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Suppose we have two identical oppositely charged spheres separated by some short $ x $ distance, then if we say that superposition principle holds, then at all points in space outside the conductors, the net field is the sum of field due to each. This is simple to calculate because if we look at the spheres from a region outside boundary, they sphere set up is same as a dipole.

Now, here is the problem, what is the field inside one of the spheres? Let's take the positively charged sphere, by itself it has zero field due to being a conductor and then we add the field inside that region due to negatively charged sphere.

However, this seems wrong because the positively charged conducting sphere must always zero field inside.

Hence, my question, does superposition theorem fail for regions inside conductors?

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    $\begingroup$ Maybe think about the charge distribution on the surface of the spheres. For a single isolated sphere we'd expect the distribution to be uniform, but in this case it will be affected by the presence of the other sphere. A perfect arrangement could cancel all the fields inside the spheres while maintaining all results we'd expect from superposition. $\endgroup$ – Quantum Mechanic Jun 1 at 16:45
  • $\begingroup$ 1. How do you proof such an arrangement exists? 2. So, super position holds everywhere except inside conducting surfaces? @QuantumMechanic $\endgroup$ – Buraian Jun 1 at 16:52
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    $\begingroup$ That's just my gut feeling - superposition holds everywhere, so I expect things to only make sense once we know the correct distribution of charges that will arise from this particular arrangement. I know for sure that the charges will not be arranged uniformly - I haven't dont the calculation to see what it will look like $\endgroup$ – Quantum Mechanic Jun 1 at 16:55
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    $\begingroup$ Quick search brought me to this: doi.org/10.1098/rspa.2012.0133 - the two spheres will most likely attract each other in the end, even if they are both positively charged! $\endgroup$ – Quantum Mechanic Jun 1 at 17:00
  • $\begingroup$ Related, possible duplicate: What would be electric potential due to induced charge sphere? $\endgroup$ – Michael Seifert Jun 1 at 18:03
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Now, here is the problem, what is the field inside one of the spheres? Let's take the positively charged sphere, by itself it has zero field due to being a conductor and then we add the field inside that region due to negatively charged sphere.

However, this seems wrong because the positively charged conducting sphere must always zero field inside.

Hence, my question, does superposition theorem fail for regions inside conductors?

Careful. Maxwell’s equations are linear as is Ohm’s law. So superposition holds everywhere in this scenario. However you are misapplying it in your analysis.

In your analysis you are changing the conductors, not just the charges. From a mathematical standpoint you are changing the boundary conditions as well as the sources.

To apply the principle of superposition here you would take two conductors, the first positive and the second neutral and solve that. Then you would take two conductors, the first neutral and the second negative and solve that. Then your scenario is the sum of both of those.

In each half-solution the field inside each conductor is zero. When you add them the superposition remains zero. So superposition does hold inside the conductors, when properly done. You just can’t change the boundary conditions

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Superposition principle is based on the fact that the force of attraction or repulsion between given two charges is not affected by the presence of a third charge.

Now coming to the question it seems you are assuming static distribution of charges on the conductor. Whereas in reality both the conductor changes the distribution of charges on their surfaces in such a fashion that net electric field inside each becomes 0. And hence superposition principle is not violated.

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