Poynting Vector: Energy Flow in Electromagnetic Fields

Scenario

Field controls

Field strength E₀ 1.0

Animation speed 1.0x

Vector density Medium

Show field layers

Electric field E Magnetic field B Energy flow S (Poynting)

Live readouts

|E| at probe

0.00

|B| at probe

0.00

|S| = |E × B| / μ₀ at probe

0.00

Energy flows radially into the gap as the capacitor charges.

How it works

The Poynting vector gives the direction and density of electromagnetic power flow:

S = (1/μ₀) E × B   [W/m²]

Charging capacitor — the field between the plates grows, so by the Ampere–Maxwell law (displacement current) a circular B field appears:

∮B·dl = μ₀ε₀ dΦ_E/dt

Crossing the axial E with the azimuthal B gives S pointing radially inward — the stored field energy visibly streams in from the surrounding space, not along a wire.

Current-carrying wire — the same cross-product trick applied to a resistor is the classic surprise of electrodynamics: the circular B field around the wire (Ampère's law) crossed with the small axial E field (the resistive voltage drop) gives S pointing into the wire's surface. The Joule heat I²R dissipated in the resistor arrives through the field around it, not by flowing down the copper.

Propagating wave — in a plane EM wave E and B oscillate in phase, perpendicular to each other and to the direction of travel. Their cross product never changes sign, so S always points forward along the wave, pulsing as sin²(kz − ωt) with a time-average intensity ⌈S⌉ = ½ε₀cE₀².

Scenario buttons — switch between the three canonical field geometries.
Field strength — scales the driving E₀ amplitude in every scenario.
Animation speed — scales how fast the fields evolve in time.
Vector density — how many sample points the field is drawn at.

All quantities are shown in normalized units (μ₀ = ε₀ = c = 1) so the geometry of the flow, not the SI magnitude, is the point.

Electric field E Magnetic field B Energy flow S

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