Comprehensive Notes on Static Electricity, DC Electricity & Electromagnetism

Electric Charge Basics

Electric charge is an intrinsic property of matter, analogous to mass. There are two kinds:

• Positive (carried by protons)

• Negative (carried by electrons)

Fundamental numerical values (elementary charge):

$e = 1.6\times10^{-19}\,\text{C}$

$q<em>{\text{electron}} = -e \qquad q</em>{\text{proton}} = +e$

• $e$ : Elementary charge (measured in Coulombs, C)

• $q$ : Charge of electron/proton (measured in Coulombs, C)

Because charge is quantised, every observable charge is an integer multiple of $\pm e$.

Ions and Ionisation

• A neutral atom has equal numbers of protons and electrons, so its net charge $Q = 0$.

• Removing an electron (oxidation) makes a positive ion: $Q = +ne$ where $n$ is the deficit of electrons.

• Adding an electron (reduction) makes a negative ion: $Q = -ne$.

• Ionisation alters only electron count; protons remain fixed in the nucleus.

Electrostatic Forces

• Like charges repel; unlike charges attract.

• The interaction originates from the electric field each charge produces.

• Everyday, neutral objects contain vast but balanced quantities of $+$ and $-$ charge so the net force is normally zero.

Charging Objects & Static vs Current Electricity

• An object becomes charged when electrons are transferred:

  • Gain electrons \rightarrow net negative.

  • Lose electrons \rightarrow net positive.

• Rubbing two insulators (e.g.rubber\text{-}soled shoes on nylon carpet) brings surfaces close enough for electron tunnelling.

• Static electricity: charges are stationary; observed on insulators where charges remain localised.

• Current electricity: charges move through conductors (metals, ionic liquids) producing a continuous flow.

Quantifying Charge

Smallest transferable unit: one electron charge. The net charge on $N$ excess/deficit electrons is

$Q = N e$

• $Q$ : Net charge (measured in Coulombs, C)

• $n$ or $N$ : Number of excess or deficit electrons (dimensionless integer)

Example Water droplet: $Q = -3.0\times10^{-14}\,\text{C}$

$N = \frac{|Q|}{e} = \frac{3.0\times10^{-14}}{1.6\times10^{-19}} \approx 1.9\times10^{5}\text{ electrons}$

Conservation of Charge

Electric charge is neither created nor destroyed. Any charging process merely redistributes electrons among bodies; the algebraic sum of charge in an isolated system remains constant.

Electric Fields

• Definition: a region in which a charge experiences a force.

• Vector quantity; direction is the force on a positive test charge.

• Represented by field lines:

  • Emanate from $+$, terminate on $-$ .

  • Closer spacing \Rightarrow stronger field.

Field Strength

$E = \frac{F}{q} \quad (\text{units: }\text{N C}^{-1}\text{ or V m}^{-1})$

• $E$ : Electric field strength (measured in Newtons per Coulomb, N C$^{-1}$ or Volts per meter, V m$^{-1}$ )

• $F$ : Electric force (measured in Newtons, N)

• $q$ : Charge (measured in Coulombs, C)

Point Charge vs Uniform Field

• Single point charge \rightarrow radial 3-D field, strength diminishes with distance.

• Parallel opposite plates \rightarrow uniform field between plates; lines are parallel and evenly spaced except at edges (fringing).

Calculating Uniform Field

$E = \frac{V}{d}$

where $V$ is potential difference between plates separated by $d$.

• $V$ : Potential difference (measured in Volts, V)

• $d$ : Distance between plates (measured in meters, m)

Example Two plates: $V=2000\,\text{V}$, $d=0.10\,\text{m}$ \rightarrow $E=2.0\times10^{4}\,\text{V m}^{-1}$.

Electric Potential Energy & Voltage

• Moving charge against an electric field requires work, stored as electrical potential energy $\Delta E_{p}=Eqd$.

• $\Delta E_{p}$ : Change in electrical potential energy (measured in Joules, J)

• Potential difference (voltage): energy converted per coulomb of charge:

  $V = \frac{\Delta E}{q} \quad (1\,\text{V}=1\,\text{J C}^{-1})$

  • $V$ : Potential difference/Voltage (measured in Volts, V)

  • $\Delta E$ : Energy converted (measured in Joules, J)

  • $q$ : Charge (measured in Coulombs, C)

• Batteries supply energy by chemical separation of charge; rating (e.g. $1.5\,\text{V}$) tells energy per coulomb.

