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import * as THREE from 'three';
import { number, format as formatNumber, type Parameters, type TestId } from '../experiments';
import { scene } from '../i18n';
import { type Charge, type Point } from '../physics/electricField';
import { pairEnergy } from '../physics/models';
import { assemblyState, centralState, conservationState, motionState, movingCharge, progress, rotationState } from '../test4';
import { arrow, box, colours, grid, label, line, sphere } from './primitives';
import type { Animate } from './experiments';
const format = (value: number) => formatNumber(Math.abs(value) < 1e-10 ? 0 : value)
.replaceAll(',', '{,}').replace(/e([+-]?\d+)$/, String.raw`\times10^{$1}`);
function charge(group: THREE.Group, position: Point, q: number, name: string, radius = 0.21) {
const colour = q === 0 ? colours.muted : q > 0 ? colours.positive : colours.negative;
const mesh = sphere(group, position, radius, colour);
label(group, String.raw`$${name} = ${q >= 0 ? '+' : '-'}${format(Math.abs(q) * 1e9)}\,\mathrm{nC}$`, { ...position, y: position.y + 0.6 }, colour, 1.2);
return mesh;
}
function pulse(mesh: THREE.Mesh): Animate {
let time = 0;
return delta => { time += delta; mesh.scale.setScalar(1 + 0.035 * Math.sin(time * 2)); };
}
function field(group: THREE.Group, strength: number) {
if (strength === 0) return;
for (const x of [-4, -2, 0, 2, 4]) for (const y of [-1, 1, 3]) for (const z of [-2, 2]) {
arrow(group, { x, y, z }, { x: 1, y: 0, z: 0 }, 0.6, colours.field);
}
label(group, String.raw`$\vec E\to +x$`, { x: 3.8, y: 3.5, z: -2 }, colours.field, 1.6, 600);
}
function endpoint(group: THREE.Group, name: string, point: Point) {
const marker = sphere(group, point, 0.12, colours.gold);
(marker.material as THREE.MeshStandardMaterial).wireframe = true;
label(group, name, { ...point, y: point.y - 0.55 }, colours.gold, 1);
}
function motion(group: THREE.Group, p: Parameters, angled: boolean): Animate {
grid(group, -3.5);
const m = motionState(p);
field(group, m.field);
const a = { x: -m.dx / 2, y: 0.6 - m.dy / 2, z: 0 };
const b = { x: m.dx / 2, y: 0.6 + m.dy / 2, z: 0 };
const curved = !angled && p.path === 'curved';
const at = (t: number): Point => ({ x: a.x + m.dx * t, y: a.y + m.dy * t + (curved ? 2 * Math.sin(Math.PI * t) : 0), z: curved ? 1.3 * Math.sin(Math.PI * t) : 0 });
line(group, Array.from({ length: 101 }, (_, i) => at(i / 100)), colours.gold);
if (curved) line(group, [a, b], colours.muted, 0.4);
endpoint(group, String.raw`$a:\; U=0$`, a);
endpoint(group, 'b', b);
const point = at(progress(p));
const probe = charge(group, point, movingCharge(p), 'q_0');
const f = movingCharge(p) * m.field;
arrow(group, { ...point, y: point.y + 0.1, z: point.z + 0.35 }, { x: f, y: 0, z: 0 }, 1, colours.gold);
if (f !== 0) label(group, String.raw`$\vec F_e$`, { x: point.x + Math.sign(f) * 0.6, y: point.y + 0.25, z: point.z + 0.65 }, colours.gold, 0.8);
const projection = { x: b.x, y: a.y, z: -0.18 };
line(group, [{ ...a, z: -0.18 }, projection], colours.negative);
if (angled) {
line(group, [projection, { ...b, z: -0.18 }], colours.muted, 0.5);
arrow(group, { ...a, z: 0.2 }, { x: m.dx, y: m.dy, z: 0 }, m.s, colours.gold);
// The arc is measured from the +x direction even when cos alpha is negative.
