Electric charge

Electric charge is quantized, that is, exists in discrete quantities which are integer multiples of the elementary charge $e = 1.6022 \times 1 0^{- 19} C \approx 1.6 \times 1 0^{- 19} C$
- 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 $N$ )
- $q_{1}$ and $q_{2}$ are the magnitudes of the charges (in $C$ )
- $r$ is the distance between the charges (in $m$ )
- $k = \frac{1}{4 π ϵ _{0}} = 8.99 \times 1 0^{9} N \cdot m^{2} / C^{2}$ is Coulomb’s constant
- $ϵ_{0} = \frac{1}{4 πk} = 8.85 \times 1 0^{- 12} C^{2} /N \cdot m^{2}$ is the permittivity of free space
limitations and assumptions of Coulomb’s Lawtodo
- point charges
- objects are at rest (electrostatics force)
- electric force

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: $E = \frac{F}{q}$ )
- $E$ is the electric field that a charge $q$ experiences (in $N/C$ )
- $F$ is the force on a charge (in $N$ )
- $q$ is the test charge (in $C$ )
$E = - \frac{V _{ba}}{d}$ is the electric field (uniform $E$ )
- $V_{ba}$ is the potential difference between points $a$ and $b$ (in $V$ )
- $d$ is the distance between the points (in $m$ )

Electric field of a single point charge

$E = k \frac{Q}{r ^{2}}$ (vector form: $E = k \frac{Q}{r ^{2}} \hat{r}$ )
- $P$ is the point in space where the electric field is being calculated
- $Q$ is the point charge creating the electric field (in $C$ )
- $E$ is the electric field created by a point charge $Q$ (in $N/C$ )
- $r$ is the distance between the charge and the point in space (in $m$ )
- $\hat{r}$ is the unit vector pointing from the charge to the point in space

\documentclass{standalone}
\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}

There is no electric charge at point $P$ . But there is an electric field there. The only real charge is $Q$ .

Superposition principle

the total force on a charge is the vector sum of the forces exerted by the other charges

todo

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

Electric potential energy

Electric potential energy is the energy stored in a system of charges due to their positions relative to each other
$E$ is an uniform electric field,
$q$ is a tiny positive charge first placed at point $a$ near the positive plate and then released from rest and accelerates toward the negative plate.
- $W = F d = qE d$ is the work done by the electric field $E$ to move a charge $q$ a distance $d$
- $Δ PE = PE_{b} - PE_{a} = - W$ is the change in electric potential energy when a charge $q$ moves from point $a$ to point $b$
$W_{ext} = Q_{2} V = k \frac{Q _{1} Q _{2}}{r}$ is the work done by an external force to move a charge $Q_{2}$ from infinity to a distance $r$ from a charge $Q_{1}$
- $W_{ext}$ is the work done by an external force (in $J$ )
- $Q_{1}$ is the charge creating the electric field (in $C$ )
- $Q_{2}$ is the charge being moved (in $C$ )
- $r$ is the distance between the charges (in $m$ )

Electric energy

Electric energy is the capacity to do work
$E = \int_{t_{0}}^{t} P d t = \int_{t_{0}}^{t} V I d t$
- $E$ is electrical energy (in joules) absorbed or supplied by an element from time $t_{0}$ to time $t$
- $P$ is power (in watts)
- $V$ is voltage (in volts)
- $I$ is current (in amperes)

Electric Potential

Electric potential (or simply potential) is the electric potential energy per unit charge.
$V = \frac{PE _{a}}{q}$
- $V$ is the electric potential (in $V$ , volts)
- $PE_{a}$ is the electric potential energy at point $a$ (in $J$ , joules)
- $q$ is the charge (in $C$ , coulombs)
$V = k \frac{Q}{r}$ (Coulomb potential)todo
- $V$ is the electric potential due to a point charge $Q$ (in $V$ )
- $Q$ is the point charge creating the electric potential (in $C$ )
- $r$ is the distance between the charge and the point in space (in $m$ )
- $k$ is Coulomb’s constant

Voltage

Voltage (or (electrical) potential difference) is the difference in electric potential between two points in a circuit.
- $V = V_{a} - V_{b} = \frac{PE _{a} - PE _{b}}{q}$ is the voltage between points $a$ and $b$ .
  - $V$ is the work done in moving unit charge from $b$ to $a$ (in $V$ )
  - $V_{a}$ and $V_{b}$ are the electric potentials at points $a$ and $b$ (in $V$ )
- The SI unit of voltage is the volt (V), defined as $1 V = 1 J/C$ .
- We often use ground (0 V) or infinity as a reference point.
- Given a point $a$ with a higher potential $V_{a}$ and a point $b$ with a lower potential $V_{b}$ :
  - A negative charge placed at $b$ has higher potential energy than at $a$ , so it will move from $b$ to $a$ (when released) and decrease its potential energy.
    - $PE_{a} < PE_{b} ⟹ Δ PE < 0$
  - A positive charge placed at $a$ has higher potential energy than at $b$ , so it will move from $a$ to $b$ (when released) and decrease its potential energy.
    - $PE_{a} > PE_{b} ⟹ Δ PE > 0$
  - In both cases, $Δ V = V_{a} - V_{b} > 0$ .
$V_{ba} = - d E$ is the electric field (uniform $E$ )
- $V_{ba}$ is the potential difference between points $a$ and $b$ (in $V$ )
- $d$ is the distance between the points (in $m$ )
- $E$ is the electric field (in $N/C$ )

