of the sphere. This solution is valid for any charge distribution on the surface of the sphere.

## 2. How is the general solution derived?

The general solution is derived by considering the electric potential at a point inside the sphere due to a small element of charge on the surface. This is then integrated over the entire surface of the sphere to obtain the total potential at that point. This process is repeated for all points inside the sphere, resulting in the general solution.

## 3. Is the general solution applicable to all types of charge distributions?

Yes, the general solution is applicable to all types of charge distributions on the surface of the sphere. This includes both continuous and discrete charge distributions.

## 4. What are the limitations of the general solution?

The general solution is only valid for points inside the sphere. It cannot be used to calculate the potential at points outside the sphere. Additionally, it assumes that the sphere is a perfect conductor, meaning that the charge is evenly distributed on the surface and there are no internal charges.

## 5. How is the general solution used in practical applications?

The general solution is often used in electrostatics problems involving spherical conductors, such as capacitors and charged particles near a conducting sphere. It allows for the calculation of the electric potential at any point inside the sphere, which can then be used to determine the electric field and other important parameters.

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