General Physics 25

Charge

concept and properties

• electric charge is attribute of body. Unlike charges attract, like charges repel.

• charge is quantized. Any observed charge q turns out to be multiples of a certain elementary charge e.

  $$e = 1.60217733 \times 10^{-19} C \\ q = 0, \pm e, \pm2e,\pm3e \cdots$$

Notes

1. Two elementary charged particles inside atoms, proton and electron, make charge quantized.
2. e is very small and of no sign.
3. Neutrons and Protons consist of quarks( $-\frac{1}{3}e$ or $\frac{2}{3}e$).

Applications

Electrostatic paint spraying, Powder coating, Fly-ash precipitation, Nonimpact ink-jet printing and Photocopying.

Coulomb’s Law,1785

The repulsive force between two small spheres charged with same sort of electricity is in the inverse ratio of the squares of the distances between the centers of the spheres.

$$F \propto{\frac{q_1q_2}{r^2}}$$

The force from 1 acting on 2 is:

$$\vec{F}_{12} = \frac{kq_1q_2}{r_{12}^2}\hat{r_{12}} = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r_{12}}$$

Units

	vector / scalar	unit	notes
$r_{12}$	scalar	meter
$\hat{r_{12}}$	vector	1	unit vector pointing from 1 to 2
$\vec{F_{12}}$	vector	Newton	same as $\hat{r_{12}}$
$q$	scalar	Coulomb	signed positive/negative
$k$	scalar	$N\cdot m^2 /C^2$	a special constant, $k = 9 \times 10^9N\cdot m^2 /C^2$
$\epsilon_0$	scalar	$C^2/(N\cdot m^2)$	an elementary constant, $\epsilon_0 = 8.854 \times 10^{-12}C^2/(N\cdot m^2)$

Notes

1. Coulomb’s Law generally holds only for charged objects whose size are much smaller than the distance between them. $i.e.$ only for point charges

2. Assume that $F = k\frac{q_1q_2}{r^\lambda}$, experiments show that $\lambda = 2.01$(by Torsion balance) and $\lambda = 2 \pm 10^{-6}$(by indirect experiment).

3. Law of gravitation, $\vec{F} = m\vec{a}$ , defines inertial mass$m$ and determines gravity constantG.
   Coulomb’s law defines k first, then the basic unit of charge is determined. $1 C = 6 \times 10^{18}e$.

4. Newton’s law of gravitation could be considered an approximation of theory of relativity, while Coulomb’s law is an exact result (not an approximation from higher laws) and remains valid in quantum limit.

5. At level of electrons, Electric force is much stronger than gravitational force.

   $$\frac{F_{elec}}{F_{grav}} = \frac{q_1q_2}{m_1m_2}\frac{k}{G} = 4.17 \times 10^{42}$$

   For an electron,

   $$q = 1.6 \times 10^{-19} C\\ m_e = 9.1 \times 10^{-31} kg$$

Continuous Charge Distribution

Though the charge is quantized in essence, the charge’s distributions are considered continuous in macroscopic EM. For discrete charge distributions, the superpositions of vectors may work. For continuous distributions, we divide the charge into infinitesimal charged elements and each element is regarded as a point charge.

Charge density

• Linear charge density $\lambda = \frac{\rm d\it{q}}{\rm d\it{x}}$
• Surface charge density $\sigma = \frac{\rm d\it{q}}{\rm d\it{A}}$
• Volume charge density $\rho = \frac{\rm d\it{q}}{\rm d\it{V}}$

$$\rm d{\it\vec{F}_{q_0}} = \frac{1}{4\pi\epsilon_0}\frac{q_0(\rho\rm d\it V)}{\vert\vec{r} - \vec{r}'\vert^3} (\vec{r} - \vec{r}') \\ \vec{F}_{q_0} = \iiint_{q}\rm d{\it\vec{F}_{q_0}}$$

Examples

Ring

A ring of radius $R$ is charged $q$ uniformly. A point charge $q_0$ is at distance $z$ from the center of the ring.

