4 files changed, 483 insertions, 52 deletions
diff --git a/docs/Test_4.md b/docs/Test_4.md
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+# Test 4
+
+**Test 4A:** Questions 1–3.  
+**Test 4B:** Questions 4–6.  
+**Max point:** 100 + 100.
+
+**NB!** Small sketches may improve the answers in questions 1, 3, 4, 5, 6.
+
+## 1. (30p)
+
+What is the work done by the electric force, when $q_0$ is moved from $a$ to $b$?
+
+## 2. (15p)
+
+Please write the condition for conservation of the total energy.
+
+## 3. (55p)
+
+When $q_0$ moves from $a$ to $b$ in an E-field from a point charge $q$ (“Central field”).
+
+**a. (30p)** Please derive the work done by the electric force.
+
+**b. (25p)** The work in above is equal to the drop in the electric potential energy
+$U_{\mathrm{e\,pot}\,a} - U_{\mathrm{e\,pot}\,b}$.
+What is $U_{\mathrm{e\,pot}\,a}$ and $U_{\mathrm{e\,pot}\,b}$ if $b \to \infty$?
+
+## 4. (25p)
+
+Bringing charges together.
+
+**a. (15p)** What is the total electric potential energy if $q_1, q_2, q_3$ are brought together from infinite?
+
+**b. (10p)** What is the additional electric potential energy if $q_0$ is brought together with $q_1, q_2, q_3$?
+
+## 5. (25p)
+
+When $q_0$ moves from $a$ to $b$ in a uniform E-field.
+
+**a. (15p)** Please derive the work done by the electric force.
+
+**b. (10p)** What is $U_{\mathrm{pot}\,a}$ and $U_{\mathrm{pot}\,b}$?
+
+## 6. (50p)
+
+The electric dipole in a uniform E-field.
+
+**a. (10p)** Please derive the torque vector.
+
+Prove, that the magnitude is
+
+$$
+\tau = p_e E \sin(\theta),
+$$
+
+where $\theta = \angle(\vec{p}_e, \vec{E})$.
+
+**b. (30p)** Please derive the work $W_{ab}$ done by the electric force when the electric dipole rotates from $\theta_a$ to $\theta_b$ and what is $U_{\mathrm{e\,pot}\,a}$ and $U_{\mathrm{e\,pot}\,b}$?
+
+You don’t have to derive $dW = -\tau\,d\theta$.
+
+**c. (10p)** Explain why $\theta_a = 90^\circ$ can be used as a reference position for the electric potential energy and write the result for $U_{\mathrm{e\,pot}\,a}$.
+
+Now find the general expression of $U_{\mathrm{e\,pot}}$ for the electric dipole, also written as a scalar product.
diff --git a/docs/Test_Answers.md b/docs/Test_Answers.md
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+++ b/docs/Test_Answers.md
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+# Test 4 – Answers
+
+We assume electrostatic fields in a vacuum and use:
+
+$$
+k=\frac{1}{4\pi\varepsilon_0}.
+$$
+
+Here, $U$ denotes electric potential energy, $V$ denotes electric potential, and $q_0$ is the charge being moved. 4Te_pw14 (1)(1)
+
+## 1. Work done by the electric force from $a$ to $b$
+
+The work is found by integrating the force along the path of the charge:
+
+$$
+W_{ab}=\int_a^b\vec F_e\cdot d\vec r.
+$$
+
+Since $\vec F_e=q_0\vec E$, we obtain:
+
+$$
+\boxed{W_{ab}=q_0\int_a^b\vec E\cdot d\vec r}.
+$$
+
+The electrostatic force is conservative. The work therefore equals the decrease in potential energy:
+
+$$
+\boxed{W_{ab}=U_a-U_b=q_0(V_a-V_b)}.
+$$
+
+Positive work means that the potential energy decreases.
+
+## 2. Condition for conservation of energy
+
+The total energy of an **isolated system** is constant:
+
+$$
+\boxed{\Delta E_{\mathrm{total}}=0}.
+$$
+
+If only the electric force does work, the sum of kinetic and electric potential energy is conserved:
+
+$$
+\boxed{K_a+U_a=K_b+U_b}.
