Every definition and working formula from the twelve modules, gathered for revision. Fractions are written stacked, exactly as you should on paper. Conventional current, and fields defined on a positive test charge, throughout.
0 · Conventions & the rules of direction
◦ Field lines leave + charge, enter −; density ∝ strength; they never cross.
◦ ⊙ = out of the page, ⊗ = into the page (arrow tip vs tail feathers).
◦ Right-hand grip: thumb along I, fingers curl along B (wire); fingers curl along I, thumb gives B (loop/solenoid).
◦ Force direction: F = qv×B — fingers v, curl to B, thumb F (positive charge; flip for negative).
◦ Lenz's law: induced effects oppose the change in flux that made them — energy conservation in disguise.
◦ EM wave travel is along E × B; E, B and v are mutually perpendicular.
1 · Electric field & potential
Coulomb's law — the force between two point charges, along the line joining them, inverse-square in the separation. Field E = force per unit positive test charge.
Dipole — charges ±q a distance d apart; moment p = qd points − → +. Axial field is twice the equatorial field at the same distance.
Potential V — work per unit charge brought from infinity; a scalar. E points down the steepest fall of V and is everywhere ⊥ to equipotentials. Inside a conductor E = 0; the surface is one equipotential.
F = q₁q₂4πε₀r²E = FqV = kqrE = −dVdraxial E = 2kpr³ · equatorial kpr³τ = pE sinθ · U = −pE cosθ
2 · Gauss's law
Electric flux Φ — field lines threading a surface; counts E across area, weighted by the tilt cosθ.
Gauss's law — the total flux through ANY closed surface equals the enclosed charge over ε₀; charge outside contributes nothing. Choose surfaces that match the symmetry (sphere, cylinder, pillbox).
Φ = EA cosθ∮E·dA = qencε₀wire E = λ2πε₀rsheet E = σ2ε₀ · conductor σε₀shell: 0 inside, kQr² outside · solid sphere inside kQrR³
3 · Capacitors
Capacitance C = Q/V — charge stored per volt; set by geometry alone. A dielectric (constant K) multiplies C by K by polarising and weakening the internal field.
Battery connected: V fixed, Q and U rise with K. Isolated: Q fixed, V and U fall. Energy lives in the field itself, density ½ε₀E².
C = ε₀Ad → Kε₀AdU = 12CV² = Q²2Cu = 12ε₀E²series 1C = Σ1Cᵢ · parallel C = ΣCᵢsphere C = 4πε₀R
4 · Electric current in conductors
Drift — electrons crawl at vd (~mm/s) though the signal moves near light speed. Ohm's law: V ∝ I when ρ is constant; ρ rises with temperature in metals, falls in carbon/semiconductors.
EMF ε — work per charge by the source; terminal voltage V = ε − Ir droops under load. Max power transfer at R = r (efficiency then only 50%).
Kirchhoff: junction (charge) + loop (energy). Wheatstone bridge balances at P/Q = R/S — the galvanometer reads zero. Potentiometer: compares EMFs by balancing lengths, drawing no current.
Joule heating — collisions turn electrical work into heat; the basis of fuses, heaters, bulbs.
Thermoelectricity — a junction pair at different temperatures drives a thermo-emf (Seebeck), parabolic in θ with a neutral temperature (max) and inversion temperature (sign flip). Peltier and Thomson are its reversible cousins.
Electrolysis — Faraday: mass deposited ∝ charge passed, and ∝ chemical equivalent (M/z). F = 96,500 C deposits one gram-equivalent.
H = I²RtE = aθ + 12bθ²θn = −ab · θi = 2θn − θcoldm = M I tz F
6 · Magnetic field & force
Lorentz force — F = qE + qv×B. The magnetic part is ⊥ v, so it does no work — it only bends the path.
Circular motion — a charge ⊥ B runs in a circle; the period is independent of speed (the cyclotron's trick). v at an angle adds a drift → helix. Crossed E and B pass only v = E/B (velocity selector).
Currents feel it too — force on a wire, torque on a loop with moment m = NIA; a spring restores the moving-coil galvanometer to make deflection ∝ current.
