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Consider a uniformly charged sphere with a spherical hole of smaller radius inside, the goal is to find the electric field inside the hole.

I understand the common approach which uses superposition of two fields, but my confusion arises when it comes to applying Gauss’s law inside the hole.

Say using a spherical Gaussian surface of radius smaller than that of the hole, the charge enclosed is evidently 0, which suggests 0 electric field according to Gauss’s law. Also, it seems unintuitive that, without an external field, there is field existing inside a cavity surrounded by uniform charges. I am wondering what is missing here in my logic.

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  • $\begingroup$ Why do you think it is different than you described here? $\endgroup$ Nov 20, 2023 at 7:30
  • $\begingroup$ Because zero electric field is not what’s described by most of the sources I found, example of a reference youtube.com/watch?v=tR_GIrYya9g $\endgroup$
    – km12180202
    Nov 20, 2023 at 8:03

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Your reasoning is correct to some degree:

the charge enclosed is evidently 0, which suggests 0 electric field according to Gauss’s law

One point is missing in your reasoning: Gauss's law tells you about the integral of the electric field over the surface you chose, so "0 electric field" has to be understood as an integral over the spherical Gaussian surface. So, from Gauss's law alone, any distribution of the field would be fine if it sums up to zero.

But one additional information is crucial: "a uniformly charged sphere with a spherical hole", and, as far as I understand, the hole need not be centered in the sphere. If it were centered, the whole geometry would be rotation-symmetric, meaning the the field should be the same all over the Gaussian surface, and then the only solution would be a zero field.

But this symmetry isn't given, and so a direct application of Gauss's law isn't helpful.

To cite Wikipedia:

Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field.

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