+$$
+
+For a particle of mass $m$:
+
+$$
+\frac12mv_a^2+U_a=\frac12mv_b^2+U_b.
+$$
+
+This follows from the work–energy theorem:
+
+$$
+\Delta K=W_{ab}=-\Delta U.
+$$
+
+## 3. Motion in the field of a point charge
+
+### 3a. Derivation of the work
+
+A fixed point charge $q$ creates the field:
+
+$$
+\vec E(r)=k\frac{q}{r^2}\hat r.
+$$
+
+The force on $q_0$ is:
+
+$$
+\vec F_e=k\frac{qq_0}{r^2}\hat r.
+$$
+
+Let $r_a$ and $r_b$ be the distances from $q$ to $a$ and $b$, respectively. Since $\hat r\cdot d\vec r=dr$, we obtain:
+
+$$
+W_{ab}
+=kqq_0\int_{r_a}^{r_b}\frac{1}{r^2}\,dr.
+$$
+
+Thus:
+
+$$
+W_{ab}
+=kqq_0\left[-\frac1r\right]_{r_a}^{r_b}.
+$$
+
+Therefore:
+
+$$
+\boxed{W_{ab}=kqq_0\left(\frac1{r_a}-\frac1{r_b}\right)}.
+$$
+
+The result depends only on the initial and final distances, not on the path taken.
+
+### 3b. Potential energy with a reference at infinity
+
+We choose:
+
+$$
+U(\infty)=0.
+$$
+
+As $r_b\to\infty$, the work becomes:
+
+$$
+W_{a\infty}=\frac{kqq_0}{r_a}.
+$$
+
+Since $W_{a\infty}=U_a-U_\infty$, we obtain:
+
+$$
+\boxed{U_a=\frac{kqq_0}{r_a}},
+\qquad
+\boxed{U_b=U_\infty=0}.
+$$
+
+The general expression for the potential energy is therefore:
+
+$$
+\boxed{U(r)=\frac{kqq_0}{r}}.
+$$
+
+Charges with the same sign give positive potential energy; charges with opposite signs give negative potential energy.
+
+## 4. Assembling point charges
+
+Let $r_{ij}$ be the distance between charges $q_i$ and $q_j$. The zero of potential energy is chosen to be when all charges are infinitely far apart.
+
+### 4a. Total potential energy of $q_1,q_2,q_3$
+
+The charges can be assembled one at a time:
+
+- $q_1$ requires no work, since no other charges are present yet.
+- $q_2$ adds the energy $kq_1q_2/r_{12}$.
+- $q_3$ adds the energy $kq_1q_3/r_{13}+kq_2q_3/r_{23}$.
+
+The total potential energy is:
+
+$$
+\boxed{
+U_{123}
+=k\left(
+\frac{q_1q_2}{r_{12}}
++\frac{q_1q_3}{r_{13}}
++\frac{q_2q_3}{r_{23}}
+\right)
+}.
+$$
+
+Each pair of charges is counted once.
+
+### 4b. Additional potential energy when adding $q_0$
+
+With the three existing charges held in place, adding $q_0$ gives:
+
+$$
+\boxed{
+\Delta U
+=k\left(
+\frac{q_0q_1}{r_{01}}
++\frac{q_0q_2}{r_{02}}
++\frac{q_0q_3}{r_{03}}
+\right)
+}.
+$$
+
+This can also be written as:
+
+$$
+\boxed{\Delta U=q_0V_{123}},
+$$
+
+where $V_{123}$ is the potential due to $q_1,q_2,q_3$ at the position where $q_0$ is placed.
+
+## 5. Motion in a uniform electric field
+
+### 5a. Derivation of the work
+
+In a uniform field, $\vec E$ is constant:
+
+$$
+W_{ab}=q_0\int_a^b\vec E\cdot d\vec r
+=q_0\vec E\cdot(\vec r_b-\vec r_a).
+$$
+
+Thus:
+
+$$
+\boxed{W_{ab}=q_0\vec E\cdot(\vec r_b-\vec r_a)}.
