PX157 - B6b - electric field around a conductor

• eg: a spherical conductor with charge, $Q$
  • outside the sphere:
    
  $$\subset \supset \iint \overrightarrow{E}d\overrightarrow{S}=\frac{Q_{encl.}}{{ϵ}_0}⟹\overrightarrow{E}=\frac{Q}{4\pi {ϵ}_0{r}^2}\hat{r}$$
  • inside the sphere:
    
  $$\overrightarrow{E}=\overrightarrow{0}$$
  • surface charge density, $\sigma =\frac{Q}{4\pi a^2}$, outside:

• eg: a conducting infinite plane with a finite width
  
  • the excess charges will be on the top and the bottom planes, each with a charge density, $\frac{\sigma }{2}$
  • the gaussian surface is a cylinder that dissects one face of the plane:

- ${\iint }_{curved}\overrightarrow{E}d\overrightarrow{S}=0$ as $\overrightarrow{E}\parallel d\overrightarrow{S}$
- ${\iint }_{bottom}\overrightarrow{E}d\overrightarrow{S}=0$ as it is inside the conductor
$$\oiint \vec E,d\vec S = \iint E \hat z \cdot \hat z , dS = E\iint dS = EA$$
- for enclosed charge, $Q_{encl.}=\frac{\sigma }{2}A$: