Learning #35 – Electromagnetism & Maxwell’s Equations

Prerequisites and Learning Goals

This guide assumes you are comfortable with basic calculus (derivatives, integrals) and vectors (dot product, cross product, gradient, divergence, curl). A first course in mechanics is helpful but not required. By the end you should be able to:

State and apply the four Maxwell equations in both integral and differential form
Derive the electric and magnetic fields of simple charge and current distributions using Gauss’s and Ampère’s laws
Apply Faraday’s law and Lenz’s law to induction problems
Derive the electromagnetic wave equation and compute wave speed from μ₀ and ε₀
Compute the Poynting vector and radiation pressure
Understand qualitatively: AC generators, transformers, LC oscillators, waveguides

Module 1: Electric Fields and Coulomb’s Law

Electric charge and the Coulomb force. All electromagnetic phenomena begin with electric charge. Charge is quantised (elementary charge e = 1.602×10−19 C) and conserved. Like charges repel; unlike charges attract. Coulomb’s law gives the force between two point charges q₁, q₂ separated by distance r:

F 12 = k q 1 q 2 /r² &rcirc; 12 k = 1/(4πε 0) \approx 8.99 \times 10&sup9; N\cdotm²/C² ε 0 = 8.854 \times 10-12 F/m (permittivity of free space)

The force is along the line joining the charges (central force), falls as 1/r², and obeys the superposition principle: the total force on a charge equals the vector sum of all pairwise Coulomb forces.

The electric field concept. Instead of thinking about force between pairs, introduce the electric field E at a point P as the force that would be exerted on a unit positive test charge placed at P (in the absence of the test charge disturbing the source charges). For a point charge Q at the origin:

E (r) = kQ/r² &rcirc; = Q/(4πε 0 r²) &rcirc; Force on charge q: F = q E

The electric field is a vector field defined everywhere in space. Field lines start on positive charges and end on negative charges (or go to infinity), and the field strength is proportional to line density.

Gauss’s law and symmetry. The total electric flux through any closed surface equals the enclosed charge divided by ε₀:

&oiint; E \cdotd A = Q enc /ε 0 (integral form) \nabla\cdot E = ρ/ε 0 (differential form, ρ = charge density)

For a sphere of charge Q: choose a spherical Gaussian surface of radius r > R. By symmetry, E is uniform on the surface and radially outward, so: E × 4πr² = Q/ε₀, giving E = Q/(4πε₀r²) &rcirc;. Gauss’s law is most powerful for distributions with spherical, cylindrical, or planar symmetry.

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Static Electricity Simulator: place positive and negative charges on a canvas and watch the electric field lines and equipotential surfaces update in real time. Try arranging a dipole (+q, −q) and verify that field lines start on + and end on −.

Module 2: Electric Potential and Energy

Electric potential. Because the electrostatic force is conservative (∇×E = 0 in static case), we can define a scalar potential V such that E = −∇V. The potential at a distance r from a point charge Q:

V(r) = kQ/r = Q/(4πε 0 r) For a charge distribution: V = k \int dq/r' (superposition) Work to move charge q from A to B: W = q(V A - V B)

Equipotential surfaces are perpendicular to field lines. No work is done moving a charge along an equipotential.

Capacitors and energy storage. A capacitor stores charge Q on two conductors at potential difference V: Q = CV (capacitance C in farads). A parallel-plate capacitor with plate area A and separation d has C =&ε₀A/d. Energy stored in the electric field:

U E = ½CV² = Q²/(2C) = ½ε 0 E² \times (volume) Energy density: u E = ½ε 0 E² [J/m³]

The energy density in the field is a general result — it applies everywhere, not just inside capacitors.

Module 3: Magnetic Fields and Biot-Savart

The magnetic force on moving charges. A charged particle moving with velocity v in a magnetic field B experiences the Lorentz force:

F = q v \times B For a current element Id l : d F = Id l \times B

Crucially, the magnetic force is always perpendicular to the velocity, so it does no work. It alters the direction of motion but not the kinetic energy. A charged particle in a uniform field moves in a circle at the cyclotron frequency ω_c = qB/m (Larmor radius r_L = mv/qB).

Sources of magnetic fields: Biot-Savart law. Moving charges create magnetic fields. The contribution to B from a current element Idl at position r from the element:

d B = (μ 0 /4π) Id l \times &rcirc;/r² For a long straight wire at distance R: B = μ 0 I / (2πR) (circling the wire)

Magnetic field lines form closed loops — they have no beginning or end.

Ampère’s law. For a closed Ampèrian loop C enclosing current I_enc:

\oint B \cdotd l = μ 0 I enc (integral form) \nabla\times B = μ 0 J (differential form, J = current density) Solenoid (n turns/m): B = μ 0 nI (inside, uniform)

Just as Gauss’s law exploits symmetry for E, Ampère’s law is most powerful for symmetric current distributions: infinite wire, solenoid, toroid.

