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Lets say that we have a point charge $q$. Let's draw a Gaussian sphere at a distance $d$ from the charge, as no charge is enclosed inside the sphere the integral $$ \oint_A \vec E \cdot \text d \vec A = 0 $$ vanishes. Hence electric flux at that point is zero. Now I understand that flux through an external point is zero but doesn't that mean that $\vec{E}$ is also $0$, which is wrong. Could someone tell me why this happens and how to use Gauss law?

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  • $\begingroup$ I don't think I understand what you are saying "Hence electric flux at that point is zero." and "flux through an external point is zero". Flux is not defined at a point. Do you mean flux density? And what is "that point"? Where is it? Can you clarify those statements? $\endgroup$ – garyp Aug 4 at 14:04
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No, if Gauss' law gives $$\int_{\partial V}\vec E\cdot \text d\vec S=0 \tag{1}$$ this means that the total flux of the electric field through the surface is zero and it then follows that the total charge contained within that surface must also be zero.

Note that in order for this integral to be zero, $\vec E$ doesn't have to be zero, what we're saying is that if you take every point on the surface $\partial V$ and add up the dot product of the outward pointing vector with the electric field at each point the result will be zero. What we're saying is that at some points the flux will be positive, at other points it will be negative, but the total flux will cancel to zero.

Just to further this point, look at this sum: $$\sum_i x_i=0 \tag{2},$$ all we know is that the total sum of all values of $x_i$ is zero, it doesn't necessarily mean that every value of $x_i$ is zero.

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No it does not.

This is because dS is a vector quantity whose direction,by convention, is radially outwards for a closed body (sphere,here).

enter image description here

In this diagram,the field of q is towards left going towards the sphere. For the left half of the sphere the direction of dS is towards left.Roughly speaking, the field due to q and dS are parallel or their dot product is positive

However for the right of the sphere,the direction of dS is towards right.But the field is still towards left.So they are anti parallel.Their dot product is negative.

Both these fluxes hence cancel out even though E is not zero.


P.S.

  1. I am not following strict mathematical treatment for directions for the sake of simplicity.
  2. Bold characters denote vector quantities.
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