How Does Charge Distribution Affect Electric Field Calculation with Gauss's Law?

[Petar Mali]

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 R\rightarrow \infty compared with a score of Gaussian theorem.

Homework Equations

Gauss theorem

\int_S \vec{E}\cdot\vec{dS}=\frac{q}{\epsilon_0}

The Attempt at a Solution

I calculate first part of assignment.

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

dS=\rho d\rho d\varphi

r=\sqrt{\rho^2+z^2}

\vec{r}=z\vec{e}_z-\rho\vec{e}_{\rho}

and get

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

When R\rightarrow \infty \alpha_0\rightarrow \frac{\pi}{2}

So when R\rightarrow \infty

\vec{E}_A=\frac{\sigma}{2\epsilon_0}sgnz \vec{e}_z

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

Homework Statement

Homework Equations

The Attempt at a Solution

 

[hikaru1221]

When R -> infinity, the plate -> something. What is it?

 

[Petar Mali]

When R -> infinity, the plate -> something. What is it?

infinite plane?

 

[hikaru1221]

Correct
And what does the Gauss theorem give for E of an uniformly charged infinite plane?