Gauss’s law for magnetism. Unlike electric field lines which start/end on charges, magnetic field lines always close on themselves:

&oiint; B \cdotd A = 0 (no magnetic monopoles) \nabla\cdot B = 0 (differential form)

Magnetic flux through any closed surface is zero. The absence of magnetic monopoles is one of the fundamental asymmetries between electricity and magnetism.

Module 4: Faraday Induction and Lenz’s Law

Magnetic flux and Faraday’s law. Define the magnetic flux through a surface S:

Φ B = \int\int S B \cdotd A Faraday’s law: ϵ = -dΦ B /dt N-turn coil: ϵ = -N dΦ B /dt

The induced EMF drives a current in the coil. Note: you do not need to have a physical circuit — a changing B creates a circulating E field in empty space (from the integral form: ∮E·dl = −dΦ_B/dt).

Lenz’s law — the sign matters. The minus sign in Faraday’s law enforces Lenz’s law: the induced current always flows in the direction that opposes the change in flux. If flux is increasing through a loop, the induced current creates a field opposing the increase. This is a direct consequence of energy conservation: if the induced current aided the flux change, you could create energy from nothing.

Physical examples: a magnet dropped through a copper tube falls slowly (eddy currents brake it); regenerative braking in electric cars (the motor acts as a generator, opposing wheel rotation); transformer heating in iron cores.

Motional EMF and generators. When a conductor moves in a magnetic field, the free electrons experience F = qv×B, creating a charge separation and an EMF. For a rod of length L moving at velocity v perpendicular to field B:

ϵ = BLv (motional EMF) Rotating coil (N turns, area A, angular frequency ω): ϵ(t) = NBAω sin(ωt) (AC generator)

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Electromagnetic Induction Simulator: choose the bar-magnet mode and drag the magnet toward and away from the coil. Notice the galvanometer deflects only while the magnet is moving, and reverses direction when the motion reverses. Switch to AC Generator mode to see ϵ(t) = NBAω sin(ωt) traced in real time as the coil spins.

Module 5: Maxwell’s Displacement Current

The missing term in Ampère’s law. Ampère’s law (∇×B = μ₀J) is inconsistent when applied to a charging capacitor. Maxwell introduced the displacement current density:

J D = ε 0 \partial E /\partialt Corrected Ampère-Maxwell: \nabla\times B = μ 0 J + μ 0 ε 0 \partial E /\partialt

Physical meaning: a time-varying electric field acts as a source of magnetic field, even with no real current present. This is the key insight that leads to self-sustaining electromagnetic waves.

Module 6: The Full Maxwell Equations

The four Maxwell equations in differential form in SI units:

(I) \nabla\cdot E = ρ/ε 0 [Gauss electric] (II) \nabla\cdot B = 0 [Gauss magnetic] (III) \nabla\times E = -\partial B /\partialt [Faraday] (IV) \nabla\times B = μ 0 J + μ 0 ε 0 \partial E /\partialt [Ampère-Maxwell]

Together with the Lorentz force law (F = q(E + v×B)), these equations completely describe all classical electromagnetic phenomena for given charge and current distributions.

In a linear, isotropic, homogeneous material with relative permittivity ε_r and relative permeability μ_r, replace ε₀ by ε = ε_rε₀ and μ₀ by μ =μ_rμ₀. The wave speed in the medium becomes v = c/n where the refractive index n = √(ε_rμ_r).

Module 7: Electromagnetic Waves

Deriving the wave equation. In vacuum, take the curl of Faraday’s law (III) and substitute (IV):

\nabla\times(\nabla\times E) = -\nabla\times(\partial B /\partialt) Using identity \nabla\times(\nabla\times E) = \nabla(\nabla\cdot E) - \nabla² E and \nabla\cdot E = 0 (in vacuum): \nabla² E = μ 0 ε 0 \partial² E /\partialt² Wave speed: c = 1/\sqrt(μ 0 ε 0) = 2.9979 \times 10&sup8; m/s

Plane wave solution. The simplest solution is a monochromatic plane wave (angular frequency ω, wave vector k = ω/c) propagating in the +x direction:

E y (x,t) = E 0 cos(kx - ωt + φ) B z (x,t) = (E 0 /c) cos(kx - ωt + φ) Key relations: E ⊥ B ⊥ propagation direction Amplitude ratio: E 0 /B 0 = c B = (1/c) kˆ \times E

The electric and magnetic energy densities are equal (½ε₀E² = B²/(2μ₀)).