F = qvB sinθr = mvqBT = 2πmqBselector v = EBF = BIL sinθτ = NIAB sinφ · U = −mB cosφ
7 · Magnetic field due to a current
Biot–Savart — each current element writes a small dB, ⊥ to both the element and the line to the point; integrate for wires, loops, arcs.
Ampère's law — ∮B·dl = μ₀Ienc: the circulation of B counts the threading current. Gives the solenoid (uniform inside, ~0 outside) and toroid instantly.
Parallel currents attract when parallel, repel when anti-parallel — the old definition of the ampere.
wire B = μ₀I2πdloop centre B = μ₀I2Raxis B = μ₀IR²2(R²+x²)3/2solenoid B = μ₀nIFL = μ₀I₁I₂2πd
8 · Permanent magnets
Bar magnet = dipole of moment M; cut it and each piece is a full magnet with M/2 — poles never come alone. Its field mirrors the electric dipole (axial twice equatorial).
Earth's field — described by declination (compass vs true north), dip δ (tilt below horizontal) and the horizontal component BH. An oscillating needle and a tangent galvanometer both measure through BH.
axial B = μ₀4π2Mr³τ = MB sinθT = 2π√IMBtanδ = BVBHtangent galv. I = K tanθ
9 · Magnetic properties of matter
H, M, B — H is what the coil applies, M is the material's response (dipole moment per volume), B is the total field. Susceptibility χ = M/H classifies matter.
Dia (χ small, negative — repelled) · para (χ small, positive; Curie law χ = C/T) · ferro (χ huge; domains; spontaneous M below the Curie temperature TC).
Hysteresis — B lags H around a loop; retentivity (B left at H = 0), coercivity (reverse H to kill B); loop area = energy lost per cycle. Soft iron: thin loop → cores; steel/alnico: fat loop → permanent magnets.
Faraday — a changing flux Φ = BA cosθ induces an EMF equal to its rate of change; Lenz's minus sign sets the direction (oppose the change).
Motional EMF — a rod sweeping flux at speed v generates BLv; on rails it drives a current, feels a retarding force, and the mechanical power in equals the I²R heat out.
Inductance — a coil's flux per ampere: NΦ = LI. It resists changes in current, stores ½LI² in the field, and sets the LR time constant. Eddy currents — induced swirls in bulk metal: brakes and induction stoves (friend), core losses (foe — laminate!).
RMS — the steady DC that heats identically; meters read rms, insulation survives the peak (√2 higher). Mean over a cycle is zero.
Reactance — frequency-dependent opposition: XL = ωL (V leads I by 90°), XC = 1/ωC (V lags by 90°) — remember CIVIL. Add voltages as phasors, never arithmetically.
Resonance — at ω₀ the reactances cancel, Z = R, current peaks; sharpness Q, bandwidth Δω = R/L. Only the in-phase current does work: power factor cosφ; a pure reactance draws wattless current. The transformer trades volts for amps at constant power.
Displacement current — a changing electric flux acts like a current, id = ε₀ dΦE/dt; it completes Ampère's law and closes the circuit across a capacitor gap.
The wave — E and B, mutually ⊥, in phase, transverse, regenerating each other at c; E/B = c at every instant. Energy is split exactly half electric, half magnetic.
Momentum & pressure — absorbed energy U carries momentum U/c; reflection doubles the push. Spectrum (λ falling): radio · micro · IR · visible (400–700 nm) · UV · X · γ — the name records the source.
id = ε₀dΦEdtc = 1√μ₀ε₀B₀ = E₀c⟨u⟩ = 12ε₀E₀² · I = ⟨u⟩cpressure Ic (absorb) · 2Ic (reflect)
Constants: k = 1/4πε₀ = 9×10⁹ N m²C⁻² · ε₀ = 8.85×10⁻¹² SI · μ₀ = 4π×10⁻⁷ SI · e = 1.6×10⁻¹⁹ C · c = 3×10⁸ m/s · F = 96,500 C mol⁻¹ Compiled from the Vedatom Physics EM Lab · every formula has a live, draggable simulation on its module page. ZAP!