$$\lambda = \frac{q}{2\pi R}$$

$$\rm d {\it q} = \lambda (\it R \rm d \theta) = \frac{\it q \rm d \phi}{2\pi}$$

$$\rm d {\it F} = \frac{1}{4\pi\epsilon_0}\frac{\it q_0\rm d \it q}{\it z^2 \rm + \it R^2} = \frac{1}{8\pi^2\epsilon_0}\frac{\it q_0q \rm d \phi}{\it z^2 \rm + \it R^2}\\ F_{xy} = 0$$

$$F_z = \int_{0}^{2\pi}{\cos\theta\rm d\it F} = \int_{0}^{2\pi}{\frac{1}{8\pi^2\epsilon_0}\frac{\it zq_0q \rm d \phi}{(\it z^2 \rm + \it R^2)^{\frac{3}{2}}}} = \frac{1}{4\pi\epsilon_0}\frac{\it zq_0q}{(\it z^2 \rm + \it R^2)^{\frac{3}{2}}} \rightarrow \frac{1}{4\pi\epsilon_0}\frac{\it q_0q}{\it z^2}\quad(\frac{R}{z} \rightarrow 0)$$

In vector form:

$$\vec{F} = \vec{F}_z = \frac{1}{4\pi\epsilon_0}\frac{\it zq_0q}{(\it z^2 \rm + \it R^2)^{\frac{3}{2}}}\hat{k} \rightarrow \frac{1}{4\pi\epsilon_0}\frac{\it q_0q}{\it z^2}\hat{k}\quad(\frac{R}{z} \rightarrow 0)$$

Disk

A disk of radius $R$ is charged $q$ uniformly. A point charge $q_0$ is at distance $z$ from the center of the disk.

$$\sigma = \frac{q}{\pi R^2}$$

Divide the disk into infinitesimal charged rings, which has been calculated above.

$$\rm d {\it q} = \sigma (2\pi \omega \rm d \omega) = \frac{2\it q\omega\rm d\omega}{R^2}$$

In vector form:

$$\vec{F} = \vec{F}_z = \frac{1}{4\pi\epsilon_0}\frac{\it zq_0q}{(\it z^2 \rm + \it R^2)^{\frac{3}{2}}}\hat{k} \rightarrow \frac{1}{4\pi\epsilon_0}\frac{\it q_0q}{\it z^2}\hat{k}\quad(\frac{R}{z} \rightarrow 0)$$

$$\rm d {\it\vec{F}} = \frac{1}{4\pi\epsilon_0}\frac{\it zq_0\rm d \it q}{(\it z^2 \rm + \it \omega^2)^{\frac{3}{2}}}\hat{k} = \frac{1}{2\pi\epsilon_0}\frac{\it zq_0q\omega\rm d \omega}{\it R^2(\it z^2 \rm + \omega^2)^{\frac{3}{2}}}\hat{k}$$

$$\vec{F} = \int_{0}^{R}{\rm d\it \vec{F}} = \int_{0}^{R}{\frac{\it zq_0q}{4\pi\epsilon_0\it R^2}\frac{2\omega\rm d \omega}{(\it z^2 \rm + \omega^2)^{\frac{3}{2}}}}\hat{k} = \frac{q_0q}{2\pi\epsilon_0\it R^2}(1 - \frac{z}{\sqrt{\ z^2 + R^2}})\hat{k} = \frac{q_0\sigma}{2\epsilon_0}(1 - \frac{z}{\sqrt{\ z^2 + R^2}})\hat{k}$$

Using $(1+x)^\alpha \rightarrow 1+\alpha x$​ when $x$​ gets close to $0$​,

$$\vec{F} \rightarrow \frac{1}{4\pi\epsilon_0}\frac{\it q_0q}{\it z^2}\quad(\frac{R}{z} \rightarrow 0)$$

Charge Conservation

In any physical process happened in a closed system, the algebraic sum of the total charges remains invariant.

Vecs and Scals

Notation

Vectors are specified by magnitude(length) and direction, and written like $\vec{F},\vec{v},\hat{r}$.
The magnitude is a scalar quantity, $\vert\vec{F}\vert = F$ .
The unit vector is denoted by ^, indicating only a direction (it has no units!). $\hat{r} = \frac{\vec{r}}{\vert \vec{r}\vert}$

Composition: Vectors can be composited into $x,y$ and $z$ components. $\vec{F} = F_x\hat{x} + F_y\hat{y} + F_z\hat{z}$ .

Superposition

$$\vec{F} = \sum{\vec{F_i}}$$

Conductors, Insulators …

	Once charged, the charges
Insulators	Not free to move (averagely $\leq 1e/cm^3$)	Glass, Plastic, Dry wood
Semiconductors	(averagely $10^{10} \sim 10^{12}e/cm^3$)	Silicon, Germanium
Conductors	All free to move ((averagely $ 10^{23}e/cm^3$)	Aluminum, Copper, Iron, Silver
Superconductors	R = 0,B = 0 under some pressure and temperature conditions