Gauss's law

Reading: Chapter 22, GAUSS'S LAW

Study guide: Chapter 23

  1. Finding the Electric Field for continuous sources (vector integrals)
  2. Fields of a few special geometric shapes
  3. Gauss's Law relating field to charge enclosed by a surface
  4. Fields around surfaces and conductors

Vector integrals

In Lab we found electric fields....


On a line perpendicular to a charged rod (line charge) with charge density

$$\lambda\equiv \frac{Q}{L}$$

It was exactly

$$\myv E(x) = \frac{kQ}{x\sqrt{x^2+(L/2)^2}}\,\uv x$$

By symmetry you figured out that $E_y = 0$.

When $x\gg L$: $$E \to k\frac{Q}{x^2}$$

As $L$ gets large compared to $x$: $$ E\to k\frac{2Q}{xL}=k\frac{2}{x}\lambda$$ compared to $1/r^2$ dependence for point charge.

Then we found the charge near an infinite plane with surface charge density $\sigma=Q/A$ to be $$\myv E(z) = 2\pi k\sigma \uv z$$ ..which doesn't depend on $z$ at all!

and it only hurt for a little while!


Gauss's Law

Any charge in the box? How could we tell??

Flux $\Phi$ of electric field through a surface ${\cal S}$: $$\Phi=\int_{\cal S} \myv E \cdot \uv n \,dA=\int_{\cal S} E_\perp dA$$

Gauss's Law : $$\oint E_\perp\,dA = \frac{q_\text{enc}}{\epsilon_0}=4\pi k q_\text{enc}$$

"The flux of $\myv E$ through any closed surface is equal to the total charge inside the surface (divided by $\epsilon_o$)."

Image Credits

Deborah Leigh