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Reading: Chapter 22, GAUSS'S LAW
Study guide: Chapter 23
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!
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 $\epsilon_o$)."