Using Gauss' Law to Calculate electric field near rod.

[Nathan B]

Homework Statement

No variables, just a conceptual question.

Homework Equations

Flux = EA = Q/ε

The Attempt at a Solution

Given a uniformly charged rod of FINITE length, could we use Gauss' law for electric flux to calculate the field at a point p a distance x away from the rod, so long as the whole rod is enclosed and x lies on the surface area of the enclosing gaussian surface? I tried it with the equation E = λ L/(ε A), but it didn't work. I also found multiple different A's could be used, but none of them gave the right answer. Could someone please explain to me where I'm going wrong with this?

 

[NFuller]

Gauss's Law states that
$$\oint\mathbf{E}\cdot d\mathbf{a}=\frac{Q_{enc}}{\epsilon}$$
Normally, there is some kind of symmetry argument which can be made that allows us to know the direction of $\mathbf{E}$. If the rod was infinitely long, then you could use mirror symmetry and translational symmetry to argue that only the radial component of $\mathbf{E}$ is non-zero at all points. In that case, the dot product $\mathbf{E}\cdot d\mathbf{a}=E\hat{r}\cdot da\hat{r}=Eda$ and we can evaluate the integral. In the case of a finite rod, do these symmetry arguments hold? If they don't then can you evaluate $\mathbf{E}\cdot d\mathbf{a}$?