Electric power

Electric power is the rate of transfer of electrical energy within a circuit
$P = \frac{d E}{d t} = \frac{d E}{d q} \frac{d q}{d t} = V I$
- $E$ is electrical energy (in joules)
- $t$ is time (in seconds)
- $P$ is power (in watts)
- $V$ is voltage (in volts)
- $I$ is current (in amperes)
- $q$ is charge (in coulombs)

Electric current

Electric current is the flow of electric charge
$I = \frac{d q}{d t}$
- $I$ is the electric current of the circuit (in $A$ )
- $q$ is the electric charge has passed through some surface in the circuit up to time $t$ (in $C$ )
- $t$ is interval during which the charge flows (in $s$ )
The SI unit of current is the ampere (abbreviated $amp$ or $A$ ), defined as $1 A = 1 C/s$
A current is positive if it flows from the positive terminal of a voltage source to the negative terminal

Resistance & Conductance

\usepackage{circuitikz}
\begin{document}
\begin{circuitikz}[american, thick]
\draw (0,0) to [V, l={$V$}, invert](0,2) to[short, >, i=$I$](2,2) to [R, l={$R$}](2,0) to (0,0);
\end{circuitikz}
\end{document}

the electric current through a conductor between two points is directly proportional to the voltage across the two points
- Ohm’s Law holds for ohmic materials (like most metals) but not for non-ohmic materials (like diodes, transistors, and other semiconductors)
- The unit of resistance is the $ohm$ ( $Ω$ ) defined as $1 Ω = 1 V/A$
- The reciprocal of resistance is called the electrical conductance (in $S$ , siemens, which is $1 S = 1 Ω^{- 1}$ )
- $V = I R$
  - $V$ is the voltage (in $V$ )
  - $I$ is the current (in $A$ )
  - $R$ is the resistance (in $Ω$ )
Electrical resistivity (or specific resistance) denoted by $ρ$ , is a measure of how strongly a material opposes the flow of electric current
- $R = ρ \frac{L}{A}$
  - $R$ is the resistance of the conductor (in $Ω$ )
  - $ρ$ is the resistivity of the material (in $Ω \cdot m$ )
  - $L$ is the length of the conductor (in $m$ )
  - $A$ is the cross-sectional area of the conductor (in $m^{2}$ )
- The reciprocal of the resistivity, called the electrical conductivity (or specific conductance) is $σ = \frac{1}{ρ}$ (in $S/m$ , siemens per meter, or $(Ω \cdot m)^{- 1}$ )

Resistor

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american, thick]
\draw (0,0) to [R, l={$R$}](2,0);
\end{circuitikz}
\end{document}

Kirchhoff’s Laws

Kirchhoff’s Current Law

The sum of the currents flowing into any node is equal to the sum of the currents flowing out

Kirchhoff’s Voltage Law

The voltage drop is the decrease in electric potential along the path of a current flowing in a circuit
In one traversal of any closed loop, the sum of the voltage rises equals the sum of the voltage drops.
given a circuit with a voltage source $V$ and resistors $R_{1}, R_{2}, \dots, R_{n}$ :
- $I = \frac{V}{\sum R _{i}}$ is the current through the circuit
- $V_{0}, V_{1}, V_{2}, \dots, V_{n}$
- $I \cdot R_{i}$ is the voltage drop across the $i$ -th resistor
- $V_{i}$ is equal to $V_{i - 1}$ minus the voltage drop across $R_{i}$
  - $V_{0} = V$
  - $V_{n} = 0$