const arc = Array.from({ length: 41 }, (_, i) => ({ x: a.x + 0.8 * Math.cos(m.angle * i / 40), y: a.y + 0.8 * Math.sin(m.angle * i / 40), z: 0.3 }));
line(group, arc, colours.field);
label(group, String.raw`$\alpha = ${number(p, 'angle')}^\circ$`, { x: a.x + 1, y: a.y + 0.5, z: 0.4 }, colours.field, 1.4);
}
label(group, String.raw`$\Delta x = ${format(m.dx)}\,\mathrm m$`, { x: 0, y: a.y - 1, z: 0 }, colours.negative, 2);
label(group, String.raw`$W_{ab} = ${format(m.work * 1e9)}\,\mathrm{nJ}$`, { x: 0, y: -2.7, z: 2 }, colours.gold, 2.8);
return pulse(probe);
}
function conservation(group: THREE.Group, p: Parameters): Animate {
grid(group, -3.3);
const s = conservationState(p);
const source = { x: -3.5, y: 1.8, z: 0 };
const point = { x: source.x + s.r, y: 1.8, z: 0 };
charge(group, source, 3e-9, 'q');
const probe = charge(group, point, movingCharge(p), 'q_0');
line(group, [source, { x: 2, y: 1.8, z: 0 }], colours.muted, 0.5);
if (s.speed > 0) arrow(group, { ...point, y: 2.8 }, { x: 1, y: 0, z: 0 }, 0.8 + s.speed, colours.gold);
label(group, String.raw`$r = ${format(s.r)}\,\mathrm m,\quad v = ${format(s.speed)}\,\mathrm{m/s}$`, { x: 0, y: 3.8, z: 0 }, colours.gold, 3.5);
const baseline = -2.7;
for (const [i, [name, joules, colour]] of ([['U', s.u, colours.negative], ['K', s.kinetic, colours.gold], ['K + U', s.total, colours.field]] as [string, number, string][]).entries()) {
const x = -2.2 + i * 2.2, height = joules / s.total * 2.6;
if (height > 0.001) box(group, { x, y: baseline + height / 2, z: 1 }, { x: 0.85, y: height, z: 0.6 }, colour, 0.85);
line(group, [{ x: x - 0.55, y: baseline, z: 1 }, { x: x + 0.55, y: baseline, z: 1 }], colours.muted);
label(group, String.raw`$${name} = ${format(joules * 1e9)}\,\mathrm{nJ}$`, { x, y: baseline - 0.45, z: 1 }, colour, 1.3);
}
line(group, [{ x: -2.8, y: -0.1, z: 1 }, { x: 2.8, y: -0.1, z: 1 }], colours.field, 0.4);
label(group, scene('energyConstant'), { x: 0, y: -4.2, z: 1 }, colours.field, 3.5);
return pulse(probe);
}
function central(group: THREE.Group, p: Parameters): Animate {
grid(group, -4.6);
const s = centralState(p);
const source = { x: -3.3, y: 1.4, z: 0 };
charge(group, source, s.q, 'q');
if (s.q !== 0) for (let i = 0; i < 24; i++) {
const angle = i * Math.PI / 12;
for (const radius of [1, 2.4, 3.6]) {
const dir = { x: Math.cos(angle), y: Math.sin(angle) * 0.6, z: Math.sin(angle) * 0.8 };
arrow(group, { x: source.x + radius * dir.x, y: source.y + radius * dir.y, z: radius * dir.z }, { x: Math.sign(s.q) * dir.x, y: Math.sign(s.q) * dir.y, z: Math.sign(s.q) * dir.z }, 0.25, colours.field);
}
}
const visibleEnd = Number.isFinite(s.rb) ? s.rb : 6;
const at = (r: number) => ({ x: source.x + r, y: source.y, z: 0 });
line(group, [at(s.ra), at(visibleEnd)], colours.gold);
endpoint(group, String.raw`$a:\;r_a=${s.ra}\,\mathrm m$`, at(s.ra));
if (Number.isFinite(s.rb)) endpoint(group, String.raw`$b:\;r_b=${s.rb}\,\mathrm m$`, at(s.rb));
else {
arrow(group, at(5.6), { x: 1, y: 0, z: 0 }, 1.3, colours.gold);
