Electromagnetism is the most successful physical theory ever developed. It unites electricity and magnetism into a single, elegant framework that explains everything from lightning bolts to radio waves to the light from distant stars. At its heart are Maxwell’s four equations—mathematical poetry that describes how electric and magnetic fields interact, propagate, and create electromagnetic waves.

Let’s explore this beautiful unification of forces that powers our technological civilization.

Electric Fields and Charges

Coulomb’s Law

The force between charges:

F = (1/4πε₀) × (q₁q₂)/r²

Where ε₀ = 8.85 × 10^-12 C²/N·m² is the permittivity of free space.

Electric Field

Force per unit charge:

E = F/q = (1/4πε₀) × Q/r² (for point charge)

Field lines show direction and strength of electric field.

Gauss’s Law

Electric flux through closed surface:

∮ E · dA = Q_enc/ε₀

Relates field to enclosed charge. Simpler than Coulomb’s law for symmetric charge distributions.

Electric Potential

Work per unit charge:

V = -∫ E · dl

For point charge: V = (1/4πε₀) × Q/r

Capacitance

Charge storage ability:

C = Q/V

Parallel plates: C = ε₀A/d

Magnetic Fields and Currents

Magnetic Force on Moving Charges

Lorentz force:

F = q(v × B)

Direction given by right-hand rule.

Ampère’s Law

Circulation of magnetic field:

∮ B · dl = μ₀ I_enc

Where μ₀ = 4π × 10^-7 T·m/A is permeability of free space.

Biot-Savart Law

Magnetic field from current element:

dB = (μ₀/4π) × (I dl × r̂)/r²

Calculates B field from arbitrary current distributions.

Magnetic Flux

Field through surface:

Φ_B = ∮ B · dA

Faraday’s law relates changing flux to induced EMF.

Maxwell’s Equations: The Complete Picture

Gauss’s Law for Electricity

∇ · E = ρ/ε₀

Electric field divergence equals charge density.

Gauss’s Law for Magnetism

∇ · B = 0

No magnetic monopoles—magnetic field lines are closed loops.

Faraday’s Law

∇ × E = -∂B/∂t

Changing magnetic field induces electric field (electromagnetic induction).

Ampère-Maxwell Law

∇ × B = μ₀ J + μ₀ε₀ ∂E/∂t

Magnetic field curl equals current plus displacement current.

The Displacement Current

Maxwell’s crucial addition:

Displacement current: I_d = ε₀ dΦ_E/dt

Predicts electromagnetic waves in vacuum.

Electromagnetic Waves

Wave Equation

From Maxwell’s equations:

∇²E - (1/c²) ∂²E/∂t² = 0
∇²B - (1/c²) ∂²B/∂t² = 0

Where c = 1/√(μ₀ε₀) = 3 × 10^8 m/s

Plane Wave Solutions

Traveling waves:

E = E₀ sin(kx - ωt)
B = B₀ sin(kx - ωt)

With E₀ = c B₀ (speed of light relationship)

Poynting Vector

Energy flow direction:

S = (1/μ₀) E × B

Magnitude gives power per unit area.

Spectrum of EM Waves

From radio to gamma rays:

Radio: λ > 1 mm
Microwave: 1 mm > λ > 1 μm
Infrared: 1 μm > λ > 700 nm
Visible light: 700 nm > λ > 400 nm
Ultraviolet: 400 nm > λ > 10 nm
X-rays: 10 nm > λ > 0.01 nm
Gamma rays: λ < 0.01 nm

Light as Electromagnetic Wave

Polarization

Electric field direction:

Linear polarization: E in single plane
Circular polarization: Rotating E field
Elliptical polarization: Elliptical rotation

Reflection and Refraction

Snell’s law:

n₁ sinθ₁ = n₂ sinθ₂

Where n = √(εμ) is refractive index.

Interference

Superposition of waves:

Constructive: Path difference = nλ
Destructive: Path difference = (n + ½)λ

Diffraction

Wave bending around obstacles:

Single slit: sinθ = λ/a
Double slit: d sinθ = nλ

Electromagnetic Induction

Faraday’s Law

Induced EMF equals rate of magnetic flux change:

ε = - dΦ_B/dt

Lenz’s law: Induced current opposes change causing it.

Inductance

Magnetic flux linkage:

Φ = L I
L = N Φ_B / I

Self-inductance: EMF = -L dI/dt

Transformers

Voltage transformation:

V₂/V₁ = N₂/N₁ = I₁/I₂

Energy conservation in ideal transformer.

Electromagnetic Energy and Momentum

Energy Density

Stored in fields:

u_E = (1/2) ε₀ E²
u_B = (1/2) (B²/μ₀)
Total: u = u_E + u_B

Stress-Energy Tensor

Momentum density:

Momentum density = (ε₀/ c²) S

Where S is Poynting vector. Light carries momentum!

Radiation Pressure

Force from electromagnetic waves:

P_rad = I/c (normal incidence)

Explains comet tails, solar sails.

Applications in Modern Technology

Antennas and Wireless Communication

Dipole antenna radiation pattern:

Power pattern: sin²θ
Directivity: 1.5 (relative to isotropic)

Microwave Ovens

Magnetron generates 2.45 GHz microwaves:

Frequency chosen to match water absorption
Wavelength: 12.2 cm
Penetration depth: ~1-2 cm

Fiber Optics

Total internal reflection:

Critical angle: θ_c = arcsin(n₂/n₁)

Enables low-loss long-distance communication.

Medical Imaging

MRI uses nuclear magnetic resonance:

Larmor frequency: ω = γ B₀
γ = 42.58 MHz/T for hydrogen

Creates detailed anatomical images.

Quantum Electrodynamics

Photon-Electron Interactions

Photoelectric effect:

hν = K_max + φ

Compton scattering:

Δλ = h(1-cosθ)/(m_e c)

Quantum Field Theory

Electromagnetism as quantum field:

Interactions via photon exchange
Feynman diagrams visualize processes
Renormalization handles infinities

Conclusion: The Unified Force

Maxwell’s equations unified electricity and magnetism into a single electromagnetic force. This unification predicted electromagnetic waves and explained light as an EM phenomenon. The theory has been spectacularly successful, describing everything from household electricity to cosmic radio sources.

Electromagnetism shows us that fields are as real as particles, that waves can carry energy and momentum, and that the dance of electric and magnetic fields creates the light by which we see the universe.

The electromagnetic symphony continues to play.

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Electromagnetism teaches us that electric and magnetic fields are two sides of the same phenomenon, that light is an electromagnetic wave, and that fields can carry energy and momentum like particles.

What’s the electromagnetic phenomenon that fascinates you most? 🤔

From charges to waves, the electromagnetic journey continues… ⚡