+$$
+
+If the displacement has magnitude $s$ and the angle between the displacement and the field is $\alpha$:
+
+$$
+\boxed{W_{ab}=q_0Es\cos\alpha}.
+$$
+
+Choosing the $x$-axis along the direction of the field gives:
+
+$$
+\boxed{W_{ab}=q_0E(x_b-x_a)}.
+$$
+
+### 5b. Potential energy at $a$ and $b$
+
+Since $W_{ab}=U_a-U_b$:
+
+$$
+U_b-U_a=-q_0E(x_b-x_a).
+$$
+
+A potential energy that satisfies this is:
+
+$$
+U(x)=-q_0Ex+C.
+$$
+
+Therefore:
+
+$$
+\boxed{U_a=-q_0Ex_a+C},
+\qquad
+\boxed{U_b=-q_0Ex_b+C}.
+$$
+
+The constant $C$ is determined by the chosen zero of potential energy. Choosing $U_a=0$ gives:
+
+$$
+\boxed{U_a=0},
+\qquad
+\boxed{U_b=-q_0E(x_b-x_a)}.
+$$
+
+For a positive charge, the potential energy decreases when it moves in the direction of the field.
+
+## 6. Electric dipole in a uniform electric field
+
+A dipole consists of charges $+q$ and $-q$, separated by the vector $\vec d$, which points from the negative charge to the positive charge.
+
+The dipole moment is defined as:
+
+$$
+\boxed{\vec p_e=q\vec d}.
+$$
+
+The angle between $\vec p_e$ and $\vec E$ is denoted by $\theta$.
+
+### 6a. Derivation of the torque
+
+The forces on the charges are:
+
+$$
+\vec F_+=q\vec E,
+\qquad
+\vec F_-=-q\vec E.
+$$
+
+The net force is zero, but the forces form a couple.
+
+Taking the midpoint of the dipole as the origin, the position vectors are:
+
+$$
+\vec r_+=\frac{\vec d}{2},
+\qquad
+\vec r_-=-\frac{\vec d}{2}.
+$$
+
+The total torque is:
+
+$$
+\begin{aligned}
+\vec\tau
+&=\vec r_+\times\vec F_+
++\vec r_-\times\vec F_-\\
+&=\frac{\vec d}{2}\times q\vec E
++\left(-\frac{\vec d}{2}\right)\times(-q\vec E)\\
+&=q\vec d\times\vec E.
+\end{aligned}
+$$
+
+Thus:
+
+$$
+\boxed{\vec\tau=\vec p_e\times\vec E}.
+$$
+
+The magnitude of the cross product gives:
+
+$$
+\boxed{\tau=p_eE\sin\theta}.
+$$
+
+The torque tends to align the dipole moment with the field.
+
+### 6b. Work during rotation from $\theta_a$ to $\theta_b$
+
+The torque acts in the direction of decreasing $\theta$. With $\tau=p_eE\sin\theta$ as its magnitude:
+
+$$
+dW=-\tau\,d\theta.
+$$
+
+The work is therefore:
+
+$$
+W_{ab}=-p_eE\int_{\theta_a}^{\theta_b}\sin\theta\,d\theta.
+$$
+
+Integration gives:
+
+$$
+\boxed{
+W_{ab}=p_eE(\cos\theta_b-\cos\theta_a)
+}.
+$$
+
+Since $W_{ab}=U_a-U_b$, the potential energy can be written as:
+
+$$
+U(\theta)=-p_eE\cos\theta+C.
+$$
+
+Therefore:
+
+$$
+\boxed{U_a=-p_eE\cos\theta_a+C},
+$$
+
+$$
+\boxed{U_b=-p_eE\cos\theta_b+C}.
+$$
+
+### 6c. Reference at $90^\circ$ and the general expression
+
+The zero of potential energy can be chosen freely. We choose:
+
+$$
+\boxed{U_a=U(90^\circ)=0}.
+$$
+
+At $90^\circ$, the two charges are at the same potential in the uniform field, so their contributions to the total potential energy cancel each other.
+
+Since $\cos90^\circ=0$, this choice of reference gives $C=0$. Therefore:
+
+$$
+\boxed{U(\theta)=-p_eE\cos\theta}.
