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Gauss's Law states that the flux is equal to the charge inside divided by a constant. and is also equal to the surface integral of the electric field. So, if there is no charge inside the closed surface, the flux is null but why do we deduct that the electric field is null at all points inside the surface? It doesn't make sense.

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    $\begingroup$ Who are we? We who? $\endgroup$
    – Qmechanic
    Jan 18, 2021 at 17:17
  • $\begingroup$ It is not true that zero flux through a closed surface implies the field is zero. $\endgroup$
    – garyp
    Jan 18, 2021 at 18:04

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Neither the flux through a portion of the surface nor the electric field within the closed surface has to be zero. But the total flux is zero and the electric field generation is zero.

For instance: If the closed surface is placed within an external electric field (an electric field caused by charges from outside the surface), then you can measure a non-zero fieldstrength of that electric field at points within the surface.

Think of the electric field lines like sun rays. If the closed surface is a glass box, then you can easily have light shining through due to the external source (the Sun or your flashlight). So you do have points of non-zero "lightfield"-strength within the closed surface.

But the field lines that "enter" will also "leave" through the surface. This is what Gauss' Law tells us. There may be field lines passing through the surface, but there is no electric field generated within the surface. Thus there can be no charge (a charge is an electric field generator) within the surface.

In the same way any flux entering will also leave the closed surface. The flux across one part of the surface is not necessarily zero (there is a positive flux through the frontal part of the glass box through which the light shines in). But the flux calculated for all parts of the surface must in total be zero. That's what you calculate when you find the surface integral where you (presumably) include the entire closed surface. We should thus not say that there's no flux but rather no net flux or no total flux across the surface.

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Yes, you are right; it does not make sense. It does not make sense because it is just false. Take a point charge at the origin. The field extends everywhere, i.e., the field is $\neq 0$ everywhere. However, the total flux over a closed surface not including the origin is null.

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