ElectromagnetismIntermediate

Concept module

Maxwell's Equations Synthesis

See what each Maxwell equation says physically, how sources and circulation differ, and why changing electric and magnetic fields together unify electricity, magnetism, and light.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 4.00 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s4.00 s

Maxwell's Equations Synthesis

One compact synthesis stage keeps the source laws, circulation laws, and light bridge on the same state instead of splitting them into disconnected formula cards.

Live synthesis

Gauss for E

Charge makes electric flux

Enclosed charge sets the net electric flux through a closed surface. Positive charge sends the field outward; negative charge pulls it inward.

Qenc1.1 arb.

oint E dA1.1 arb.

readingoutward flux

Use this card to separate field sources from circulation. Charge changes the electric flux balance directly.

Gauss for B

Magnetic flux still balances

Magnetic field lines close on themselves. You can have stronger or weaker magnetic patterns locally, but the net magnetic flux through a closed surface stays zero.

net B flux0 arb.

closed loops0.7 arb.

readingfield lines close

This card is the no-monopoles reminder. Magnetic patterns can intensify without creating a net source term.

Faraday

Changing B makes circulating E

A changing magnetic flux does not just change a number on a graph. It creates a circulating electric field whose direction flips when the magnetic change flips.

dPhi_B/dt0 arb.

oint E dl0 arb.

readingnear zero

Watch the center glyph and the ring together. The sign of dPhi_B/dt sets the E-circulation sense.

Ampere-Maxwell

Current and changing E make circulating B

Magnetic circulation is not sourced only by conduction current. Maxwell's displacement-current term means a changing electric field also feeds the same B circulation story.

Ienc0.7 arb.

dPhi_E/dt0 arb.

oint B dl0.7 arb.

Compare the two source terms: conduction current and changing electric flux. Both land on one shared B loop.

Unification

When both changes keep feeding each other, light appears

Faraday and Ampere-Maxwell are the handoff pair. If changing E and changing B are both present in the same story, the field update can keep propagating instead of dying locally.

cycle rate0.85 Hz

period1.18 s

wave cue0 arb. (no light cue)

pairnot reinforcing

The bridge is strongest when both changing-field terms stay active together. This is the compact intuition behind why Maxwell's equations unify electricity, magnetism, and light.

Live readout

t0 s

Qenc1.1 arb.

Ienc0.7 arb.

dPhi_E/dt0 arb.

dPhi_B/dt0 arb.

oint B dl0.7 arb.

oint E dl0 arb.

Gauss-B stays at zero net flux while closed loops strengthen to 0.7 arb..

Ampere-Maxwell is currently current dominated, and the light bridge reads no light cue.

Synthesis reading

Lowering the cycle rate lengthens the handoff period to 0.63 s up to the current 1.18 s window, so the graph and stage stay on the same oscillation clock.

This page keeps the four equations compact: flux laws on top, circulation laws below, and the light bridge only when the changing-field pair can keep feeding itself.

Graphs

Switch graph views without breaking the live stage and time link.

Flux laws

Compares the electric source-law reading with the closed-surface magnetic-flux reading so the two flux laws stay visibly different.

time (s): 0 to 4flux: -2 to 2

Net electric fluxNet magnetic flux

Hover or scrub to link the graph back to the stage.time (s) / flux

Controls

Adjust the physical parameters and watch the motion respond.

Enclosed charge

Q_{enc}

1.1 arb.

Sets the electric source term for the closed-surface flux reading.

Conduction current

I_{enc}

0.7 arb.

Sets the current contribution to the Ampere-Maxwell loop story.

Changing electric field

\frac{d Φ _{E}}{d t}

0.9 arb.

Changes the size and sign of the changing-electric contribution over the shared cycle.

Changing magnetic field

\frac{d Φ _{B}}{d t}

0.9 arb.

Changes the size and sign of the changing-magnetic contribution over the shared cycle.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

Cycle rate

f

0.85 Hz

Controls how quickly the shared changing-field clock runs.

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

MisconceptionPrompt 1 of 2

The magnetic card is not saying there is no magnetic field. It is saying the net closed-surface flux stays zero even while magnetic loops are strong locally.

Try this

Watch the closed-loop readout while the lower loop cards are active. Local magnetic structure is still present even when the net flux law stays at zero.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

Q_{enc}

Enclosed charge

1.1 arb.

Changes the electric source term directly, so the electric-flux card can switch between outward, inward, and balanced net flux.

Graph: Flux lawsOverlay: Charge surface

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use one prompt at a time. Each one points at a distinction the live synthesis stage and graphs are already making.

MisconceptionPrompt 1 of 2

Graph: Flux laws

The magnetic card is not saying there is no magnetic field. It is saying the net closed-surface flux stays zero even while magnetic loops are strong locally.

Try this

Watch the closed-loop readout while the lower loop cards are active. Local magnetic structure is still present even when the net flux law stays at zero.

Why it matters

This keeps Gauss for B from being misread as magnetism disappearing.

Graph: Flux lawsOverlay: Magnetic closure

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

5 visible

Overlay focus

Charge surface

Highlights the closed surface around enclosed charge so the electric source law stays visible.

