Updated Animation Specification

1. Animation Description & Purpose

Create a concise 3‑D Manim animation (≈26 s) that visualises a magnetic trap formed by four infinite straight current‑carrying wires arranged at the corners of a square in the $x\!y$-plane. The animation should:

• Establish the coordinate system ($x$, $y$ axes in the plane, $z$ pointing out of the screen).
• Show the four wires with their current directions.
• Illustrate the magnetic field generated by each wire (circular field lines) and the resulting net field in the plane.
• Introduce a positively charged particle $+q$ released from the origin with an initial velocity $\mathbf{v}_0$ along the positive $y$-axis.
• Depict the particle’s trajectory only for the first 6 s of its motion in the $x\!y$-plane under the Lorentz force $\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}$.
• Emphasise the physical concepts: Biot–Savart law, superposition of magnetic fields, and magnetic confinement.

2. Mathematical Elements & Formulas

• Magnetic field of a single infinite wire (Biot–Savart):
  $$\mathbf{B}_i(\mathbf{r}) = \frac{\mu_0 I_i}{2\pi r_i}\,\hat{\boldsymbol{\phi}}_i$$
  where $r_i$ is the perpendicular distance from the wire and $\hat{\boldsymbol{\phi}}_i$ is the azimuthal unit vector given by the right‑hand rule.
• Superposition for the total field in the plane:
  $$\mathbf{B}(x,y) = \sum_{i=1}^{4} \mathbf{B}_i(x,y)$$
• Lorentz force on the particle (gravity ignored):
  $$\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}$$
• Equation of motion (non‑relativistic):
  $$m\,\frac{d\mathbf{v}}{dt}=q\,\mathbf{v}\times\mathbf{B}(x,y)$$
  The animation will display a pre‑computed trajectory limited to the first 6 s of motion.

3. Visual Elements

Element	Representation	Color / Style
Coordinate axes	Three arrows: $x$ (right), $y$ (up), $z$ (out of screen)	$x$: bright red, $y$: bright green, $z$: bright blue; semi‑transparent axis lines
Wires	Thin vertical cylinders intersecting the $x\!y$-plane at $(\pm a, \pm a)$	Metallic gray; radius small relative to spacing
Current direction	Small arrow loops wrapped around each wire (right‑hand rule)	Same as wire color, arrowheads white
Magnetic field lines	Concentric circles around each wire, lying in the $x\!y$-plane, animated to rotate in the direction given by the right‑hand rule	Semi‑transparent cyan for wires with current +z, magenta for ‑z; line width 1.5 pt
Net magnetic field vectors	Short arrows placed on a grid (e.g., 0.5 a spacing) showing the vector sum $\mathbf{B}$	White arrows with orange tails, length proportional to magnitude (clamped)
Charged particle	Small solid sphere (radius 0.08 a)	Bright yellow
Particle velocity vector	Arrow attached to the sphere, pointing along instantaneous $\mathbf{v}$	Lime green, semi‑transparent tail
Trajectory trace	Fading curve following the particle’s path, stops after 6 s	Yellow‑orange gradient, fading after a few seconds
Optional labels (wire numbers)	Text "1", "2", "3", "4" near each wire	White text on a dark semi‑transparent rectangle background

4. Animation Timing & Transitions (total ≈26 s)

Time (s)	Action
0.0 – 3.0	Fade‑in the 3‑D coordinate axes. Camera starts with a top‑down view (looking down the $-z$ direction) at a distance of ~6 a, then slowly tilts to a 30° elevation to reveal depth.
3.0 – 6.0	Appear the four wires at the square corners ($\pm a, \pm a$). Simultaneously display current direction arrows around each wire (clockwise for $-z$ current, counter‑clockwise for $+z$ current).
6.0 – 12.0	Animate magnetic field circles for each wire: circles expand outward while rotating, giving the impression of a steady azimuthal field. Fade in a sparse grid of net‑field arrows that smoothly transition from the individual contributions.
12.0 – 14.0	Introduce the particle at the origin ($0,0,0$) with a brief “pop” effect. Show an initial velocity arrow pointing along the positive $y$-axis.
14.0 – 20.0	Play the pre‑computed trajectory for 6 s: the particle moves, leaving a fading trace. The velocity arrow updates each frame; a brief instantaneous force arrow (perpendicular to $\mathbf{v}$ and $\mathbf{B}$) appears at the particle’s location to illustrate $\mathbf{F}=q\mathbf{v}\times\mathbf{B}$.
20.0 – 23.0	Pause the motion; zoom slightly to focus on a representative segment of the path. Overlay a semi‑transparent box containing the Lorentz‑force formula for a few seconds.
23.0 – 26.0	Fade out all auxiliary graphics, return to the static view of axes, wires, and net‑field arrows. End with a brief title overlay "Magnetic Trap & Charged‑Particle Motion" (text on opaque background).

5. Camera Angles & Perspectives

• Initial view: Orthographic top‑down (looking down $-z$).
• During field‑line animation: Slowly rotate around the $z$-axis (≈15°) while maintaining a 30° elevation to give a 3‑D feel.
• Particle motion segment: Keep the same 30° elevation; optionally dolly in slightly (zoom) when the particle reaches the outer region of the square to keep it in view.
• Final pause: Slightly pull back to show the whole configuration.

6. Additional Details & Assumptions

• Choose a convenient numerical value for $a$ (e.g., $a = 2$ units) to keep the scene well‑scaled.
• All currents have equal magnitude $I$; the sign (direction) is encoded by the rotation of the magnetic circles.
• The magnetic field magnitude is visualised proportionally; absolute scaling is not critical for the conceptual demonstration.
• The particle’s mass $m$ and charge $q$ are set to 1 (in arbitrary units) for simplicity; the trajectory is computed accordingly and truncated after 6 s.
• No textual labels are used except for the optional wire numbers and the final title; any text appears with an opaque dark background for readability.
• The entire animation fits within a single Manim Scene class.