Physics · Electromagnetism

Maxwell's Equations Synthesis

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Wrap-up

What you learned

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Read next: Electromagnetic WavesSee the propagation consequence

Key takeaway

Gauss for E links enclosed charge to net electric flux through a closed surface.
Gauss for B keeps the net magnetic flux through a closed surface at zero without making local magnetic fields disappear.
Faraday turns changing magnetic flux into circulating electric field.
Ampere-Maxwell lets conduction current or changing electric flux feed magnetic circulation.
Aligned changing electric and magnetic fields are the bridge from Maxwell's equations toward electromagnetic waves.

Common misconception

Do not memorize the four equations as disconnected formulas. First ask whether the equation is about net flux through a closed surface or circulation around a loop.

The four equations do different jobs. Two are source laws and two are circulation laws, so a good reading starts by separating net-flux statements from loop-response statements.

Carry the synthesis into electromagnetic waves

Keep this idea moving

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See the propagation consequenceElectromagnetism

Electromagnetic Waves

See how changing electric and magnetic fields travel together as one rightward wave, with the local field pair, source-to-probe delay, and propagation cue all tied to the same compact live stage.

Carry the synthesis into lightOptics

Light as an Electromagnetic Wave

Connect electromagnetic waves to visible light, color, frequency, and the broader spectrum while one compact stage keeps the spectrum rail, field-pair sketch, and medium-linked wavelength changes tied together.

Reconnect B to forceMagnetism

Magnetic Force on Moving Charges and Currents

See why a magnetic field bends a moving charge sideways, why faster charges can make wider arcs, and how the same cross-product direction rule pushes a current-carrying wire.

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Reference and support

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Equation snapshotGauss for E · Gauss for BShow 3 equations

Use Gauss for E and Gauss for B to split source from closure, then use Ampere-Maxwell as the loop-law upgrade; Faraday joins it in the guided bridge step.

Gauss for E
$\oint E \cdot d A = \frac{Q _{enc}}{ε _{0}}$
Enclosed charge sets the net electric flux through a closed surface.
Gauss for B
$\oint B \cdot d A = 0$
Magnetic field lines still close on themselves, so a closed surface has no net magnetic source term.
Ampere-Maxwell law
$\oint B \cdot d l = μ_{0} I_{enc} + μ_{0} ε_{0} \frac{d Φ _{E}}{d t}$
Magnetic circulation can come from conduction current or from a changing electric field.

Why it behaves this way

Explanation

Maxwell's equations become much easier to read when you sort them into two jobs. Gauss's laws are source laws: enclosed charge sets the net electric flux through a closed surface, while magnetic field lines still close on themselves so the net magnetic flux through a closed surface is zero. Faraday's law and the Ampere-Maxwell law are circulation laws: changing magnetic flux drives circulating electric fields, and conduction current plus changing electric flux drive circulating magnetic fields.

This page keeps that synthesis explanation-first. One shared stage shows the two flux laws, the two circulation laws, and a light-bridge cue on the same live state. The same enclosed charge, conduction current, changing-electric term, changing-magnetic term, and cycle rate drive the stage, graphs, prediction prompts, worked examples, and quick test, so Maxwell's equations stay tied to one field-update story rather than feeling like four disconnected formulas.

Key ideas

01Gauss for E is a source law: positive enclosed charge gives outward net electric flux, negative enclosed charge gives inward net electric flux, and zero enclosed charge removes the net source term.

02Gauss for B is a closure law: magnetic patterns can be strong locally, but the net magnetic flux through a closed surface still stays zero because magnetic field lines loop back.

03Faraday's law says changing magnetic flux creates a circulating electric field. A magnetic field that is merely present is not enough.

04Ampere-Maxwell says magnetic circulation can come from conduction current or from a changing electric field, so wire current is not the only source term in the loop story.

05When changing electric and magnetic fields both stay active together, Maxwell's equations explain how electricity, magnetism, and light fit into one unified field picture.

Worked examples

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Read the source-law and circulation-law state directly from the live surface. The same enclosed charge, conduction current, changing-electric term, changing-magnetic term, and cycle rate that drive the stage also drive the algebra below.

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Example 1 of 2

Frozen valuesUsing frozen parameters

At the current synthesis time t = 0\,\mathrm{s}, what do the two flux laws say about enclosed charge and magnetic closure?

Q_{enc}

Enclosed charge

1.1 arb.

\oint E \cdot d A

Net electric flux

1.1 arb.

B_{loop}

Closed-loop magnetic pattern

0.7 arb.

f

Cycle rate

0.85 Hz

1. Read the enclosed charge

The current enclosed charge is

Q_{enc} = 1.1 arb.

, so the source is positive.

2. Apply Gauss for E

That gives

\oint E \cdot d A = 1.1 arb.

, so the net electric flux is outward.

3. Compare with Gauss for B

At the same instant the stage still gives

\oint B \cdot d A = 0 arb.

even though the local closed-loop strength is

0.7 arb.

Source-law reading

\oint E \cdot d A = 1.1 arb., \oint B \cdot d A = 0 arb.

Positive enclosed charge sets an outward net electric source term, while the magnetic-flux law still stays at zero because magnetic lines close back on themselves.

Quick test

Loading saved test state.

Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows five compact cards on one shared synthesis surface. Two top cards summarize the flux laws for electric and magnetic fields, two lower cards summarize the circulation laws, and a wider bridge card summarizes whether the current changing-field pair supports a light-like bridge cue.

A live readout lists time, enclosed charge, conduction current, changing-electric term, changing-magnetic term, magnetic circulation, and electric circulation. Optional overlays highlight the charge surface, magnetic closure reminder, Faraday loop, Ampere-Maxwell loop, and the light bridge cue.

Graph summary

The flux-laws graph keeps the two source-law readings separate: net electric flux can change with enclosed charge, while net magnetic flux through a closed surface stays zero.

The Ampere-Maxwell graph shows how conduction current and the changing-electric term feed magnetic circulation, and the Faraday-and-bridge graph shows how changing magnetic flux drives electric circulation and how the changing-field pair contributes to the light-bridge cue.

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