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## Homework Statement

Circular plate radius R is uniformly charged and the charge of plate is Q. Find the electric field in arbitrary point perpendicular to the plate that passes through the center. Case [tex]R\rightarrow \infty[/tex] compared with a score of Gaussian theorem.

## Homework Equations

Gauss theorem

[tex]\int_S \vec{E}\cdot\vec{dS}=\frac{q}{\epsilon_0}[/tex]

## The Attempt at a Solution

I calculate first part of assignment.

[tex]\vec{E}_A=\frac{1}{4\pi\epsilon_0}\int_S\frac{\sigma dS}{r^3}\vec{r}[/tex]

[tex]dS=\rho d\rho d\varphi[/tex]

[tex]r=\sqrt{\rho^2+z^2}[/tex]

[tex]\vec{r}=z\vec{e}_z-\rho\vec{e}_{\rho}[/tex]

and get

[tex]\vec{E}_A=\frac{\sigma}{2\epsilon_0}\frac{z}{|z|}(1-cos\alpha_0)[/tex]

When [tex]R\rightarrow \infty[/tex] [tex]\alpha_0\rightarrow \frac{\pi}{2}[/tex]

So when [tex]R\rightarrow \infty[/tex]

[tex]\vec{E}_A=\frac{\sigma}{2\epsilon_0}sgnz \vec{e}_z[/tex]

I don't know how can I do the second part with Gauss theorem? Thanks for your help!