This interactive tutorial visualizes Maxwell's Equations in four modes: EM wave propagation (Faraday), Gauss's Law for electricity (point charge and flux), Gauss's Law for magnetism (dipole loops, net flux zero), and Ampère–Maxwell (wire current and displacement current). E is shown in blue and B in red.

 

Mathematical foundation

1. Wave propagation (Faraday)

A plane EM wave along x has E in y and B in z. ∇ × E = −∂B/∂t. In the sim: E = E₀ sin(kx − ωt) (blue), B = B₀ cos(kx − ωt) (red), 90° phase shift. Use the Mechanical Curl Probe (Curl Probe X): a paddlewheel samples B at left and right; the difference (∂B/∂x) is the curl and drives rotation. At the crest the paddle stops; at the zero-crossing it spins fastest—showing that E is generated by the change in B from place to place.

2. Gauss's Law (electric)

∮ E·dA = Q/ε₀. For a point charge, E ∝ 1/r²; flux through any sphere is constant. White/cyan arrows pierce the Gaussian surface (outward or inward by charge sign). Slice View cuts the sphere in half; a subplot at the bottom shows surface flux magnitudes in sequence.

3. Gauss's Law (magnetic)

∮ B·dA = 0 (no magnetic monopoles). A bar magnet (dipole) sits at the origin; B is computed from the dipole formula. White arrows = flux exiting (North side), cyan = entering (South side); the sliced circumference is highlighted. A subplot shows B·n̂ in sequence with equal outward/inward counts.

4. Ampère–Maxwell

∇ × B = μ₀J + μ₀ε₀∂E/∂t. A current I in a wire creates circular B (right-hand rule, B ∝ I/(2πr)). A changing E (e.g. between capacitor plates) also creates B (displacement current). Use Pulse to see E and B respond together.

 

Usage

Use this simulation to explore Maxwell’s Equations in four modes:

1. Mode: Switch between Wave Prop, Gauss (E), Gauss (B), and Ampère-Maxwell. The equation box and controls update for each mode.
2. Wave Prop: Blue arrows = E (vertical); red arrows = B (horizontal), 90° phase shift. Use Amplitude and Frequency; Step Fwd/Step Bwd advance or rewind time (stop first if running); Run/Stop toggles animation.
3. Gauss (E): Point charge at origin; radial E arrows (red = strong, blue = weak). Semi-transparent cyan Gaussian surface; white/cyan piercing arrows show flux direction. Charge sign (Positive/Negative) flips arrow direction. Slice View cuts the sphere and shows the slice circumference; a bottom subplot shows surface flux magnitudes as vertical arrows.
4. Gauss (B): Bar magnet (North red, South blue); dipole B piercing arrows on the sphere (white = exiting, cyan = entering). Same cyan Gaussian surface as Gauss (E). Slice View cuts the sphere and highlights the circumference; no Charge sign (no magnetic monopoles). Bottom subplot shows B·n̂ in sequence with equal white (outward) and cyan (inward) counts. Surface radius slider moves the sphere.
5. Ampère-Maxwell: Vertical wire; red B arrows circle the wire (right-hand rule, B ∝ I/(2πr)). Check Capacitor to show plates and E-field between them. Click Pulse to pulse the E-field; B-field circles “spawn” in sync (displacement current).
6. 3D view: Home (reset to default), Iso, Front, Top, Side, Z+/Z−, and pan ↑ ↓ ← → on the canvas. Drag to rotate, scroll to zoom. Arrow keys also pan when the canvas has focus.

Tips: In Gauss (B), move the surface and see flux stay zero; use Slice View to see the magnet inside the half-sphere. In Ampère-Maxwell, use Pulse with Capacitor on to see ∂E/∂t create B.

Parameters

• Charge: Wave: E and B strength. Gauss (E): charge strength. Gauss (B): dipole strength. Ampère-Maxwell: not used for wire current (use Current slider).
• Frequency (Wave) / Surface radius (Gauss) / Current (Ampère-Maxwell): Second slider label and range change by mode.
• Charge sign (Gauss E only): Positive or negative point charge. Hidden in Gauss (B) (no magnetic monopoles).
• Slice View (Gauss E and Gauss B): Cuts the Gaussian sphere in half and moves the camera to the cut; the slice circumference is drawn as a solid white line. In Gauss (B) the bar magnet is visible inside the half-sphere.
• Bottom subplot (Gauss E and Gauss B): Shows surface flux in sequence as vertical arrows—Gauss (E): flux magnitudes; Gauss (B): B·n̂ with equal outward (white) and inward (cyan) counts.
• Capacitor (Ampère-Maxwell): Toggle capacitor plates and E-field arrows between them.
• Pulse (Ampère-Maxwell): Triggers a pulse of changing E-field; B-field circles intensify in sync.
• Curl Probe X (Wave only): Position of the Mechanical Curl Probe (paddlewheel) along the X-axis. Drag the slider to move it. The paddlewheel samples B at its left and right blades; the difference (∂B/∂x) is the curl and sets the rotation speed. At the wave crest the paddle stops (curl = 0); at the zero-crossing it spins fastest (max curl). The blue E arrow at the center shows that this rotation generates E (Faraday).
• Step Fwd / Step Bwd / Run–Stop: Step Fwd and Step Bwd advance or rewind the wave (Wave mode only); they stop the animation first if it is running. Run–Stop toggles continuous animation.

Mathematical notes

Wave: Faraday ∇ × E = −∂B/∂t. Gauss (E): ∯ E·dA = Q_enc/ε₀; E ∝ 1/r² for point charge. Gauss (B): ∯ B·dA = 0 (no monopoles); dipole B loops. Ampère-Maxwell: ∇ × B = μ₀J + μ₀ε₀∂E/∂t; B ∝ I/(2πr) around wire; displacement current μ₀ε₀∂E/∂t also creates B.