I am having trouble understanding Gauss's Law. Suppose we wish to find the electric field strength between two parallel plates with charge density $\sigma$. I know it should be $E=\sigma/(2\epsilon_0)$.
But if we place a cylinder between the plates with base parallel to the plates such that the cylinder does not intersect the plates, then the charge enclosed is $0$ and the flux is $2EA$ where $A$ is the area of the base of the cylinder. Thus by Gauss's Law, $2EA=0$, which gives $E=0$. What is the error in my understanding?
Best Answer
Your equation is wrong. The flux is not $2EA$. Rather, is $-EA + EA = 0$. The reason is because Gauss's law involves the dot product of $E$ and $dA$. The direction of $dA$ is the way out of the cylinder. When the electric field goes through the bottom of the cylinder, the electric field is in the opposite direction of the area vector, so the electric flux is $-EA$. When the electric field vector exits the cylinder, it is in the same direction as the area vector, so the electric flux is $EA$. Hence, the net electric flux is $-EA + EA = 0$, not $2EA$. Hence, $0 = 0$, not $2EA = 0$. To solve for the electric field, take a cylinder which goes through one plate. Then, the electric field goes outside the cylinder through the inner base, so the electric field is $EA$. The charge enclosed is $\sigma A$, so $EA = \sigma A/\epsilon_0 $, and thus $E = \sigma /\epsilon_0$.