Voltage\text{\textendash}Field Relationship in Parallel Plates

Combining $E = V/d$ with $F = Eq$ yields force on a charge in a uniform field:

$F = Eq = \frac{Vq}{d}$

• $F$ : Electric force (measured in Newtons, N)

• $E$ : Electric field strength (measured in Newtons per Coulomb, N C$^{-1}$ or Volts per meter, V m$^{-1}$ )

• $q$ : Charge (measured in Coulombs, C)

• $V$ : Potential difference (measured in Volts, V)

• $d$ : Distance (measured in meters, m)

Acceleration of a particle of mass $m$:

$a = \frac{F}{m} = \frac{Vq}{md}$

• $a$ : Acceleration (measured in meters per second squared, m s$^{-2}$ )

• $F$ : Force (measured in Newtons, N)

• $m$ : Mass of the particle (measured in kilograms, kg)

• $g$ : Acceleration due to gravity (measured in meters per second squared, m s$^{-2}$ ) (used in Millikan's gravity force $mg$ )

Example Proton: $V=2000\,\text{V}, d=0.10\,\text{m} \Rightarrow a \approx 1.9\times10^{12}\,\text{m s}^{-2}$.

Potential Gradient & Voltage Divider

• Potential gradient (slope $\Delta V/\Delta d$) inside a conductor equals its internal electric field.

• A uniform resistance wire across a battery forms a linear voltage gradient; tapping at different positions provides selectable voltages ext{\textendash} the principle behind potentiometers, volume controls & light dimmers.

Millikan Oil-Drop Experiment & Elementary Charge

• Balanced gravitational weight $mg$ with electric force $Eq$ in a uniform field $E = V/d$.

• Found all measured droplet charges were integer multiples of $1.6\times10^{-19}\,\text{C}$, confirming quantisation and establishing $e$.

Motion of Charged Particles in Electric Fields

• Force parallel (for $+$) or antiparallel (for $-$) to field lines.

• Constant acceleration \rightarrow parabolic trajectory when initial velocity has transverse component (analogue of projectile under gravity).

• Applications: cathode-ray oscilloscope deflection plates, smoke precipitators, mass spectrometer (time-of-flight acceleration through known $V$ gives velocity, hence mass).

DC Circuits: Current, Voltage, Series & Parallel

Electric Current

$I = \frac{q}{t}$

• $I$ : Electric current (measured in Amperes, A)

• $q$ : Charge (measured in Coulombs, C)

• $t$ : Time (measured in seconds, s)

• Unit: ampere (A) = coulomb per second.

• Conventional direction: from $+$ to $-$ , opposite to electron flow in metals.

Measuring

• Ammeter in series, voltmeter in parallel.

Series Circuit Rules

• Current identical everywhere.

• Voltages add: $V<em>{\text{cell}} = V</em>1 + V_2 + \dots$

• $V_{\text{cell}}$ : Total voltage supplied by the cell (measured in Volts, V)

• $V<em>1, V</em>2, \dots$ : Voltages across individual components (measured in Volts, V)

Parallel Circuit Rules

• Voltage same across each branch: $V<em>{\text{cell}} = V</em>1 = V_2 = \dots$.

• Currents split/recombine: $I<em>{\text{in}} = \sum I</em>{\text{out}}$

• $I_{\text{in}}$ : Total current entering a junction (measured in Amperes, A)

• $I_{\text{out}}$ : Total current leaving a junction (measured in Amperes, A)

Resistance & Ohm\text{\textquotesingle}s Law

• Resistance is opposition to current caused by collisions between electrons and lattice ions.

• Ohm\text{\textquotesingle}s law (constant-temperature ohmic conductors): $V = IR$

• $V$ : Voltage/Potential difference (measured in Volts, V)

• $I$ : Current (measured in Amperes, A)

• $R$ : Resistance (measured in Ohms, $\Omega$ )

• Units: ohm ($\Omega$). 1 $\Omega$ means $1\,\text{A}$ flows under $1\,\text{V}$.

Combining Resistors

• Series: $R<em>{\text{T}} = R</em>1 + R_2 + \dots$ (increase total $I$ decreases).

• $R_{\text{T}}$ : Total equivalent resistance in series (measured in Ohms, $\Omega$ )

• $R<em>1, R</em>2, \dots$ : Resistances of individual resistors (measured in Ohms, $\Omega$ )

• Parallel: $\dfrac{1}{R<em>{\text{T}}}=\dfrac{1}{R</em>1}+\dfrac{1}{R_2}+\dots$ (decrease total, current capacity increases).

• $R_{\text{T}}$ : Total equivalent resistance in parallel (measured in Ohms, $\Omega$ )

• $R<em>1, R</em>2, \dots$ : Resistances of individual resistors (measured in Ohms, $\Omega$ )

Ohmic vs Non-Ohmic

• Ohmic: $V\text{\textendash}I$ graph straight line through origin, constant slope = $R$.

• Non-ohmic (e.g. filament lamp): curved $V\text{\textendash}I$, resistance rises with temperature.

Electric Power & Energy

• Energy transferred: $E = V I t$ (J).

