field-potential-curves

# Field & Potential Curves (E(r), V(r) vs. distance) Specialized skill for the **curve panel** of electrostatics figures: plotting electric field $E(r)$ and potential $V(r)$ as functions of distance $r$ (or $\rho$) for a charge distribution. This is the right-hand panel that pairs with a model diagram (see the general `physics-model-figures` skill for layout, presets, and conventions — this skill narrows in on the E/V specifics). The general skill covers *any* model+curves figure. Reach for **this** skill when the curves are specifically $E$ and $V$ vs. distance — i.e. when you need the piecewise functions, the dual-y-axis recipe, the ρ₀ reference, or the canonical formula set below. ## When to use - Right panel shows $E(r)$ and/or $V(r)$ vs. distance for any charge geometry: point charge, spherical shell, solid sphere, concentric shells, infinite line charge, cylindrical shell, … - You need the **standard piecewise forms** for any of these geometries. - You need the **dual-y-axis** E/V layout (blue E on left, green V on right). Do NOT use for: pure model diagrams with no curves (use `physics-model-figures` single-panel variant), or non-electrostatics curves. ## The canonical charge geometries All forms use $k = 1/(4\pi\varepsilon_0)$. For line/cylindrical geometries the grouping constant is $\dfrac{\lambda}{2\pi\varepsilon_0}$. | Geometry | $E(r)$ | $V(r)$ | |---|---|---| | **Point charge $q$** | $kq/r^2$ | $kq/r$ | | **Spherical shell (radius $R$, charge $Q$)** | $\begin{cases}0 & r<R \\ kQ/r^2 & r\geq R\end{cases}$ | $\begin{cases}kQ/R & r\leq R \\ kQ/r & r>R\end{cases}$ | | **Solid sphere (radius $R$, charge $Q$)** | $\begin{cases}kQr/R^3 & r<R \\ kQ/r^2 & r\geq R\end{cases}$ | $\begin{cases}\dfrac{kQ}{2R}(3-r^2/R^2) & r\leq R \\ kQ/r & r>R\end{cases}$ | | **Concentric shells ($Q_a$@$a$, $Q_b$@$b$)** | $\begin{cases}0 & r<a \\ kQ_a/r^2 & a\leq r<b \\ k(Q_a+Q_b)/r^2 & r\geq b\end{cases}$ | $\begin{cases}kQ_a/a+kQ_b/b & r<a \\ kQ_a/r+kQ_b/b & a\leq r<b \\ k(Q_a+Q_b)/r & r\geq b\end{cases}$ | | **Infinite line charge $\lambda$** | $\lambda/(2\pi\varepsilon_0\rho)$ | $\dfrac{\lambda}{2\pi\varepsilon_0}\ln(\rho_0/\rho)$ | | **Cylindrical shell (radius $R$, $\lambda$)** | $\begin{cases}0 & \rho<R \\ \lambda/(2\pi\varepsilon_0\rho) & \rho\geq R\end{cases}$ | $\begin{cases}\dfrac{\lambda}{2\pi\varepsilon_0}\ln(\rho_0/R) & \rho<R \\ \dfrac{\lambda}{2\pi\varepsilon_0}\ln(\rho_0/\rho) & \rho\geq R\end{cases}$ | ### Continuity rules (sanity-check your plot) - $E(r)$: **discontinuous** at a **surface** charge (shell, cylindrical shell) — jumps by $\sigma/\varepsilon_0$. **Continuous** for a **volume** charge (solid sphere) and at the inner edge of a shell (no surface there). - $V(r)$: **always continuous** everywhere (potential can't jump). - $E(R^+) \neq E(R^-)$ is *expected* for shells — don't "fix" it by forcing a join. If your plotted E looks continuous at a shell surface, the function is wrong. ### The ρ₀ reference $V$ for line/cylindrical geometries needs a **reference distance** $\rho_0$ where $V(\rho_0)=0$ (potential is only defined up to a constant). Pick $\rho_0$ inside the plotted range and draw a vertical dashed line at it. A common choice: $\rho_0 = 2R$ or $\rho_0$ at the right edge of the plot. ## The