Electrical Circuits

elements of an electric circuit:
- A branch represents a single two-terminal element (such as a voltage source or a resistor)
- A node is the point of connection between two or more branches
  - Nodes that are connected by perfectly conducting wires are considered to be the same node
classification of elements:
- terminals number:
  - One-port elements (two terminals) - dioes, resistors, capacitors, inductors
  - Two-port elements (four terminals)
  - Multiport elements
- energy source:
  - passive elements do not have a source of energy - dioes, resistors, capacitors, inductors
  - active elements (or sources) have a source of energy - voltage sources, current sources - dependent sources
- linearity:
  - linear elements have a linear relationship between voltage and current
    - resistors, capacitors, inductors
  - nonlinear elements are elements in which the relation between voltage and current is a nonlinear function
    - dioes
sources
- An ideal independent source is an active element that provides a specified voltage or current that is completely independent of other circuit elements
A short circuit is a circuit element with resistance approaching zero, so $I = R \to 0 lim \frac{V}{R} = \infty$
An open circuit is a circuit element with resistance approaching infinity, so $I = R \to \infty lim \frac{V}{R} = 0$
todo
- electric current (aka: conventional current) is a flow of electric charge
  - electric current from $+$ to $-$ is equivalent to a negative electron flow from $-$ to $+$
  - conventional current combines the effects of electron, ion, proton, and hole flows all into one number.
  - conventional current is not the opposite of electron current
  - Electron current is a subset of conventional current.
- A 19th-century convention, still in use, treats any electric current as a flow of positive charge from a region of positive potential to one of negative potential.
  - The real motion, however, in the case of electrons flowing through a metal conductor, is in the opposite direction, from negative to positive.
todo
- Components of an electrical circuit
  - Voltage source
    - Terminals:
      - Positive voltage terminal (higher voltage)
      - Negative voltage terminal (lower voltage)
- Passive sign convention (PSC)
  - electric power is positive if it flows out of the circuit into an electrical component
  - electric power negative if it flows into the circuit out of a component
  - Passive components (loads) will have positive power dissipation ( $p > 0$ ) and positive resistance ( $r > 0$ )
  - Active components (sources) will have negative power dissipation ( $p < 0$ ) and negative resistance ( $r < 0$ )
  - The conventional current variable $i$ :
    - If $i$ is defined such that positive current enters the device through the positive voltage terminal:
      - $p = v i$ and $r = \frac{v}{i}$
    - If $i$ is defined such that positive current enters the device through the negative voltage terminal:
      - $p = - v i$ and $r = - \frac{v}{i}$

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american]
\draw (0,0) to [V, l={$v$}](2,0) to[short, =>, i=$i$](3,0);
\end{circuitikz}
\end{document}

Light-emitting diode

A light-emitting diode (LED) is a semiconductor device that emits light when an electric current passes through it

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american, thick]
\draw (0,0) to [leDo, l={$$}](2,0);	
\end{circuitikz}
\end{document}

Capacitance

A capacitor is a device that can store electric charge, and normally consists of two condaucting objects (usually plates or sheets) placed near each other but not touching
- The capacitor was originally known as the condenser
- $Q = C V$ is the relationship between charge, capacitance, and voltage
  - $Q$ is the charge stored on the capacitor (in $C$ )
  - $C$ is the capacitance of the capacitor (in $F$ , farads)
  - $V$ is the voltage across the capacitor (in $V$ )
- $C = ϵ_{0} \frac{A}{d}$ is the capacitance of a parallel-plate capacitor
  - $C$ is the capacitance (in $F$ )
  - $ϵ_{0}$ is the permittivity of free space (or vacuum permittivity) that is $8.85 \times 1 0^{- 12} F/m$
  - $A$ is the area of the plates (in $m^{2}$ )
  - $d$ is the separation between the plates (in $m$ )

Voltage Source

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american, thick]
\draw (0,0) to [V, l={$V$}](2,0);
\end{circuitikz}
\end{document}

Notation

Symbols for independent voltage sources

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american, voltage dir=RP]
% First circuit (A)
\draw (0.5,-1) node[above] {\textsf{constant/time-varying voltage}};
\draw (0,2.5) to [short, -o] (1,2.5);
\draw (0,0) to [V, -, v={$v$}](0,2.5);
\draw (0,0) to [short, -o] (1,0);
% Second circuit (B), shifted to the right by 5 units
\begin{scope}[shift={(5,0)}]
\draw (0.5,-1) node[above] {\textsf{constant voltage (DC)}};
\draw (0,2.5) to [short, -o] (1,2.5);
\draw (0,0) to [battery1, -, v={$V$}](0,2.5);
\draw (0,0) to [short, -o] (1,0);
\end{scope}
\end{circuitikz}
\end{document}

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american, voltage dir=RP]
\draw (0,0) to [V, l={$9\ \mathsf{V}$}](0,2) to[short](2,2) to [R, l={$10\ \mathsf{k\Omega}$}](2,0) to (0,0);
\end{circuitikz}
\end{document}

or shoter version

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american]
\draw (0,0) to [short, o-, l={$9\ \mathsf{V}$}](0,-1) to[R, l={$10\ \mathsf{k\Omega}$}](0,-2) to[short](0,-2.5) node[ground] {};
\end{circuitikz}
\end{document}

Ground

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american]
\draw (0,0) node[ground] {};
\end{circuitikz}
\end{document}

