## A Charge Fills Space With a Field A charge sets up an **electric field** at every point around it — there whether or not anything feels it. Drop a **test charge** to reveal it: the field is the **force per unit charge**, `E = F / q`. <viz id="0"></viz> **Drag** the charges; **slide** `Q` and `q`; **switch** arrows / field lines. The arrows point **away** from `+`, **toward** `−`, and shrink with distance. Change `q` and the *force* changes, but the *field* `E` does not — the field belongs to the source. So what happens when the charge starts to **move**? ## Set It Moving: The Magnetic Field Appears In <ref slide="1">A Charge Fills Space With a Field</ref> the charge sat still and carried a single, radial electric field. Now set it **moving**, and something new switches on: a **magnetic field**. The electric field still fills all of space as **radial lines** streaming out in every direction. But a magnetic field is a different kind of beast — it doesn't point outward, it **curls**. Its arrows wrap *around* the line of motion, swirling through the whole space surrounding the charge like a current circling a drain. Which way they swirl is fixed by the **right-hand rule**: point your right thumb along the velocity, and your fingers curl the way the field goes. <viz id="1"></viz> **Raise the speed** and **orbit** to see the swirl; **toggle** `E` and `B`. Look closely and you'll notice the swirl is **densest and brightest near the charge** and in the plane straight through it, fading away with distance and toward the front and back — the magnetic field is strongest *perpendicular* to the motion and vanishes straight ahead. And the faster the charge moves, the stronger the whole pattern: `|B|` grows directly with speed `v`. So a moving charge carries **two** fields at once — the electric field from before, and this new magnetic field circling its path. They look like separate things. But are they? In the next slide we'll climb aboard and ride alongside the charge, and watch the magnetic field melt away — revealing that **electricity and magnetism are one and the same field, seen from different points of view.** ## Same Field, Different Frame In <ref slide="2">Set It Moving: The Magnetic Field Appears</ref> a moving charge carried two fields: a radial **electric** field and a swirling **magnetic** one. But here's the question that breaks the whole picture open: *moving relative to whom?* A magnetic field shows up because the charge is moving **past you**. So what if you stopped standing still and **chased it** — matched its speed and rode alongside? In your new frame the charge isn't moving at all. Does its magnetic field simply... vanish? <viz id="2"></viz> **Slide your speed up** to ride alongside — the magnetic field fades. Yes — the magnetic field really does disappear. Not because you destroyed anything, but because **there was never a separate magnetic thing to begin with**. What one observer calls "an electric field plus a magnetic field," another observer — moving differently — calls "just an electric field." They are describing the **same physical reality** with a different split. This is the deep truth: electricity and magnetism are not two forces but **one electromagnetic field**, and how it divides into `E` and `B` depends entirely on **how you are moving**. Magnetism is what electricity looks like from a moving frame — it is relativity in disguise. So here is the loophole, and it's the doorway to light. *Steady* motion can always be transformed away: there's always a frame where the charge sits still and the magnetic field is gone. But what about a charge whose motion **changes** — one that **accelerates**? You can never find a frame where an accelerating charge is at rest. That disturbance **cannot be transformed away by anyone**. In the next slide we let the charge change its motion — and watch the field do something it has never done before: **break off and leave.** ## What Acceleration Adds A charge at rest, or moving steadily, already fills space with a field — so a distant test charge *already* feels a force. And as we saw, *uniform* motion changes nothing real: whether the charge drifts past at a constant speed or **we** ride along beside it, there's always a frame where it sits still — no news is sent, nothing new happens. Only **acceleration** is absolute. So what does accelerating the source actually change? The answer is about **information and time**: when the charge changes its motion, the news cannot reach the test charge instantly. It travels outward at the speed of light `c`, as a **kink** in the field. <viz id="3"></viz> **Wiggle once** and **slow `c`** to watch the kink travel; **switch** `E` / `B` / Both. Here is the whole idea, seen as a slice through space. Steady motion (rest or constant velocity) sends no news — the field simply tracks the charge. But a **change** of motion *does* send news, propagating at `c`. When it reaches a point, it delivers a transverse kick that falls off only as `1/r` (not `1/r²`), so it reaches across space — and it carries an **electric** field and a **magnetic** field locked together and at right angles. You can see it directly: the field lines develop a travelling **kink**, and the magnetic glyphs flare where the field is changing, both swelling and reversing as the news sweeps by. A change of motion, propagating outward as a self-supporting pair of perpendicular `E` and `B` — that is radiation. And the strangest thing about it is *how far it reaches*: that transverse kick fades only as `1/r`, not the `1/r²` of an ordinary field. Why does the news refuse to die away so quickly? The next slide shows you why — and it's the same reason a lamp dims with distance. ## Why It Reaches So Far We just said the kick fades only as `1/r`, not the `1/r²` of an ordinary field — and that one fact is why a wiggle here can reach a charge across the room, or across the galaxy. But *why* `1/r`? There's a reason you already feel every day: it's the same reason a lamp looks dimmer the farther away you stand. <viz id="4"></viz> **Send a pulse** and **slow `c`** to watch it spread; the rings mark where its intensity is `100%`, `25%`, `11%`. One wiggle sends out a fixed **pulse of energy** — a shell expanding at `c`. That energy is **conserved**: the shell never gains