Understanding Gauss's Law: When to Apply and When it Fails

[dipole]

This is not a homework question.

I'm just having trouble understanding when it's appropriate to apply Gauss's law or not, and under what conditions does it fail.

For example, suppose I have two point charges forming a dipole. If I draw a cylinder around them, then the total charge enclosed is zero, implying that the field is zero.

However, this is obviously not true for a dipole.

So would I be using Gauss's law incorrectly in this case, or does it not applying to discontinuous charge distributions, or what?

 

[torquil]

This is not a homework question.

I'm just having trouble understanding when it's appropriate to apply Gauss's law or not, and under what conditions does it fail.

For example, suppose I have two point charges forming a dipole. If I draw a cylinder around them, then the total charge enclosed is zero, implying that the field is zero.

No, it implies that the closed surface integral of a certain projection of the electric field is zero. Not that the field is zero at each point on the enclosing surface.

 

[vela]

For example, suppose I have two point charges forming a dipole. If I draw a cylinder around them, then the total charge enclosed is zero, implying that the field is zero.

Gauss's Law would say $\oint \vec{E}\cdot d\vec{S} = 0$. This means the electric flux into a volume is balanced by the electric flux out of the volume. It doesn't mean $\vec{E}=0$, which would mean there is no electric flux at all into or out of the volume.