Switch-on bench

The battery comes on over a few nanoseconds. Drag to orbit, scroll to zoom.

0.00 ns light has reached 0.00 m

Point at the slice

|E|— |B|— |S|— V—

What to draw

Any combination; they layer.

Time

Now—

Switch-on time—

Slow motion—

The trace is the battery's EMF and the current it drives. They are not the same shape: the loop's own inductance holds the current back, and the gap between the two curves is what builds the field.

Viewpoint

Rotate (azimuth)—

Tilt (elevation)—

Zoom (distance)—

Lens (field of view)—

Rendering

Arrows through the volume Strength as colour on the slice Field lines in space Surface charge (+ / −) Sphere of radius ct Outline the slice

Arrow size—

Field line density—

With more than one field shown, the colour map takes the first of them — two over one plane is mud — while arrows and field lines layer happily, each in its own colour. The colour map is where the wavefront shows up most clearly. The sphere has radius ct about the battery, which is the only thing that changes: nothing else can respond until the news reaches it, so the field is strictly zero outside it. Wire the news has not reached is drawn dark, which is where ct has got to along the wire — exactly, where the charge marks are quantised and cannot be.

Slice plane

Offset—

Circuit

Battery—

LED resistance—

Wire resistance—

Wire radius—

Shape

Numbers

EMF now—

Current now—

Settles at—

Loop inductance—

L / R—

Light along the wire—

Loop length—

When L/R is much longer than the light-crossing time, the circuit is behaving quasi-statically and the model is on firm ground. Push the switch-on time down until the two are comparable and you are looking at an illustration rather than a derivation — see the README.

How this is computed

Ohm's law along the wire, with the induced field kept: the potential gradient is the battery's EMF, minus the resistive drop, minus ∂A/∂t. Integrated once around the loop those three give L İ + R I = e(t), and L is not assumed — it is the same geometric kernel the charge solve uses, weighted by t·t′ instead of 1.

The charge is linear in (e, I, İ), so three solves of one matrix give it for all time. Continuity then fixes how the current varies along the wire, and that variation is what radiates. The field comes from Jefimenko's equations at retarded time, so causality is not imposed anywhere: it falls out of the lookup.