EGR 153 Comprehensive Study Guide: Electrical Forces, Fields, Potentials, and Capacitors

I. Historical Foundations and Motivation

• Timeline of Early Concepts:     * ~600 BC: The Greek philosopher Thales of Miletus recorded that amber (fossilized pine sap) rubbed with fur could attract light objects like feathers. This property was initially thought unique to amber.     * Early 1600s: Francis Bacon introduced the adjective "electric" to the English language, meaning "capable of attracting objects when rubbed."     * 1600: William Gilbert published his treatise de Magnete; he coined the modern Latin term electricus ("of amber") based on the Latin electrum and the Greek elektron.     * 1646: Thomas Browne of England first used the noun "electricity" as a power to attract certain objects.
• The 1700s and Electric Machines:     * Large-scale static electricity effects were achieved using machines where one object (fur) was attached to a wheel and rapidly rubbed against a fixed object (amber). Charge was transferred to anything or anyone touching the body.     * DuFay (Paris): Postulated two types of electricity: "resinous electricity" (on amber rubbed with fur) and "vitreous electricity" (on glass rubbed with silk).
• Benjamin Franklin’s Contributions (~1747):     * Single Fluid Theory: He postulated that electricity is one fluid transferred between materials. Sparking returned the fluid.     * Arbitrary Naming: Franklin defined "positive" as an excess of fluid and "negative" as a deficit.     * Naming Convention: He chose the fluid to flow from amber to fur. In reality, electrons flow from fur to amber, which is why we now say electrons have a "negative" charge. This choice defines modern current direction as opposite to electron flow.     * The Lightning Rod (~1750): Franklin proved lightning was a giant spark caused by atmospheric charge accumulation. He scaled his benchtop experiments (using pointed wires to discharge objects) to invent the lightning rod connected to the ground to discharge buildings before sparks could form.     * Practical Anecdote: In letters to Peter Collinson of the Royal Society in London, Franklin described using electrical shocks to kill turkeys, claiming the meat was more tender.

II. Timeline of Electrical Science and Innovation

• ~1785: Charles-Augustin de Coulomb (France) measured the $1/r^2$ dependence of forces between charges.
• 1800: Alessandro Volta (Italy) invented the battery.
• ~1825–1830: Joseph Henry and others developed electromagnets.
• 1830s: Faraday, Henry, and Davenport developed early electrical motors and generators.
• 1830s–1840: Samuel Morse and others developed the telegraph.
• 1876: Alexander Graham Bell developed the telephone.
• 1879: Thomas Edison developed the first long-lasting light bulb and large-scale power systems (General Electric Company).
• ~1900: J.J. Thompson (England) measured the charge/mass ratio of the electron.
• 1909: Robert Millikan (US) measured the charge on an electron.
• 1906: Lee de Forest developed the first vacuum tube amplifier.
• ~1946: Bardeen, Brattain, and Shockley created the first transistor.
• ~1959: Robert Noyce and Jack Kilby created the first microchip (integrated circuit).

III. The Princeton Connection to Electrical Engineering

• Local Milestones:     * Edison and Francis Upton: Upton was Princeton's first Ph.D. ($<em>1877$).      Joseph Henry: A Princeton professor involved with electromagnets and the telegraph.     * Cyrus Fogg Brackett: Lit the first US classroom electrically in the 1880s.     * The School of Electrical Engineering: Founded around 1889 at Princeton as the first to offer graduate degrees in EE.     * Invention of the Transistor: John Bardeen (Princeton $<em>36$; two-time Nobel Laureate) at Bell Labs in Murray Hill, NJ.      Computing Theory: Alan Turing ($<em>36$) and Professor John von Neumann (Institute for Advanced Study/Princeton).      Industrial Research: Significant work at RCA Princeton Lab on the microchip, solar cells (Bell Labs), and color TV.     * Quantum Phenomena: Nobel Laureates Dan Tsui (EE professor) and Duncan Haldane (Physics professor).

IV. Electrical Charges and Coulomb’s Law

• Basic Properties of Charge:     * Charge ($Q$) can be positive or negative. It is measured in Coulombs ($C$).     * Proton charge ($e$): $+1.6 imes 10^{-19} C$.     * Electron charge ($q$): $-1.6 imes 10^{-19} C$.     * $1$ Coulomb consists of roughly $6 imes 10^{18}$ protons. Charge is additive.
• Coulomb Force ($F_{Coul}$):     * Experimental finding by Coulomb around 1785.     * The force between two point charges ($Q_1, Q_2$) is proportional to the product of charges and inversely proportional to the center-to-center distance ($r$) squared:     $F_{Coul.} = \frac{Q_1 imes Q_2}{(4 imes ext{pi} imes ext{epsilon}_o) imes r^2}$     * Permittivity of Free Space ($ext{epsilon}_o$): $ext{epsilon}_o = 8.8 imes 10^{-12} C^2 / (N imes m^2) = 8.8 imes 10^{-12} F / m$.     * Superposition: The total force on a test charge is the vector sum of individual forces from all surrounding charges.
• Comparison to Gravity:     * $F_{Gravity} = \frac{G imes M_1 imes M_2}{r^2}$.     * Ratio for one electron and one proton: $\frac{F_{Coul}}{F_{Gravity}} \approx 10^{39}$.     * Why gravity dominates planetary orbits: Large bodies are electrically neutral (equal protons and electrons), while mass is always additive and positive.

