1.5 Gauss's Theorem

1.5 Gauss's Theorem

The total electric flux through a closed surface is:

$$\oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$

Applications of Gauss's Law

1. Infinite line charge (linear charge density $\lambda$):

$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$

2. Infinite plane sheet (surface charge density $\sigma$):

$$E = \frac{\sigma}{2\varepsilon_0}$$

3. Uniformly charged thin spherical shell (total charge $q$, radius $R$):

$$E = \begin{cases} \dfrac{1}{4\pi\varepsilon_0}\cdot\dfrac{q}{r^2} & r > R \\[10pt] 0 & r < R \end{cases}$$

Example

A uniformly charged infinite plane sheet has $\sigma = 5 \times 10^{-6}\,\text{C/m}^2$. Find the electric field on either side.

$$E = \frac{\sigma}{2\varepsilon_0} = \frac{5 \times 10^{-6}}{2 \times 8.854 \times 10^{-12}} \approx 2.82 \times 10^5\,\text{N/C}$$