Energy transport: the Poynting vector. The electromagnetic energy flux (power per unit area) is:

S = (1/μ 0) E \times B [W/m²] Time-averaged Poynting flux: ⟨S⟩ = E 0 B 0 /(2μ 0) = E 0 ²/(2μ 0 c) Radiation pressure: P rad = ⟨S⟩/c (absorbed) or 2⟨S⟩/c (reflected)

Radiation pressure is tiny for everyday light intensities but significant in stellar interiors (supporting the Sun against gravitational collapse), near-critical laser pulses, and proposed solar sails.

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Maxwell Waves Simulator renders the oscillating E and B field vectors as an animated transverse wave. Verify that they are perpendicular and in phase, and that the Poynting vector (cross product) points along the propagation direction.

Module 8: Electromagnetic Induction Applications

Self-inductance and RL circuits. A coil opposes changes in current through itself via self-EMF ϵ = −L dI/dt. In a series RL circuit (resistance R, inductance L) driven by EMF ϵ₀:

I(t) = (ϵ 0 /R)(1 - e- t/τ) Time constant: τ = L/R (inductive time constant) Stored magnetic energy: U L = ½LI² Solenoid: L = μ 0 N²A/ℓ

LC oscillators and RLC circuits. In an ideal LC circuit (no resistance), energy oscillates between the inductor (magnetic) and capacitor (electric) at the resonant frequency:

ω 0 = 1/\sqrt(LC) (resonance) Q(t) = Q 0 cos(ω 0 t + φ) I(t) = -Q 0 ω 0 sin(ω 0 t + φ) With resistance R: ω d = \sqrt(ω 0 ² - (R/2L)²)

The LC resonator is the electromagnetic analogue of the mechanical harmonic oscillator (spring-mass). It is fundamental to radio receivers (tuning circuits), oscillators, and filters.

Transformers and impedance matching. An ideal transformer with primary turns N_p and secondary turns N_s:

V s /V p = N s /N p I s /I p = N p /N s Z in = (N p /N s)² Z load (impedance transformation)

The impedance transformation property is used in audio amplifiers to match the high output impedance of an amplifier to a low-impedance speaker (8 Ω), and in RF circuits to match antenna impedance (often 50 Ω) to transmission line impedance.

Module 9: Boundary Conditions and Waveguides

At an interface between two media, Maxwell’s equations imply specific boundary conditions on the field components. For the tangential E and normal B (from Gauss and Faraday):

E 1t = E 2t (tangential E continuous) B 1n = B 2n (normal B continuous) D 1n - D 2n = σ free (normal D discontinuous by surface charge) H 1t - H 2t = K free (tangential H discontinuous by surface current)

At a perfect conductor surface: E_tangential = 0 and B_normal = 0. These conditions confine electromagnetic waves inside metallic pipes — waveguides. A rectangular waveguide of width a and height b supports TE_mn and TM_mn modes with cutoff frequencies f_c = c/(2) √[(m/a)² + (n/b)²]. Below the cutoff frequency, waves are evanescent (exponentially decaying, no net energy transport). Waveguides are used in microwave engineering: radar, satellite communications, lab microwave measurements.

Module 10: Relativity and Electromagnetism

Maxwell’s equations are Lorentz-covariant: they have the same form in all inertial reference frames. This is why Einstein’s 1905 paper was titled “On the Electrodynamics of Moving Bodies” — special relativity was born from the inconsistency of classical mechanics with Maxwell’s equations.

Key consequences for electromagnetism:

Electric/magnetic field unification: what one observer calls a pure electric field, a boosted observer sees as a combination of E and B. The six field components form the antisymmetric electromagnetic tensor F^μν.
Magnetism as a relativistic effect: the magnetic force between two parallel current-carrying wires can be derived entirely from Coulomb’s law plus special relativity (Lorentz contraction of charge densities). Magnetism is intrinsically relativistic.
Electromagnetic 4-potential: scalar potential V and vector potential A combine into the 4-potential A^μ = (V/c, A), defined up to gauge transformations A^μ → A^μ + ∂^μχ.

Summary: the hierarchy of classical EM theory

Electrostatics (static charges: Gauss E, Coulomb) → Magnetostatics (steady currents: Gauss B, Biot-Savart, Ampère) → Faraday induction (changing B → E) → Maxwell displacement current (changing E → B) → Self-sustaining EM waves (light, radio, X-rays) → Lorentz covariance (4D unification, tensor F^μν) → QED (quantisation of the EM field, photons, virtual particles)

Recommended Simulations for This Module

Static Electricity — Coulomb force, electric field lines, potential
Electromagnetic Induction — Faraday’s law, Lenz’s law, AC generator
Maxwell Waves — plane EM wave, Poynting vector, polarisation
Plasma Oscillations — Langmuir waves, plasma frequency, Debye shielding
Antenna Pattern — dipole radiation pattern, array directivity
Diffraction & Interference — Huygens wavelets, Young’s double slit, gratings

Electromagnetism & Maxwell’s Equations — A Guided Learning Path