- #1
math4everyone
- 15
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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)
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 $$\rho$$?