Charge

Electric charge

• Electric charge is quantized, that is, exists in discrete quantities which are integer multiples of the elementary charge $e=1.6022\times 10^{-19}\mathsf{C}\approx 1.6\times 10^{-19}\mathsf{C}$
  • The charge of an electron is $-e$ and the charge of a proton is $+e$
  • The SI unit of charge is the coulomb (C)
• Conservation of charge: the total charge in an isolated system remains constant
• An object can become charged by:
  • rubbing (friction)
  • conduction (transfer of charge from one charged object to another by touching)
  • induction

Coulomb’s Law

\documentclass[varwidth]{standalone}
\usepackage{xcolor,tikz,amsmath}
\newcommand*\circled[2]{\tikz[baseline=(char.base)]{\node[shape=circle,draw,fill=#1,inner sep=2pt] (char) {#2};}}
\begin{document}
$\underset{q_1}{\circled{red!40}{$\pm$}}$ 
$\underset{\vec{\mathbf{F}}_{12}}{\longrightarrow}$ 
$\underset{\vec{\mathbf{F}}_{21}}{\longleftarrow}$ 
$\underset{q_2}{\circled{blue!40}{$\mp$}}$
\\
$\underset{\vec{\mathbf{F}}_{12}}{\longleftarrow}$
$\underset{q_1}{\circled{red!40}{$\pm$}}$
$\underset{q_2}{\circled{red!40}{$\pm$}}$ 
$\underset{\vec{\mathbf{F}}_{21}}{\longrightarrow}$
\end{document}

• $F=k\frac{q_1{q}_2}{r^2}$ is the electrostatic force (or Coulomb force) between two charges (in $\mathsf{N}$)

  • $q_1$ and $q_2$ are the magnitudes of the charges (in $\mathsf{C}$)
  • $r$ is the distance between the charges (in $\mathsf{m}$)
  • $k=\frac{1}{4\pi {ϵ}_0}=8.99\times 10^9\mathsf{N}\cdot {\mathsf{m}}^2/{\mathsf{C}}^2$ is Coulomb’s constant
  • ${ϵ}_0=\frac{1}{4\pi k}=8.85\times 10^{-12}{\mathsf{C}}^2/N\cdot {\mathsf{m}}^2$ is the permittivity of free space

• limitations and assumptions of Coulomb’s Lawtodo

  • point charges
  • objects are at rest (electrostatics force)
  • electric force

• (Superposition principle) The total force on a charge is the vector sum of the forces exerted by the other charges

• $|{\vec{F}}_E|=\frac{1}{4\pi {ϵ}_0}\frac{|q_1{q}_2|}{r^2}=k\frac{|q_1{q}_2|}{r^2}$todo

Electric field

• The electric field of is defined as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal test charge at rest at that point
• SI unit of electric field is $N/C=V/m$
• $k$ is Coulomb’s constant
• $E=\frac{F}{q}$ (vector form: $\vec{\mathbf{E}}=\frac{\vec{\mathbf{F}}}{q}$) or $\vec{\mathbf{E}}=\underset{q\to 0}{\lim }\frac{\vec{\mathbf{F}}}{q}$
  • $E$ is the electric field that a charge $q$ experiences (in $N/C$)
  • $F$ is the force on a charge (in $\mathsf{N}$)
  • $q$ is the test charge (in $\mathsf{C}$)

Electric Field due to a Point Charge

• $E=k\frac{Q}{r^2}⟸E=\frac{F}{q}=\frac{kQq/r^2}{q}$

  • $P$ is the point in space where the electric field is being calculated
  • $Q$ is the point charge creating the electric field (in $\mathsf{C}$)
  • $r$ is the distance between the point $P$ and the charge $Q$ (in $\mathsf{m}$)
  • $E$ is the electric field (at $P$) due to the source charge $Q$ (in $N/C$)
  • $\vec{\mathbf{E}}=k\frac{Q}{r^2}\hat{\mathbf{r}}$ where $\hat{\mathbf{r}}$ is the unit vector pointing from $Q$ to $P$

• (Superposition Principle) The total electric field at a point in space is the vector sum of the electric fields due to the individual charges

  • ${\vec{\mathbf{E}}}_{\text{total}}={\vec{\mathbf{E}}}_1+{\vec{\mathbf{E}}}_2+{\vec{\mathbf{E}}}_3+...$

• $\vec{E}=\frac{1}{4\pi {ϵ}_0}\int \frac{dq}{r^2}\hat{r}$todo

\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\definecolor{_red}{HTML}{D63146}
\definecolor{_pink}{HTML}{ef3875}
\definecolor{_green}{HTML}{5dc3ad}
\newcommand{\customarrow}[5]{
	\draw[ultra thick, arrows = {-Stealth[reversed, reversed]}, color=#4] (#1,#2) -- (#3,#2) node[midway, below] {#5};
}
\begin{document}
\begin{tikzpicture}
\customarrow{0}{0}{1}{_red}{$\vec{\mathbf{E}}$}
\draw[fill] (0,0) circle [radius=0.08] node[above] {$P$};
\draw[fill=_green, draw=none] (2,0) circle [radius=0.2] node[right, xshift=0.2cm] {$Q$};
\node at (2, 0) {$-$};
\node at (4, 0) {$\mathsf{negative\ charge}$};
\draw[|-|] (0,0.7) -- (2,0.7) node[midway, below] {$r$};
\customarrow{0}{-1}{-1}{_red}{$\vec{\mathbf{E}}$}
\draw[fill] (0,-1) circle [radius=0.08] node[above] {$P$};
\draw[fill=_pink, draw=none] (2,-1) circle [radius=0.2] node[right, xshift=0.2cm] {$Q$};
\node at (2, -1) {$+$};
\node at (4, -1) {$\mathsf{positive\ charge}$};
\end{tikzpicture}
\end{document}

Notes

• There is no electric charge at point $P$. But there is an electric field there. The only real charge is $Q$.
• Notice that $E$ depends only on the charge $Q$ which produces the electric field, and not on the value of the test charge $q$.
• In the figure, the electric field is positive, so it points towards a negative charge and away from a positive charge. But if the electric field is negative, it is the opposite.

Relationship between electric field and potential difference

• $E=-\frac{V_{ba}}{d}$ is the electric field (uniform $\vec{\mathbf{E}}$)
  • $V_{ba}$ is the potential difference between points $a$ and $b$ (in $\mathsf{V}$)
  • $d$ is the distance between the points (in $\mathsf{m}$)

Electric Field Lines

• $E=\frac{Q}{{ϵ}_{0}A}$ is

  • where $A$ is the area of the Gaussian surface

• Electric field lines indicate the direction of the electric field; the field points in the direction tangent to the field line at any point (note that the field lines never cross)

• The lines are drawn such that the magnitude of the electric field, $E$, is proportional to the number of lines crossing unit area perpendicular to the lines. The closer the lines, the stronger the field

• The lines start on positive charges and end on negative charges

• ${\Phi }_E=\int \vec{E}\cdot d\vec{A}$

• $\oint \vec{E}\cdot d\vec{A}=\frac{q_{enc}}{{ϵ}_0}$

• $Q_{\text{total}}=\int \rho (r)dV$

todo

• electric field lines
• electric dipole
• static electric field
• equipotential surfaces, equipotential lines