• $E$ : Energy transferred (expressed in Joules, J)

• $V$ : Voltage (expressed in Volts, V)

• $I$ : Current (expressed in Amperes, A)

• $t$ : Time (expressed in seconds, s)

• Power (rate): $P = \dfrac{E}{t} = VI$.

• $P$ : Power (expressed in Watts, W)

• Alternate forms using Ohm\text{\textquotesingle}s law:

  • $P = I^{2} R$ (useful if $I,R$ known)

  • $P = \dfrac{V^{2}}{R}$ (useful if $V,R$ known)

  • $R$ : Resistance (expressed in Ohms, $\Omega$ )

• Household safety: high-power appliances demand large $I$; fuses/circuit-breakers disconnect if current exceeds safe limit, preventing overheating.

Magnetism Fundamentals

• Magnetic materials: iron, steel, nickel, cobalt & certain ceramics.

• Magnetism originates from electron spin & orbital motion. In ferromagnets domains align, producing net field.

• Field lines run $N \rightarrow S$ externally; strongest at poles.

• Earth\text{\textquotesingle}s field acts like a tilted bar magnet; compasses align to it.

• Graphical symbols: (\times) = into page, $\bullet$ = out of page.

Current & Magnetism; Solenoids and Electromagnets

• Straight conductor with current $I$ produces concentric circular field: Right-hand-grip rule (thumb = $I$, curled fingers = $B$).

• Solenoid (coil): internal field resembles bar magnet; right-hand-grip around coil gives pole orientation.

• Field strength increases with $I$ and turn count.

• Electromagnetic locks, relays, MRI etc. exploit controllable solenoid fields.

The Motor Effect (Force on Current)

• A conductor of length $L$ carrying current $I$ in a field $B$ experiences force:

  $F = B I L$ (maximum when wire \perp field).

  • $F$ : Magnetic force (measured in Newtons, N)

  • $B$ : Magnetic field strength (measured in Tesla, T)

  • $I$ : Current (measured in Amperes, A)

  • $L$ : Length of the conductor in the magnetic field (measured in meters, m)

• Direction by right-hand slap rule: fingers ($B$), thumb ($I$), palm (force).

• Applications: loudspeakers, galvanometers, electric motors.

DC Motor Operation

1. Coil in uniform $B$; opposite sides carry currents in opposite directions \rightarrow forces form a couple, producing torque.

2. Split-ring commutator reverses connections every half-turn, maintaining continuous rotation.

3. Factors increasing torque: more turns, stronger $B$, higher $I$, larger coil area.

Charged Particles in Magnetic Fields

• Moving charge $q$ with velocity $v$ across uniform field $B$ feels force:

  $F = B q v$ (max at $90^{\circ}$).

  • $F$ : Magnetic force (measured in Newtons, N)

  • $B$ : Magnetic field strength (measured in Tesla, T)

  • $q$ : Charge (measured in Coulombs, C)

  • $v$ : Velocity of the charged particle (measured in meters per second, m s$^{-1}$ )

• Direction: right-hand slap for positive charge; for electrons reverse thumb (or use left hand).

• Force always perpendicular to velocity \Rightarrow circular motion (centripetal).

• Used in CRTs, cyclotrons, mass spectrometers; in nature causes aurorae.

Electromagnetic Induction (Generator Effect)

Moving conductor of length $L$ at speed $v$ across field $B$ induces emf:

$V = B v L$ (max at $90^{\circ}$ crossing).

• $V$ : Induced electromotive force (EMF), or Voltage (measured in Volts, V)

• $B$ : Magnetic field strength (measured in Tesla, T)

• Magnitude \text{\textuparrow} with $B, v, L$ (or coil turns/area).

• Direction from left-hand slap: fingers ($B$), palm (motion), thumb (induced $+$ end / conventional current).

• Induction explained microscopically by motor force separating charges until internal electric field balances.

Generators

• Convert mechanical \rightarrow electrical energy (inverse of motors).

• In DC generator a rotating coil with split-ring commutator yields pulsating but unidirectional voltage.

• Output emf depends on:

  • rotational speed

  • coil turns & area

  • magnetic field strength.

• Large-scale generation uses rotating magnet (rotor) inside stationary coils (stator) to simplify current collection.

Practical & Real-World Links

• Touch-screens detect field distortion by finger, converting to position data.

• Spark plugs & lightning: air breaks down when $E\ge3\times10^{6}\,\text{N C}^{-1}$ producing arcs.

• Sharks\text{\textquotesingle} electroreceptors sense fields as low as $5\times10^{-9}\,\text{N C}^{-1}$.

• Smoke precipitators charge particulates electrostatically then collect on oppositely charged plates to curb pollution.

• Magnetic-strip cards store data via aligned microscopic domains.

• Power overload protection uses fuses/magnetic circuit breakers; high $I$ evokes heat (via $I^{2}R$) and magnetic trip mechanisms.