+$$
+
+Using the dot product, we obtain:
+
+$$
+\boxed{U=-\vec p_e\cdot\vec E}.
+$$
+
+| Angle | Orientation | Potential energy | Equilibrium |
+|---|---|---|---|
+| $0^\circ$ | Dipole moment aligned with the field | $-p_eE$, minimum | Stable |
+| $90^\circ$ | Dipole moment perpendicular to the field | $0$ | No equilibrium |
+| $180^\circ$ | Dipole moment opposite to the field | $+p_eE$, maximum | Unstable |
+\ No newline at end of file
diff --git a/docs/implementation-progress.md b/docs/implementation-progress.md
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-# Implementation progress — PLAN.md
-
-The user asks for educational 3D graphics for eight subjects. PLAN.md is the
-subject specification; no separate design or implementation spec exists.
-
-Ruling: Implement the eight subjects as selectable experiments in the existing
-Three.js app, sharing a scene host and parameter/readout interface. This gives
-each topic an interactive visual immediately. The models are idealizations,
-explained in the UI, rather than general field/circuit/quantum solvers.
-
-Ruling: Continue in this directory because `.git` is an empty managed mount,
-not a usable Git repository; worktree/commit scripts cannot run here.
-
-Pre-flight: Potential and energy share Coulomb models; capacitance and
-permittivity share a parallel-plate model; current and semiconductors share
-carrier constants. All physics functions use SI units and rendering handles
-unit conversion and illustrative scales.
-
-- Baseline: existing 3 physics tests pass.
-- [x] Electrostatics: charge/force/field, flux, energy, potential/gradient.
-- [x] Matter/circuits: capacitance, dielectrics, current, semiconductors.
-- [x] Eight experiment definitions and educational UI.
-- [x] Eight 3D scenes, interaction, animation, cleanup.
-- [x] Tests, browser checks, build, README.
-
-Implementation:
-- Electrostatics complete: force/field, exact enclosed-charge flux (surface
-  singularity disclosed), interaction energy/work, potential and probe gradient.
-- Matter/circuits complete: parallel plates, fixed-V/fixed-Q dielectrics,
-  Ohmic wire with signed drift, intrinsic/n/p silicon equilibrium/bands.
-- UI and eight scenes complete: independent settings, deep links/history,
-  reset, pause/reduced motion, responsive layout and WebGL fallback.
-- Renderer rebuilds are coalesced; obsolete GPU resources are disposed.
-- Visual inspection: field, potential, capacitance and semiconductor screenshots
-  checked. Labels changed to stable screen sizes for readability.
-
-Final review: fresh read-only reviewer approved the current implementation;
-no Critical or Important findings.
-- Model direction-only capacitor-arrow disclosure was already added during
-  implementation and resolves the reviewer's corresponding suggestion.
-- Final: minor (deferred): browser pause test verifies the state/button;
-  pixel comparisons proving actual animation cessation are not yet included.
-- Final: minor (deferred): finite-difference potential-gradient unit test covers
-  one charge along x; add multiple sources/off-axis components when expanding.
-- Build has a non-fatal Three.js bundle-size advisory (~598 kB minified,
-  ~153 kB gzip). No deployment or code-splitting requirement in PLAN.md.
-
-Final verification: 20/20 unit tests, 10/10 Chromium browser tests, strict
-TypeScript and Vite production build pass after the final label adjustment.
-All eight scenes were checked for rendered geometry and browser exceptions.
-The app remains served on http://localhost:5173. No Git integration action
-was available or required in this non-repository workspace.
diff --git a/docs/superpowers/plans/2026-10-24-test4.md b/docs/superpowers/plans/2026-10-24-test4.md
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+++ b/docs/superpowers/plans/2026-10-24-test4.md
@@ -0,0 +1,59 @@
+# Test 4 in 3D Implementation Plan
+
+> **For agentic workers:** Use superpowers:executing-plans to implement this plan task by task.
+
+**Goal:** Represent every question and answer in `docs/Test_4.md` and `docs/Test_Answers.md` through an educational, interactive 3D lesson.