Electric Battery

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american]
 
% battery symbol `battery1`
%\draw (2,0) to [battery1, l={$\mathsf{}$}] (3,0);
 
% `battery1`, polarity marks below, without text
\draw (0,0) to [battery1, l={$\mathsf{}$}, v_=$\;$] (1,0);
 
\end{circuitikz}
\end{document}

A Galvanic cell (or voltaic cell) is an electrochemical cell that converts chemical energy into electrical energy, consisting of:
- two electrodes: conductors through which electric current enters or leaves the cell
  - anode ( $-$ ): the electrode where oxidation occurs (loses electrons)
  - cathode ( $+$ ): the electrode where reduction occurs (gains electrons)
  - the part of each electrode outside the solution is called the terminal which is used to connect the cell to a circuit
- An electrolyte is a substance that conducts electricity by allowing ions to move and can exist either as a liquid (wet cell) or a paste (dry cell)
A battery is a collection of electric cells connected together
The total voltage of a series (connected end-to-end positive to negative) connection is the sum of the voltages of the individual cells

Example: Bunsen cell (zinc-carbon & dilute sulfuric acid)

Electrodes:

Zinc anode: Zinc metal dissolves into the electrolyte as zinc ions ( $\ce Z n^{2 +}$ ), leaving behind electrons on the zinc electrode, which becomes negatively charged.

Carbon cathode: The sulfuric acid electrolyte pulls electrons from the carbon electrode, making it positively charged.

Electrolyte: The dilute sulfuric acid ( $\ce H 2 SO 4$ ) serves as the electrolyte, allowing ions (e.g., $\ce Z n^{2 +}$ ) to move between the electrodes while completing the internal charge balance.

Chemical Reaction:

At the zinc anode: Zinc undergoes oxidation ( $\ce Z n - > Z n^{2 +} + 2 e^{-}$ ), releasing electrons into the electrode and producing zinc ions.

At the carbon cathode: Electrons flow from the zinc anode to the carbon cathode through an external circuit, where reduction reactions can take place.

Voltage: The zinc electrode becomes negatively charged, and the carbon electrode becomes positively charged, creating a potential difference (voltage) between the two terminals.

Current:

(closed circuit) Electrons flow through the external circuit from the zinc anode to the carbon cathode, creating an electric current.

(open circuit) When the terminals are not connected, only a small amount of the zinc is dissolved,

todo
- terms
  - dry cell
  - half-cell
  - salt bridge
  - standard electrode potential
  - volatge regulator
  - State of charge (SoC)
  - State of health (SoH)
  - battery management system (BMS)
- why does electric cars typically have only one gear?

Digital electronics

transistor
- A transistor is a semiconductor device that can amplify or switch current
  - BJT (bipolar junction transistor)
    - The base is the control terminal
    - The collector is the high-voltage terminal
    - The emitter is the low-voltage terminal
    - Type of BJT:
      - NPN (negative-positive-negative)
      - PNP (positive-negative-positive)
  - FET (field-effect transistor)
a logic gate is a device that implements a logic function

Figure: An NPN transistor as a switch

todo

\usepackage{color,graphicx,circuitikz}
\begin{document}
\begin{circuitikz}[american, scale=0.7, every node/.style={transform shape}]
\draw (0,0) node[npn](npn){};
\draw (npn.B) to[R, -o] ++(-2,0) node[left] {$V_{\mathrm{in}}$};
\draw (npn.C) to[short, -o] ++(0,0) node[above] {$V_{\mathrm{cc}}$};
\draw (npn.E) to[R, -] ++(0,-2) node[ground] {};
\draw (npn.E) to[short, -o] ++(0.5,0) node[right] {$V_{\mathrm{out}}$};
\end{circuitikz}	
\end{document}

Digital signal processing

The difference between the original continuous analog signal and its digital approximation is called the quantization error
The resolution (or bit depth) is the number of bits or values for the voltage of each sample (=measurement)
The sampling rate is the number times per second the original analog voltage is measured (“sampled”)
ADC (analog-to-digital converter) is a device that converts a continuous analog signal to a discrete digital signal
DAC (digital-to-analog converter) is a device that converts a digital signal to an analog signal
bandwidth, Spectral band, frequency bandtodo

Explorer

Electricity

Electric charge

Coulomb’s Law

Electric field

Electric field of a single point charge

Superposition principle

todo

Electric potential energy

Electric energy

Electric Potential

Voltage

Electric power

Electric current

Resistance & Conductance

Resistor

Kirchhoff’s Laws

Kirchhoff’s Current Law

Kirchhoff’s Voltage Law

Electrical Circuits

Light-emitting diode

Capacitance

Voltage Source

Ground

Electric Battery

Digital electronics

Figure: An NPN transistor as a switch

Digital signal processing

Table of Contents

Backlinks