V. Electric Fields ($ext{epsilon}$)

• Concept of the Electric Field:     * Defined to simplify calculations involving zillions ($10^{23}$ atoms/$cm^3$) of charges.     * $ext{epsilon}(x) = \frac{F_{Total, test}(x)}{Q_{test}}$.     * Units: Newtons/Coulomb ($N/C$) or Volts/meter ($V/m$).     * If the electric field is known, the force on any second test charge is $F_{Total} = Q_{test2} imes ext{epsilon}(x)$.
• Isolated Charge Field:     * Strength decreases as $1/r^2$ from the central charge.     * Field points radially outward from a positive charge and radially inward toward a negative charge.
• Electric Field Lines:     * Represent paths of field vectors laid end to end.     * Lines "start" at positive charges and "end" at negative charges.

VI. Electrical Potential Energy and Voltage

• Analogy to Gravity:     * Potential Energy ($U$) in gravity: $U_{Gravity} = M imes g imes H(x)$.     * Potential Energy in electricity: $U_{Electric, Q}(x) = Q imes ext{phi}(x)$.     * Electric Potential ($ext{phi}(x)$): Analogous to height ($H$). It is a scalar measured in Volts ($V$).     * $1 ext{ Volt} = 1 ext{ Joule} / ext{ Coulomb}$.
• Charge Behavior:     * Positive charges move toward lower electric potential (downhill).     * Negative charges move toward higher electric potential (uphill).
• Electric Field and Gradient:     * The electric field is the negative derivative of electric potential with respect to distance:     $ext{epsilon}(x) = -\frac{d ext{phi}}{dx}$     * The electric field points from high potential to low potential and is perpendicular to lines of constant potential (equipotential contours).

VII. Batteries and Voltage Sources

• Voltage Supply (Power Supply):     * A device that maintains a fixed potential difference ($V_{target}$ or $V_{21}$) between two wires by "shoveling" charge as needed.     * Analogous to a "smart pump" maintaining the water level difference between two lakes.
• Alessandro Volta and Batteries (~1800):     * Converts chemical potential energy to electrical potential energy.     * Chemical reactions pump positive charge from the low potential end (Cathode or negative terminal) to the high potential end (Anode or positive terminal).     * A dead battery occurs when the chemicals are fully reacted; re-chargeable batteries can force the reaction backward by applying higher external voltage.

VIII. Capacitors and Charge Storage

• Structure: Two parallel metal plates separated by distance $d$ with a gap area $A$.
• Gap Electric Field ($ext{epsilon}<em>{gap}$):     * Constant throughout the gap if the gap is small: $\text{epsilon}</em>{gap} = \frac{V_C}{d}$.     * There is zero electric field inside the metal plates (no current flow).
• Capacitance ($C$):     * Defined as the ratio of charge ($Q$) to voltage ($V_C$): $C = \frac{Q}{V_C}$.     * Units: Farads ($F$). $1 ext{ Farad} = 1 ext{ Coulomb} / ext{ Volt}$.     * For parallel plates: $C = \frac{A imes ext{epsilon}_o}{d}$.
• Energy Storage:     * Energy stored in a capacitor: $U_{Cap} = \frac{1}{2} C imes V_C^2 = \frac{1}{2} Q imes V_C$.     * Bits in computers are stored as voltages on capacitors ($5 V = ext{logic 1}$, $0 V = ext{logic 0}$).
• The Dielectric Constant ($K$):     * Insulators (dielectrics) placed between plates increase capacitance via polarization.     * Polarization is the slight separation of charge in atoms due to an applied field.     * $C = \frac{A imes K imes ext{epsilon}_o}{d}$.     * Silicon dioxide ($SiO_2$) has $K = 3.9$.

IX. Gauss’s Law

• Concept: "What is generated inside must come out."
• Mathematical Form: The total electric flux through an imaginary closed surface equals the net charge enclosed divided by the dielectric constant:     $\text{Total Flux} = \int \text{epsilon}<em>{out,perp} dA = \frac{Q</em>{inside}}{\text{epsilon}_o}$
• Applications:     * Isolated Point Charge: Results in the Coulomb formula field strength.     * Metal Plate: Field outside a plate with charge per area ($Q/A$) is constant with distance: $\text{epsilon} = \frac{Q}{A \times \text{epsilon}_o}$.     * Infinite Line of Charge: Field magnitude depends on $\frac{1}{R}$.     * Inside Metal: Field is zero in steady-state (charges move to the surface to cancel internal fields).