label(group, String.raw`$r_b\to\infty,\quad U_b=0$`, { x: 3.5, y: 2.8, z: 0 }, colours.gold, 2.3);
}
const radius = s.ra + progress(p) * (visibleEnd - s.ra);
const probe = charge(group, at(radius), s.q0, 'q_0');
const forceSign = Math.sign(s.q * s.q0);
arrow(group, { ...at(radius), z: 0.4 }, { x: forceSign, y: 0, z: 0 }, 0.9, colours.gold);
const y0 = -2.4, z0 = 1.5;
const plot = (r: number): Point => ({ x: source.x + r, y: y0 + THREE.MathUtils.clamp(pairEnergy(s.q, s.q0, r) * 1e9 / 30, -2.5, 2.5), z: z0 });
line(group, Array.from({ length: 108 }, (_, i) => plot(0.65 + i * 0.05)), colours.negative);
line(group, [{ x: -3, y: y0, z: z0 }, { x: 3.6, y: y0, z: z0 }], colours.muted, 0.6);
line(group, [{ x: -3, y: y0 - 1.6, z: z0 }, { x: -3, y: y0 + 2.3, z: z0 }], colours.muted, 0.6);
sphere(group, plot(radius), 0.12, colours.gold);
label(group, String.raw`$U(r)$ · 30 nJ / height unit`, { x: 0.8, y: -0.7, z: z0 }, colours.negative, 3);
label(group, String.raw`$U = 0\to$`, { x: 3.8, y: y0 - 0.3, z: z0 }, colours.muted, 1.4);
label(group, String.raw`$r\;(\mathrm m)\to$`, { x: 2, y: -4.4, z: z0 }, colours.muted, 1.4);
return pulse(probe);
}
function assembly(group: THREE.Group, p: Parameters) {
grid(group, -1.4);
const s = assemblyState(p);
const names = ['q_1', 'q_2', 'q_3', 'q_0'];
s.charges.forEach((q, i) => {
if (i < s.stage) charge(group, q, q.coulombs, names[i]);
else {
const ghost = sphere(group, q, 0.19, colours.muted);
const material = ghost.material as THREE.MeshStandardMaterial;
material.wireframe = true; material.transparent = true; material.opacity = 0.35;
label(group, String.raw`$${names[i]}$ · not placed`, { ...q, y: q.y + 0.6 }, colours.muted, 1.2);
}
});
for (let i = 0; i < s.stage; i++) for (let j = i + 1; j < s.stage; j++) {
const a: Charge = s.charges[i], b = s.charges[j];
const colour = j === s.stage - 1 ? colours.gold : colours.field;
line(group, [a, b], colour, 0.8);
const r = Math.hypot(a.x - b.x, a.y - b.y, a.z - b.z);
const pair = `${i === 3 ? '0' : i + 1}${j === 3 ? '0' : j + 1}`;
label(group, String.raw`$U_{${pair}} = ${format(pairEnergy(a.coulombs, b.coulombs, r) * 1e9)}\,\mathrm{nJ}$`, { x: (a.x + b.x) / 2, y: (a.y + b.y) / 2 - 0.3, z: (a.z + b.z) / 2 }, colour, 1);
}
label(group, `${s.stage} charges · ${s.stage * (s.stage - 1) / 2} unique pairs`, { x: 0, y: 4.5, z: 0 }, colours.field, 3);
label(group, String.raw`$U = ${format(s.current * 1e9)}\,\mathrm{nJ}$`, { x: 0, y: -1.9, z: 1 }, colours.gold, 2.4);
}
function dipole(group: THREE.Group, p: Parameters): Animate {
grid(group, -4);
const s = rotationState(p), d = number(p, 'length'), q = number(p, 'charge') * 1e-9;
field(group, s.field);
const centre = { x: 0, y: 1.5, z: 0 };
const at = (theta: number, sign: number) => ({ x: sign * d / 2 * Math.cos(theta), y: centre.y + sign * d / 2 * Math.sin(theta), z: 0 });
const minus = at(s.theta, -1), plus = at(s.theta, 1);
// Initial/final orientation marks help relate the scrubber to theta_a and theta_b.
