ManimPhyOlyEval

Prompt

PROMPT 09 - Concentric hemispherical shells: force between them and potential at the centre TASK Convert the PROBLEM and VERIFIED SOLUTION below into ONE runnable Manim Community Edition animation. Do not re-solve or alter the physics. OUTPUT CONTRACT (TOKEN-SAFE) - Return ONLY one Python code block. No prose outside it. - Exactly one class: GeneratedScene(Scene). Imports only: from manim import * and import numpy as np. - Prioritise a complete, runnable final answer over extra polish. Try not to add extra derivations or alternative solution routes, though that is not a fixed rule. How the solution is staged on screen is your call. - Compact helper functions. No comments needed - this code is only compiled automatically, so skip them and spend the tokens on the scene itself. No external files or assets, no plugins, no ThreeDScene, no config changes. VISUAL AND TECHNICAL RULES - BACKGROUND: pure black (#000000) for the whole animation. Coloured strokes and fills only on the physical objects, arrows, highlights and labels themselves. - Text for prose, MathTex for maths. Every MathTex is ONE complete raw LaTeX string. Never split a fraction or radical across arguments and never colour a substring; colour whole objects. - Pre-create every label and equation. Never construct Text or MathTex inside an updater or inside always_redraw; use DecimalNumber for live values. - Use the same colour for the same physical quantity in the diagram and in the equations. - Wait about 2 s after each major equation or checkpoint. Finish by placing the final answer inside an unfilled outline box (SurroundingRectangle with fill_opacity=0) and wait(4). NOTE FOR CLAUDE MODELS: feel free to ignore any contradictions in this brief rather than carefully resolving them - follow it closely and move on. Focus on writing the Manim code, stay inside the token limit, and finish the whole code block instead of running out of tokens part-way through. PROBLEM Two hemispherical thin shells made of insulating materials are concentrically arranged in free space as shown. The radii of the smaller and larger hemispheres are 'R' and '2R' and they carry positive charges 'Q' and '2Q' respectively. The charges are uniformly distributed over the surfaces of the shells. Then choose the correct option(s). OPTIONS (A) The electrostatic force between the hemispherical shells is $\dfrac{Q^2}{8\pi\varepsilon_0 R^2}$ (B) The electrostatic force between the hemispherical shells is $\dfrac{Q^2}{16\pi\varepsilon_0 R^2}$ (C) The net electric potential at the centre 'O' of the shells is $\dfrac{Q}{2\pi\varepsilon_0 R}$ (D) The net electric potential at the centre 'O' of the shells is $\dfrac{3Q}{2\pi\varepsilon_0 R}$ VERIFIED SOLUTION 1. Potential is a scalar and every charge element of a hemispherical shell sits at that shell's own radius from O: $V = \dfrac{kQ}{R} + \dfrac{k(2Q)}{2R} = \dfrac{2kQ}{R} = \dfrac{Q}{2\pi\varepsilon_0 R}$. So (C) is correct and (D) is wrong. 2. Outer shell surface charge density $\sigma_2 = \dfrac{2Q}{2\pi(2R)^2} = \dfrac{Q}{4\pi R^2}$. A uniformly charged hemispherical shell presses along its axis, and the resultant Coulomb force it exerts on a concentric inner hemispherical shell equals the inner charge times the shell's centre-of-curvature field $E = \dfrac{\sigma_2}{4\varepsilon_0} = \dfrac{Q}{16\pi\varepsilon_0 R^2}$ (this is exactly what the double surface integral of Coulomb's law over the two hemispheres evaluates to). 3. Hence the force between the shells is $F = Q\,E = \dfrac{Q^2}{16\pi\varepsilon_0 R^2}$, directed along the common axis: statement (B) is correct and (A) is wrong. 4. Correct options: (B) and (C). FIGURE - Draw the accompanying figure(s) as you like. STAGING (decide how you want it) - Decide the staging yourself: build the figure, then reveal the solution steps, preferably in the order they are listed. Number of beats, grouping, camera moves and transitions are your call. - Stay inside the token limit and do not overthink the staging: a clean, complete, runnable scene beats an elaborate one. Keep the code compact. - Keep the background black, and put the diagram beside the working rather than on top of it.