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?