for (const [theta, name] of [[s.a, 'a'], [s.b, 'b']] as [number, string][]) {
line(group, [at(theta, -1), at(theta, 1)], colours.muted, 0.25);
const end = at(theta, 1);
label(group, String.raw`$\theta_${name} = ${format(theta * 180 / Math.PI)}^\circ$`, { x: end.x * 1.6, y: centre.y + (end.y - centre.y) * 1.6 + 0.2, z: -0.6 }, colours.muted, 1.1, 600);
}
line(group, [minus, plus], colours.muted);
charge(group, minus, -q, 'q_-');
const positive = charge(group, plus, q, 'q_+');
arrow(group, minus, { x: plus.x - minus.x, y: plus.y - minus.y, z: 0 }, d, colours.field);
label(group, String.raw`$\vec p_e:\; -\to+$`, { x: -1.7, y: centre.y + 0.4, z: 0.65 }, colours.field, 1.5);
if (s.field !== 0) {
arrow(group, { ...plus, z: 0.2 }, { x: 1, y: 0, z: 0 }, 1.15, colours.gold);
arrow(group, { ...minus, z: 0.2 }, { x: -1, y: 0, z: 0 }, 1.15, colours.gold);
label(group, String.raw`$\vec F_+$`, { x: plus.x + 1.1, y: plus.y + 0.2, z: 0.2 }, colours.gold, 0.8);
label(group, String.raw`$\vec F_-$`, { x: minus.x - 1.1, y: minus.y - 0.3, z: 0.2 }, colours.gold, 0.8);
}
const torqueNormalized = s.field ? s.torqueZ / (s.moment * s.field) : 0;
if (Math.abs(torqueNormalized) > 1e-6) {
arrow(group, centre, { x: 0, y: 0, z: torqueNormalized }, Math.abs(torqueNormalized) * 1.8, colours.negative);
label(group, String.raw`$\vec\tau\to -z$`, { x: 2, y: 1.7, z: -1.9 }, colours.negative, 1.3);
}
const arc = Array.from({ length: 61 }, (_, i) => ({ x: 0.7 * Math.cos(s.theta * i / 60), y: centre.y + 0.7 * Math.sin(s.theta * i / 60), z: 0.25 }));
line(group, arc, colours.gold);
label(group, String.raw`$\theta = ${format(s.theta * 180 / Math.PI)}^\circ$`, { x: 2.5, y: 3.6, z: 0 }, colours.gold, 2);
const plot = (theta: number): Point => ({ x: -3 + theta / Math.PI * 6, y: -2 - (s.field === 0 ? 0 : Math.cos(theta)), z: 1.2 });
line(group, Array.from({ length: 101 }, (_, i) => plot(i * Math.PI / 100)), colours.negative);
line(group, [{ x: -3.3, y: -2, z: 1.2 }, { x: 3.3, y: -2, z: 1.2 }], colours.muted, 0.5);
sphere(group, plot(s.theta), 0.13, colours.gold);
for (const [x, text] of [[-3, String.raw`$0^\circ$ · stable`], [0, String.raw`$90^\circ$ · $U=0$`], [3, String.raw`$180^\circ$ · unstable`]] as [number, string][]) {
label(group, s.field === 0 ? text.split(' · ')[0] : text, { x, y: -3.9, z: 1.2 }, colours.muted, 1.4);
}
label(group, s.field ? String.raw`$\frac{U}{p_eE} = -\cos\theta$` : String.raw`$E=0,\quad U=0$ at all angles`, { x: 0, y: -0.6, z: 1.2 }, colours.negative, 2.6);
return pulse(positive);
}
export function buildTestScene(group: THREE.Group, id: TestId, p: Parameters): Animate {
switch (id) {
case 'test-work': return motion(group, p, false);
case 'test-conservation': return conservation(group, p);
case 'test-central': return central(group, p);
case 'test-assembly': assembly(group, p); return () => {};
case 'test-uniform': return motion(group, p, true);
case 'test-dipole': return dipole(group, p);
}
}
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