# Gauss' law giving zero field where field is not zero?

Two plastic sheets with charged densities as shown: I'm trying to find the field at $B$. I obtained the correct answer by adding up the fields created by each charge density. But I realized that since the field is uniform in the region between the two sheets, I should be able to make a Gaussian surface between the sheets with the shape of a box and with one edge at $B$. Thus, the flux would be

$$\int{E \cdot dA} ~=~ EA ~=~ \frac{q_{encl}}{\epsilon_{0}}~=~0.$$

Because there would be no charged enclosed inside the surface. However, that means that $EA$ is $0$ (Note the integral reduced to $EA$ because $E$ is uniform). Since $A$ is not zero (it is the area of two sides of the box), this means that $E$ must be zero. However, $E$ is not zero there, as you can see by adding $\frac{\sigma}{2\epsilon_0}$ for each charge density.

What am I doing wrong when using Gauss' law?

• Draw your little box and draw the E field vector. The vector points into the box on one side, out of the box on the other side. When you use Gauss's law you need to dot the field with the outward pointing surface normal vector. The total flux is EA - EA = 0, that does not imply EA=0. Nov 3 '11 at 17:40
• Hall Vladimir, I guess You could not beare that Guass? :=) Nov 4 '11 at 14:48

You tell us that one surface of the box is at $B$, but you're a little vague on where the opposite face is. You do say that your surface is "between the two sheets", so I think you may mean that the surface is entirely contained in the space between the two sheets. The box does not intersect any charged surface. With that, and a uniform electric field in that region, the flux on the opposite face will have a sign opposite to the flux at the surface at $B$. The total flux is zero. The field and flux at any portion of the surface can be non-zero. Gauss speaks only to the total flux.

• Right, so my question is if the flux is zero, and the flux in this situation is EA, isn't E zero? Because E is the average field at that point no matter what charges are around it. Because the field is uniform between the sheets, the E in EA must be the exact field at the edges of the box. So by this logic E is zero. That's what I'm confused about; I understand that the flux is zero. Nov 3 '11 at 17:34
• The field is uniform but the flux over the left face of the box is negative because the field is entering the box. The flux over the right face of the box is positive because the field is leaving the box. If you sum them you get zero. In the Gauss law, the area element is a vector paralell to the field for the right face and antiparallel for the left face and hence the minus sign.
– user5800
Nov 3 '11 at 17:43
• "E dotted A" is not the same as "E times A". Therefore E dot A can equal zero for neither E or A equaling zero.
– Nic
Nov 3 '11 at 18:07

If you draw a gaussian box between the plates, the flux in the surface near to $\sigma_3$ is equal to that of the surface $\sigma_2$. In this case $\vec{E}\cdot\vec{A}$ is the same as $\vec{E}\times\vec{A}$.

If you apply the superposition principle and for each separate plate and calculate the field by Gauss law then adding the fields gives: $E= \frac{\sigma_3-\sigma_2}{\epsilon_0}$