+
+**Architecture:** Extend the existing TypeScript/Three.js lab with a separate Test 4 collection. Share the renderer and parameter UI; keep test physics, lesson content, and scene construction in focused modules. Native execution in the current checkout; the user delegated design and implementation choices.
+
+**Tech Stack:** Existing Three.js, TypeScript, Vite, Vitest, Playwright; no new dependencies.
+
+**Spec:** The two source documents above and the design below.
+
+## Design and constraints
+
+- Six lessons, with exact question scope, all subparts and original marks: 4A = 100 points, 4B = 100 points.
+- Guided lessons were selected as the implementation default after the user said “Choose the details and build it.” An exam-only view hides useful explanation; a disconnected presentation would duplicate the existing renderer.
+- Test 4 is the default entry. Existing experiment hash links and all eight experiments remain available through a collection switch.
+- Q1: uniform-field straight/curved paths with the same endpoints and work; Q2: repelling charge, energy bars and conserved K + U; Q3: radial motion, inverse-radius energy, reference at infinity; Q4: assemble three charges and add a fourth, showing pair energies once; Q5: angled displacement in a uniform field; Q6: dipole forces, torque, rotation work and energy versus angle.
+- Use adjustable progress for inspectable motion; decorative animation respects pause and reduced motion. Explicit physical units and visual scale explanations. Equations and derivations remain usable without WebGL.
+- Original questions, step-by-step answers, live numerical examples and a prediction/reveal check in each lesson. Source Markdown is linked from the built app.
+- Preserve existing unrelated work and source documents. No publication or repository integration requested.
+
+## Review focus
+
+Signed charges and work; arbitrary energy zeros; infinity represented analytically rather than as a finite point; each charge pair counted once; dipole torque sign versus magnitude and unstable equilibrium; mobile overflow; WebGL fallback; collection/history state.
+
+## Task 1: Physics models
+
+Files: `src/physics/test4.ts`, `src/physics/test4.test.ts`.
+
+- [x] Write independent numerical tests for uniform and central work, conservation, pair assembly, dipole torque/work and infinity.
+- [x] Run the tests and observe missing model failure.
+- [x] Implement SI functions with explicit energy references and domain validation.
+- [x] Run physics tests and confirm passing.
+
+## Task 2: Guided lesson content and scenes
+
+Files: `src/test4.ts`, `src/scene/test4.ts`, `src/experiments.ts`, `src/scene/experiments.ts`.
+
+- [x] Add browser checks for lesson selection, parameter changes, answers, graphics and fallback. Initial browser execution required enabling the local server and downloading Chromium into `/tmp`.
+- [x] Implement all six lesson definitions with original questions, subparts and derivations.
+- [x] Build parameter-linked 3D diagrams with endpoint labels, force/field arrows and energy plots or bars.
+
+## Task 3: Navigation, learning UI and verification
+
+Files: `src/main.ts`, `src/style.css`, `tests/test4.spec.ts`, `tests/lab.spec.ts`, `README.md`.
+
+- [x] Add Test 4/experiment collection navigation, question and solution panels, observation prompts, prediction reveal, previous/next links and retained parameter state.
+- [x] Verify direct links, collection/history navigation, mobile overflow, all scenes, resets and WebGL fallback.
+- [x] Run `npm test`, `npm run build`, `npm run test:e2e`; inspect desktop and mobile screenshots.
+- [x] Obtain an independent code review and address significant findings; document final outcome.
+
+## Execution record
+
+Baseline: 20 unit tests passed; production build passed (existing Three.js bundle-size advisory).
+
+Completed: all six questions and subparts are covered. Independent review confirmed physical and content consistency and found a mobile pagination issue. A new browser regression reproduced the issue, then passed after explicit pagination began focusing and scrolling the new heading. Review polish items were addressed with valid `aria-current="page"`, responsive framing and less crowded mobile labels. Both original Markdown links returned HTTP 200 with their original content from the production preview; assets are emitted as files rather than inlined data URLs.
+
+Final verification: 25 unit tests passed, 17 browser tests passed, strict TypeScript and production build passed, and `git diff --check` passed. No new dependencies. The existing Three.js bundle-size advisory remains. Work is left in the user's checkout without a commit or publication.