Is the Electric Flux Always 0 for a Uniform Electric Field?

[Acuben]

Homework Statement

Not really a homework problem, but I have a conceptual question

assume a uniform Electric field in a direction, meaning Electric field does not change by location
if you have any objects, the total electric reflux is always 0?

Homework Equations

Electric reflux= ClosedIntegral (E * dA) r
where r is the unit vector of direction, can often be in form of cos x where x is the angle between the normal of a plane and direction of Electric field.

The Attempt at a Solution

It seems like if I solve a problem like this. I always get 0 so I'm wondering if it's a coincidence or not

it does seem to be true for cube through
consider E, or Electric field going up.
We only consider top part of the cube and the bottom plane of the cube since the sides would be perpendicular to E and and Elec. Flux=0 for side planes
As for Top plane, it would cancel out with bottom plane since they have same Area therefore same Elec flux.

 

[fzero]

It's zero because Gauss' law requires that

$$\oint_S \vec{E}\cdot d\vec{A} = \frac{Q_S}{\epsilon_0} ,$$

where $$Q_S$$ is the net charge contained within the closed surface $$S$$. If there are no charges, $$Q_S =0$$ and the integral vanishes.