or loses any. But as it flies outward it has to cover a **sphere**, and a sphere's area grows as `r²` — double the distance and the same energy is smeared over **four times** the area. So the **intensity** — energy per unit area, what a detector actually catches — must thin out as `1/r²`. That's the familiar inverse-square law of brightness, and you can watch it on the rings: `100%`, then `25%`, then `11%`. Here's the final step. The intensity of a wave is proportional to the **square** of its field (`intensity ∝ E²`). So if intensity falls as `1/r²`, the field itself falls as its square root — **`1/r`**. That's the whole answer. A static field, by contrast, just *sits* there storing its energy; it never has to spread a fixed pulse over growing spheres, so it's free to die off faster, as `1/r²`. Radiation is energy **in flight**, forced by simple geometry to fade slowly — and that gentle `1/r` fade is exactly why light, alone among a charge's effects, can cross the emptiness between the stars and still reach your eye. ## Why It Takes Both For the last few slides the electric and magnetic fields have travelled together. Is that a necessity, or could just one of them be the wave on its own? Try removing one and see. <viz id="5"></viz> This is an **exact Maxwell solution** (the equations are shown below). Watch the arrows rise as the pulse rolls outward — each field's slope builds the other just ahead. **Switch** to Only electric / Only magnetic and it can't move: it stays pinned at the charge. A wave travels for one simple reason: **each field's change creates the other just ahead.** A changing `E` builds `B` in the next bit of space; that changing `B` builds `E` a bit further on; and so the disturbance steps outward at `c`. It's a relay — and a relay needs two runners. Take one away and the relay breaks. The lone field has no partner to hand off to, so it can't step forward — it just sits at the charge, pulsing in place. You don't get a weaker wave; you get **no wave**. So light isn't "an electric wave dragging a magnetic one along." It's the two of them creating each other, over and over, as they go — and that same coupling even fixes light's speed: `c = 1/√(μ₀ε₀)`. Two fields, each conjuring the other, sweeping outward at one fixed speed — it's time to see the whole thing at once. ## This Is Light Everything has been leading here. A charge fills space with a field. Move it and a magnetic field appears. **Wiggle** it — change its motion — and a kink breaks free and races outward at `c`, carrying an electric and a magnetic field locked together. Now see it whole, in three dimensions, and call it by its true name. <viz id="6"></viz> **Orbit** the wave; **slow `c`**; **raise the frequency** to shift the colour; **switch** `E` / `B` / Both. This wave — a transverse ripple of electric and magnetic field, perpendicular to each other, renewing each other as they go, sustaining themselves across empty space at the one speed `c` — is **light**. Not a metaphor for light, not a model of light: it *is* light. Radio from an antenna, the glow of a filament, the colours leaving this screen, the sunlight that crossed the void to reach your eye — every one of them is a charge, somewhere, that wiggled. You have just watched light being born. ## Aim the Wiggle So far the charge has bobbed straight up and down. But it can wiggle in **any** direction — and in **different ways** — and *how* it wiggles shapes the light it makes. Aim it wherever you like, try a different style of motion, and watch where the wave goes. <viz id="7"></viz> **Pick the motion**, **aim** the wiggle, and **slow `c`** to watch the pattern sweep out. For a charge bobbing in a straight line, the light it throws off is not the same in every direction. Its strength falls as `sinθ / r`, where `θ` is the angle from the wiggle axis — so it is **strongest in the plane perpendicular** to the wiggle and **exactly zero along the axis itself**. You can never see a wiggling charge "end-on": look straight down its line of motion and there is no light at all. That doughnut-shaped pattern is the **dipole radiation pattern**, and it rides with the axis — tilt the wiggle and the whole doughnut tilts with it. The direction also sets the light's **polarization**: the electric field of the wave points along the wiggle direction (projected across your line of sight). This is not an abstraction — it is why a radio antenna is a straight rod aimed a particular way, why you rotate an antenna to catch a station, and why polarized sunglasses can dim light coming from one orientation and not another. **How a charge wiggles is written into the light it sends out** — its brightness pattern, its direction, and its polarization all carry the memory of that motion. ## Aim the Wiggle in 3D The two lobes you just saw were only a **slice**. Spin that figure-8 around the wiggle axis and you get the real shape of the light a charge throws off: a glowing **doughnut** wrapped around the axis. Here you can aim the wiggle anywhere in space and see it whole. <viz id="8"></viz> **Pick the motion**, **aim the axis**, and **orbit**; **slow `c`** to watch it radiate. Each arrow is the **transverse part of the charge's retarded acceleration** — the exact thing that radiates, `E ∝ a⊥(t − r/c) / r` — its **orientation** showing the field's direction, its **brightness** the strength. For **Oscillation**, the acceleration always lies along the axis, so the field is strong around the equator and vanishes along the axis: the dipole doughnut. For **Circle**, the acceleration *rotates*, so light streams out in *every* direction — orbit to look straight down the axis and the arrows themselves **rotate**, which is **circular polarization**. **Ellipse** sits between the two — a lopsided doughnut, elliptical polarization — while **Figure-8** and **Clover** (Lissajous motions, beating two rhythms against each other) weave richer, many-lobed patterns again. And because everything uses the *retarded* acceleration, re-aiming the axis doesn't change the far field instantly — the change **sweeps outward at `c`**. Same single rule — radiate your transverse acceleration — yet wholly different light depending on **how the charge moves**. *The motion is written into the shape, direction, and polarization of the light.*