PHYS 208 Lecture 4

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Electric Field

Point charge

$\overrightarrow{E}(\overrightarrow{r})=\frac{kq}{r^2}\hat{r}$

Multiple charges

${\overrightarrow{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\sum {\overrightarrow{E}}_i=\sum _{i=0}^n\frac{k{q}_i}{{r_i}^2}{\hat{r}}_i$

Continuum of charges

${\overrightarrow{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\int \mathrm{d}\overrightarrow{E}=\underset{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\int }\frac{k\mathrm{d}q}{r^2}\hat{r}$

Example

Uniformly distributed charge $Q$ along a line of length $l$:

$\mathrm{d}q=\lambda \mathrm{d}x=\frac{Q}{l}$

Therefore, the total field at a point $P$ is:

$\overrightarrow{E}=\int \frac{kQ\mathrm{d}x}{l{r}^2}$

Electric Field Lines

Imaginary lines to graphically represent the Electric field in the space around a charge

Properties and Corollaries

• points away from a positive charge and points toward a negative charge
  • Start at positive charges and end at negative charges
• Direction aways tangent to the electric field at that point
• strength is proportional to number of lines in the area
  • parallel lines of equal density represent a uniform electric field
• Lines will never cross

The field lines between two similarly charged particles will behave asymptotically at the plane between them.

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Ch. 22: Gauss's Law

Flux of an Electric Field

the rate of transfer of fluid, particles, or energy across a given surface

Take the integral over a surface...?

$\mathrm{\Phi }=\oint \overrightarrow{E}\cdot \mathrm{d}\overrightarrow{A}$

• $\mathrm{d}\overrightarrow{A}=\mathrm{d}A\hat{n}$ is perpendicular to the surface
• Units = Nm²/C
• Open surface Electric field lines cross through a single plane
• Closed surface enters through plane and exits through plane on opposite side of object
• Proportional to the number of field lines going into a shape (influx; counted negative) and the number of field lines coming out of a shape (outflux; counted positive)
• No net charge inside surface: flux = 0

Gauss's Law

For any surface with any net charge (any distribution) inside, the flux is
$\mathrm{\Phi }=EA=\oint \overrightarrow{E}\cdot \mathrm{d}\overrightarrow{A}=\frac{Q_{\mathrm{n}\mathrm{e}\mathrm{t}}}{{ϵ}_0}$