X. Electrical Current and Resistance

• Electrical Current ($I$):     * Rate of charge flow: $I = \frac{dQ}{dt}$.     * Units: Amperes ($A$). $1 A = 1 C/s$.     * Conventionally, current flows in the direction of positive charge (Franklin's choice), which is opposite to electron flow in wires.
• Microscopic Mechanism:     * Mobile electrons in a solid scatter against vibrating atoms and impurities.     * Mobility ($\text{mu}$): The ratio of average (drift) velocity to electric field: $|v_{avg}| = \text{mu} \times |\text{epsilon}|$.     * Mobility formula: $\text{mu} = \frac{e \times \text{tau}}{m_{electron}}$ ($\text{tau}$ = time between scattering events).     * Highest recorded mobility (Princeton): $5,000 m^2/(V \times s)$.
• Conductivity ($\text{sigma}$) and Resistivity ($\text{rho}$):     * Conductivity: $\text{sigma} = e \times N_m \times \text{mu}$ ($N_m$ = mobile electron density).     * Resistivity: $\text{rho} = 1 / \text{sigma}$. Measured in Ohm-meters ($\text{Omega} \times m$).
• Resistance ($R$):     * $R = \frac{V}{I}$. Measured in Ohms ($\text{Omega} = V/A$).     * For a material of length $L$ and cross-section $W \times H$: $R = \frac{L}{W \times H} \times \text{rho}$.

XI. Circuits and Power Dissipation

• Power Consumption ($P$):     * Rate of energy transfer: $P = I \times V$. Measured in Watts ($W = J/s$).     * In a resistor, potential energy lost by electrons is converted to heat (thermal energy).     * Resistor power formulas: $P = I \times V = I^2 \times R = \frac{V^2}{R}$.
• Kirchhoff’s Laws:     * Kirchhoff’s Voltage Law (KVL): The sum of potential changes around any closed loop must be zero.     * Kirchhoff’s Current Law (KCL): The sum of currents entering a junction must equal the currents leaving (conservation of charge).
• Resistors in Combination:     * Series: $R_{total} = R_1 + R_2 + …$     * Parallel: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + …$
• Electrical Meters:     * Ammeter: Measures current; must be inserted in series; ideally zero resistance.     * Voltmeter: Measures potential difference; connected in parallel; ideally infinite resistance.     * Ohmmeter: Measures resistance of a free-standing component by applying a known voltage and measuring current.

XII. Speed and Power of Computers: RC Circuits

• RC Time Constant:     * The voltage on a capacitor cannot change instantaneously.     * Switching a bit involves charging/discharging a capacitor through a resistor (the switch's internal resistance).     * Discharge equation: $V_C(t) = V_{initial} \times e^{-t/RC}$.     * Charge equation: $V_C(t) = V_{supply} (1 - e^{-t/RC})$.     * Propagation Delay: Proportional to $R \times C$. Large $C$ (wide tanks) or large $R$ (skinny pipes) slow down computers.
• Power Consumption in Switching:     * Energy dissipated as heat per cycle: $\frac{1}{2} C \times V^2$.     * To improve computers: Reduce $V_{supply}$ and reduce capacitance ($C$) by shrinking feature sizes.

XIII. MOSFETs and Logic Gates

• MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor):     * A remote-controlled switch. A voltage on the Gate (top metal plate) controls the resistance of the Channel (lower semiconductor layer) between Source and Drain.     * n-channel MOSFET (NMOS): Turns on with a high (positive) gate voltage. Electrons are the carriers.     * p-channel MOSFET (PMOS): Turns on with a low (negative) gate voltage. "Effectively positive" holes are the carriers.     * Threshold Voltage ($V_T$): The gate voltage at which the channel begins to form (typically $0.5 V$ to $2.0 V$).
• Binary Logic Gates:     * NOT Gate (Inverter): Converts logic 1 (5V) to logic 0 (0V) and vice versa.     * NAND Gate: Output is logic 0 only if all inputs are logic 1.     * CMOS (Complementary MOS): Circuits using both NMOS and PMOS. Ideally dissipates zero power in steady-state stand-by mode because one transistor is always OFF.
• Microfabrication and Scaling:     * Early computers like ENIAC (1945) used $18,000$ unstable vacuum tubes consuming $160,000 W$.     * Integrated Circuits (ICs): Fabricating many transistors on one silicon wafer. Feature sizes have shrunk from $200 imes 10^{-6} m$ (microns) to <10 imes 10^{-9} m (nanometers).     * Scaling Miracle: Shrinking features makes gates both cheaper and faster while consuming less energy.