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I've been reading Chapter 3.2 and 3.3 of Purcell's Electricity and Magnetism, 3rd ed, and found that I don't really understand a few ideas that Purcell makes about electric fields and conducting material.

An example is given with a point charge q placed arbitrarily inside a neutral conducting spherical shell. The example explains why the electric field outside the shell is the same as a spherically symmetric field due to charge q located at the center of the shell.

In finding how +q charge is induced on the outside of the conductor, the example tells us to imagine we remove the outer +q charge so we only have the point charge q and inner surface charge -q -- they claim the combination of these charges produces zero field in the material of the conductor and outside the conductor, because 'field lines must have at least one end on a charge and can't form closed loops. In the present setup, external field lines have no possibility of touching any of the charges on the inside, because the lines can't pass through the material of the conductor to reach them, since the field is zero there. Therefore there can be no field lines outside the conductor.'

What do they mean by this? It is my understanding that the -q inner surface charge distributes itself in such a way that the superposition of its electric field and the electric field of the point charge q equals 0 anywhere in the conductor material -- and 0 anywhere outside the conductor. However, that would mean field lines have to pass the conductor material.

And in section 3.3, Purcell wants us to consider a closed metal box with some external charges outside -- he notes that there is a nonuniform distribution of charge over the surface of the box. He then claims that the field everywhere in space, including the interior of the box, is the sum of the field of the charge distribution over the surface and the fields of the external sources.

However, this again implies that external field lines pass through the conductor material, in order to sum to 0. Is it the case that I'm misinterpreting the first example? What then does it mean in that scenario that 'external field lines have no possibility of touching any of the charges on the inside ... there can be no field lines outside the conductor'?

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In general, when we speak of the "field lines" in electricity and magnetism, we almost always mean the field lines of the total electric field due to all of the charges present. One could, in both cases, draw a set of field lines due to the point charge(s) only, and a set of field lines due to the induced surface charge on the conductive surface; and (some of) these field lines would pass through the surface. But the field lines of the combined configuration (which is what Purcell is referring to here) don't pass through the surface; they must end there.

The fields, on the other hand, do reach through all of space; and at any point in space, the total field can be viewed as the superposition of the field due to the point charge(s) and the field due to the induced surface charges. There's a well-defined method by which we can add vector fields together, so it makes a bit more sense to talk about the "field of the point charge" and the "field of the induced charge" separately. But there's not a well-defined way to add two field line diagrams together, which is why it's much rarer to think about the field lines of one particular part of a charge configuration.

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  • $\begingroup$ As an aside, this is one of the reasons that I'm quite dubious about the value of "field lines" as a pedagogical tool for representing electric fields: they don't obey superposition in any meaningful way, which is one of the fundamental properties of electromagnetic fields. Perhaps some day I'll have time to write up all my thoughts on the matter. $\endgroup$ – Michael Seifert May 6 at 14:08
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I think, you actually think in a wrong way. It actually happen that any conductor material try to set it internal electric field zero. but why! this is due to every system try to minimize its energy. As we know that if any where electric field is E then there energy density is half epsilon E^2 so to minimize the electrostatic energy, conductor set it interior electric field zero. And this is accomplished by rearranging its enormous amount of electron and positive charge. Also you know that if there is no electric field in the interior then there is no charges in the interior, so all the charges are on the surface. Now I want to back your question in the first case if there is no electric field in the interior then there is no field line. That means that the field line which escape from the charge in the cavity must end up at the interior surface of the cavity so external field lines do not enter the spherical shell. As we know the number of field line which escape from the q charge must end up on -q charge. so in the interior of the shell -q charge build up and +q on the outer surface which create field like +q charge at the centre. In the second case same reasoning happen. All the which actually told are correct. There is no field line from the external charge in the conductor material and also two charges create in such a way there is no electric field in the interior. so there is no reasoning to think that there is any field line from the external charges in the interior of the conductor because there is no field line. And superposition does not tells that field line superimpose instead field line idea think as a collective way not for individual charge.

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