Proving Gauss Law using a "bad" Gaussian surface

[math4everyone]

Homework Statement

What I basically want to do is to prove Gauss Law with a cylinder perpendicular to an infinite charged wire (I know I can do this simple, but I want to do it this way)

This is what I have done so far:

Homework Equations

$$\Phi=\int \frac{dq}{4\pi \varepsilon_0 r^2} \hat{r} \cdot d\vec{A}$$

The Attempt at a Solution

So $$d=\frac{z}{tan(\theta)}$$ and therefore $$r^2=z^2 cot^2(\theta)+z^2$$. Now the flux through the top of the cylinder is $$\Phi=\int \frac{\lambda dx}{4\pi \varepsilon_0[z^2(1+cot^2(\theta))]} \widehat{r} \cdot d\vec{A}$$ where
$$\vec{A}=(\rho d\rho d\phi)\hat{z}$$ and $$\hat{r}=\frac{\vec{r}}{|r|}$$
So:
$$\Phi=\int \frac{\lambda dx}{4\pi \varepsilon_0[z^2(1+cot^2(\theta))]^{\frac{3}{2}}} (dcos(\phi),dsin(\phi),z) \cdot (\rho d\rho d\phi)(0,0,\hat{z})$$
But I don't know how to proceed... Maybe I can use cosine law to find $$\rho$$?

 

[Simon Bridge]

How do you plan to handle the flux through the sides of the cylinder?
Also: you need a clear statement of what, exactly, it is that you are proving.

 

[math4everyone]

How do you plan to handle the flux through the sides of the cylinder?
Also: you need a clear statement of what, exactly, it is that you are proving.

Using the fact that $$d\vec{A}$$ is $$\rho d\rho d\phi \hat{\rho}$$. First, I want to calculate the flux in the bottom and top of the cylinder, and I expect it will be slightly similar in the case of the sides. What I want to prove is that this flux is equal to $$\frac{q}{\varepsilon_0}$$

 

[Simon Bridge]

OK good luck ... there is a reason people don't try this: it's very difficult.
It is not uncommon to end up with integrals that cannot be evaluated analytically.

The integral you are asking about needs limits ...

... and you still should make an explicit statement of what you want to prove.
Are you just rotating the standard gaussian surface for the hell of it or are you trying to find something out?
ie. why pick a cylinder in that orientation? Why not a spheroid or a cube? Why not have the line of charge pass through the cylinder at an arbitrary angle to the cylinder axis?