What to notice

Changing the enclosed charge changes the electric source term directly, even if the circulation cards below do something different.

Why it matters

It separates electric sources from the loop laws instead of blending everything into one field-line picture.

Control: Enclosed chargeGraph: Flux lawsEquation

Q_{enc}

At t = 0 s, the enclosed charge is positive, so the net electric flux points outward while the net magnetic flux still stays 0. Ampere-Maxwell gives total B circulation 0.7 arb. from conduction current 0.7 arb. plus changing electric flux 0 arb., and Faraday gives E circulation 0 arb. from changing magnetic flux 0 arb.. Without both changing-field terms present at the same moment, the light-like bridge collapses.

Equation detailsDeeper interpretation, notes, and worked variable context.

Gauss for E

Enclosed charge sets the net electric flux through a closed surface.

\oint E \cdot d A = \frac{Q _{enc}}{ε _{0}}

Q_{enc}

Enclosed charge 1.1 arb.

Gauss for B

Magnetic field lines still close on themselves, so a closed surface has no net magnetic source term.

\oint B \cdot d A = 0

Faraday's law

Changing magnetic flux creates a circulating electric field.

\oint E \cdot d l = - \frac{d Φ _{B}}{d t}

\frac{d Φ _{B}}{d t}

Changing magnetic field 0.9 arb.

Ampere-Maxwell law

Magnetic circulation comes from conduction current and from a changing electric field.

\oint B \cdot d l = μ_{0} I_{enc} + μ_{0} ε_{0} \frac{d Φ _{E}}{d t}

I_{enc}

Conduction current 0.7 arb.

\frac{d Φ _{E}}{d t}

Changing electric field 0.9 arb.

Changing-field bridge

When both changing-field terms stay active together, the field story can keep propagating instead of remaining only local.

\partial_{t} E \leftrightarrow \partial_{t} B

\frac{d Φ _{E}}{d t}

Changing electric field 0.9 arb.

\frac{d Φ _{B}}{d t}

Changing magnetic field 0.9 arb.

f

Cycle rate 0.85 Hz

Progress

Not startedMastery: NewLocal-first

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Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Short explanation

What the system is doing

Maxwell's equations are the compact synthesis that turns electricity and magnetism from separate-looking cases into one field story. Two equations say what counts as a source: 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 stays zero. Two more equations say what changing fields do around loops: changing magnetic flux creates circulating electric fields, and conduction current plus changing electric flux create 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, overlays, worked examples, and quick test so Maxwell's equations stay tied to one honest field-update story rather than four detached 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 strengthen or weaken 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. The response is about change, not about whether a magnetic field is merely present.

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

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

Live synthesis checks

Solve the exact state on screen.

Read the current source-law and circulation-law state directly from the live synthesis surface. The same sliders and time state now driving the stage also drive the algebra below.

Live valuesFollowing current 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 source term

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. Apply 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.

Flux-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.

Field-bridge checkpoint

Imagine a region with no conduction current through the loop, but the electric field across that region is still changing. Should the magnetic circulation around the loop vanish?

Prediction prompt

Answer from Ampere-Maxwell, not from a current-only rule.

Check your reasoning

No. The magnetic circulation can still be nonzero because the changing electric field contributes through Maxwell's displacement-current term.

Ampere-Maxwell says

\oint B \cdot d l

depends on enclosed conduction current plus the changing electric-flux term. Zero wire current does not force zero magnetic circulation if

d Φ_{E} / d t

is still present.

Common misconception

Maxwell's equations are just four separate formulas to memorize, and light only means the electric field changes first while the magnetic field reacts later.

The four equations do different jobs. Two are source laws and two are circulation laws, so reading them well means separating what creates net flux from what creates loop-like response.

In the synthesis picture, changing electric and magnetic fields can both belong to the same ongoing field update. The point is not a delayed after-effect at one point, but a coupled structure that can support propagation.

Quick test

Reasoning

Question 1 of 4

Answer from the live synthesis logic, not from detached formula recall. Each question asks what the field picture must mean physically.

Which Maxwell equation is the direct source-law statement that enclosed charge changes the net field through a closed surface?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

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 compares net electric flux with net magnetic flux on the same time axis. The Ampere-Maxwell graph compares conduction current, the changing-electric term, and the resulting magnetic circulation. The Faraday-and-bridge graph compares the changing-magnetic term, the circulating electric response, and the light-bridge cue.

The accessibility takeaway is that the graphs keep source laws and circulation laws distinct while still showing how the changing-field pair can support a unified electricity-magnetism-light story.

Carry the synthesis into waves

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

See the propagation consequenceElectromagnetism

Maxwell's Equations Synthesis

Charge makes electric flux

Magnetic flux still balances

Changing B makes circulating E

Current and changing E make circulating B

When both changes keep feeding each other, light appears

See each variable before you move it.

Charge surface

What the system is doing

Solve the exact state on screen.

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

Which Maxwell equation is the direct source-law statement that enclosed charge changes the net field through a closed surface?

Keep moving through the concept map.

Electromagnetic Waves

Light as an Electromagnetic Wave

Magnetic Force on Moving Charges and Currents