Maxwell's Equations

Maxwell unified electricity, magnetism, electric fields, and magnetic fields into a single framework using four equations: divergence describes "sources", curl describes "circulation", and time variation locks electricity and magnetism into a mutually driving cycle. Gauss's law explains where fields originate, Faraday's law explains how changing magnetic fields induce electric fields, and the Ampère-Maxwell law elevates changing electric fields as sources of magnetic fields, ultimately leading to the conclusion that "light is an electromagnetic wave".

Unification of Electricity & Magnetism Displacement Current Completes Theory Predicts Electromagnetic Waves Foundation of Classical Field Theory

1. Gauss's Law (Electric Field)

∇·E = ρ / ε₀
∯S E·dA = Qin / ε₀

Electric flux is determined solely by the net charge enclosed by a closed surface. Changing the size or shape of the surface alters the local field strength distribution, but does not change the total electric flux.

2. Gauss's Law (Magnetic Field)

∇·B = 0
∯S B·dA = 0

Magnetic field lines do not "start" or "end" at points like electric field lines; they always form closed loops. To date, we have not discovered isolated magnetic monopoles.

3. Faraday's Law of Electromagnetic Induction

∇×E = - ∂B / ∂t
∮C E·dl = - dΦB / dt

Whenever magnetic flux through a loop changes, an induced electric field appears around the closed loop. The negative sign corresponds to Lenz's law, indicating the induced effect opposes the original change.

4. Ampère-Maxwell Law

∇×B = μ₀J + μ₀ε₀ ∂E / ∂t
∮C B·dl = μ₀Iin + μ₀ε₀ dΦE / dt

Electric currents produce surrounding magnetic fields. More profoundly: changing electric fields also produce magnetic fields. The displacement current term makes the theory self-consistent near charging capacitors and makes electromagnetic waves an inevitable consequence.

Interactive Experiment Area

Switch between four modes, adjust parameters, view arrows, read real-time values, and directly map abstract formulas to concrete scenarios.

From Four Local Laws to Complete Field Theory

The power of the four equations lies not just in describing individual experimental phenomena, but in how they form a complete logical chain when combined. Gauss's laws tell us "how sources generate fields", Faraday's law and Ampère-Maxwell law tell us "how fields mutually induce each other through time variation". Thus electric and magnetic fields are no longer isolated entities, but a unified structure capable of self-propagation.

Charge creates electric field Where there is charge density, electric field lines may diverge or converge. This is the relationship between divergence and sources.
Current and changing electric field create magnetic field Both conduction current in wires and displacement current in capacitor gaps cause magnetic fields to circulate around loops.
Changing magnetic field creates electric field When magnetic flux changes, a circulating electric field is induced in the loop, with direction following Lenz's law.
Coupled self-consistent propagation When E and B continuously induce each other, electromagnetic waves form that can propagate independently of local sources - light is one such wave.
This page uses normalized units, focusing on structure, direction, and relationships rather than absolute magnitudes in SI units.

位移电流为什么重要

如果安培定律只有 μ₀J，那么给充电电容器选不同曲面时，磁场环量会算出不同结果。加入 μ₀ε₀∂E/∂t 后，导线中的电流与极板间变化的电场在数学上被统一起来，方程才对所有曲面都保持一致。

What Do Divergence and Curl Mean?

Divergence concerns "whether there is a source emitting or converging here"; curl concerns "whether a small loop would be driven to rotate here". Gauss's laws tell us electric fields have sources while magnetic fields don't; the other two equations show circulation structures can be triggered by time changes.

Real-World Applications

  • Generators and transformers rely on Faraday's electromagnetic induction.
  • Radio, microwaves, and optical communications essentially manipulate time-varying electromagnetic fields.
  • Radar, antennas, waveguides, and resonant cavities can all be seen as engineering solutions to Maxwell's equations.
  • The birth of relativity is also closely related to Maxwell's equations' prediction of the speed of light.

Common Misconceptions

  • Misconception 1: Electric and magnetic fields are completely independent "things". In fact, in time-varying situations, they couple through the curl equations.
  • Misconception 2: Only real currents can produce magnetic fields. Maxwell showed that changing electric fields can also produce magnetic fields - this is the meaning of displacement current.
  • Misconception 3: Electromagnetic waves need to stay attached to wires or charges. In fact, once a self-consistent field structure forms, it can propagate independently of local sources.

Recommended Learning Sequence

  • Start with Gauss's law: Understand the one-to-one correspondence between "total flux through closed surface" and "internal net source".
  • Then study Faraday's law: The key is not how big the magnetic field is, but whether flux is changing.
  • Next understand displacement current: Why it's not a mathematical patch, but necessary for theoretical consistency.
  • Finally combine the two time-varying equations and accept that "light is an electromagnetic wave" - the complete structure then closes.