Electromagnetism Notes

Last reviewed 16 Sept 2026 · 8 min read

Electrostatics

Electric charge

Two kinds (positive, negative); like charges repel, unlike attract.
Quantised: $q = n e$ , $e = 1.602 \times 1 0^{- 19}$ C.
Conserved — total charge of an isolated system is constant.
Charging by friction, conduction and induction.

Coulomb's law

FormulaCoulomb's law

F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}} = k \frac{q _{1} q _{2}}{r ^{2}}, k \approx 9 \times 1 0^{9} N\cdotm^{2} / C^{2}

$ε_{0} = 8.854 \times 1 0^{- 12}$ C²/N·m². In a medium of relative permittivity $ε_{r}$ , force is reduced by $ε_{r}$ .

Electric field and potential

FormulaField and potential

Electric field $E = \frac{F}{q}$ (N/C or V/m); point charge $E = \frac{k q}{r ^{2}}$ Potential $V = \frac{k q}{r}$ (volts); $E = - \frac{d V}{d r}$ Potential energy of two charges $U = \frac{k q _{1} q _{2}}{r}$ Work to move charge q through potential difference ΔV: $W = q Δ V$ Electric dipole moment $p = q \cdot 2 a$ ; torque in field $τ = pE sin θ$

Field lines start on positive and end on negative charges; never intersect; closer lines mean stronger field.
Equipotential surfaces are perpendicular to field lines; no work is done moving a charge along them.

Gauss's law

\oint E \cdot d A = \frac{q _{e n c l ose d}}{ε _{0}}

Charge distribution	Field
Infinite line charge (λ per unit length)	$E = \frac{λ}{2 π ε _{0} r}$
Infinite plane sheet (σ per unit area)	$E = \frac{σ}{2 ε _{0}}$
Charged conducting sphere (outside)	As if charge at centre; zero inside

Conductors and dielectrics

In electrostatic equilibrium, the field inside a conductor is zero; charge resides on the outer surface; field is perpendicular to the surface and strongest at sharp points (principle of lightning conductors).
Electrostatic shielding (Faraday cage) — sensitive instruments, protection from lightning inside vehicles.
Dielectrics (insulators) become polarised, reducing the field; dielectric strength — maximum field before breakdown.

Capacitance

$C = Q / V$ ; parallel plate $C = \frac{ε _{0} ε _{r} A}{d}$ ; energy $\frac{1}{2} C V^{2}$ (see DC & AC Fundamentals).

Current electricity (overview)

Current $I = n A e v_{d}$ ( $n$ = free electron density, $v_{d}$ = drift velocity, very small — of the order of mm/s).
Resistivity, Ohm's law, Kirchhoff's laws, Wheatstone bridge balance $P / Q = R / S$ (basis of strain gauge circuits) — see Basic Electrical Engineering.
Joule heating $H = I^{2} R t$ .

Magnetic effects of current

Oersted (1820) showed that a current produces a magnetic field.

FormulaMagnetic fields of currents

Biot–Savart law: $d B = \frac{μ _{0}}{4 π} \frac{I d l sin θ}{r ^{2}}$ , $μ_{0} = 4 π \times 1 0^{- 7}$ T·m/A

Ampère's circuital law: $\oint B \cdot d l = μ_{0} I_{e n c l ose d}$

Configuration	Magnetic field
Long straight wire	$B = \frac{μ _{0} I}{2 π r}$
Centre of circular loop (radius R, N turns)	$B = \frac{μ _{0} N I}{2 R}$
Long solenoid (n turns per metre)	$B = μ_{0} n I$ (uniform inside)
Toroid	$B = \frac{μ _{0} N I}{2 π r}$

Direction: right-hand thumb rule.

Forces on charges and currents

FormulaLorentz force

Moving charge: $F = q (E + v \times B)$ ; magnetic part $F = q v B sin θ$ (no work done — changes direction only) Radius of circular path $r = \frac{m v}{q B}$ ; cyclotron frequency $f = \frac{q B}{2 π m}$ (independent of speed) Current-carrying conductor: $F = B I L sin θ$ (Fleming's left-hand rule) Force between parallel currents (per unit length): $\frac{F}{L} = \frac{μ _{0} I _{1} I _{2}}{2 π d}$ — attract if in same direction (basis of the former definition of the ampere) Torque on a coil: $τ = N I A B sin θ$ (moving coil galvanometer, motors)

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