dual-y-axis recipe $E$ (N/C) and $V$ (V) have different units and scales → **twin axes**, not a shared one. Single shared axis only when one curve dominates or units match (e.g. `src/point_charge_field.py` puts both on one axis because values coincide at small scales). ```python ax_e = fig.add_subplot(...) ax_v = ax_e.twinx() ax_e.set_ylabel("E (N/C)", color=BLU) ax_e.tick_params(axis="y", labelcolor=BLU) ax_e.spines["left"].set_color(BLU) ax_v.set_ylabel("V (V)", color=GRN) ax_v.tick_params(axis="y", labelcolor=GRN) ax_v.spines["right"].set_color(GRN) ``` Colour convention (used across all repo figures): **E = blue `#1f4e9b`**, **V = green `#2e8b57`**. Stick to it for visual consistency. ### Choosing y-axis ranges $E$ and $V$ ranges differ per geometry — set them per figure, not via a preset: - **Shells/solid sphere**: $E$ peaks at the surface then falls; set $E$ range to comfortably clear $E(R)$. $V$ is positive, peaks at centre; range clears $V(0)$. e.g. solid sphere $V(0)=3kQ/(2R)$ needs a taller V axis than a shell. - **Line/cylindrical**: $E\to\infty$ near $\rho\to0$ — start the x-axis at a small positive value (e.g. $0.1\rho_0$), not 0. $V$ goes **negative** for $\rho>\rho_0$ → the V y-axis must include a negative range (e.g. `set_ylim(-30, 20)`). ### Plotting piecewise functions Build separate arrays per region and `plot` each segment. Don't use a single array with `np.where` — it hides the discontinuities and can draw spurious vertical lines at boundaries. ```python r1 = np.linspace(0, R, 200) # inside r2 = np.linspace(R, 4 * R, 400) # outside ax_e.plot(r1, np.zeros_like(r1), color=BLU, lw=2.8) # E=0 inside ax_e.plot(r2, K * Q / r2**2, color=BLU, lw=2.8) # E=kQ/r² outside ``` ## Required annotations Every E/V curve panel in this repo includes these — see existing scripts for exact code: 1. **Vertical dashed line(s)** at each region boundary ($r=R$, $r=a$, $r=b$, $\rho=\rho_0$) in grey (`#888888`), with a small label. 2. **Marked points** at key values: white-filled circle (`facecolor="white"`, `edgecolor=<curve color>`) + `annotate` with the numeric value. For the central maximum $V(0)$ use a fraction form (`$V_0 = \frac{3kQ}{2R}$`) alongside the number. 3. **Asymptotes**: `axhline(0, ls="--")` with a label. Line-charge figures add $V\to -\infty$. Mark these near the right edge with `ha="right", style="italic"`. 4. **Formula boxes** (upper area of each axis): the **full piecewise expression**, one line per region, stacked with `\n`. See gotcha below for the mathtext limitation. **No numeric constants** (no `k=9e9, Q=...`) — per repo convention. 5. **Legend** (`loc="upper right"` or `"center right"`): one entry per curve, label = `$E(r)$` / `$V(r)$` (function name only, not the formula). 6. **Light grid**: `ax_e.grid(True, color="#dddddd", lw=0.6)`. 7. **Subtitle** via `ax.set_title(...)`, **no** `suptitle`. ### Marked-point label placement (avoid overlaps) When $E$ and $V$ curves cross or sit close (common near $r=R$), their marked labels collide. Offset them on **opposite sides** of the point: ```python ax_e.annotate(f"E = {e:g}", xy=(r, e), xytext=(8, +14), textcoords="offset points", ...) ax_v.annotate(f"V = {v:g}", xy=(r, v), xytext=(8, -14), textcoords="offset points", ...) ``` ## Gotchas (hard-won from 6 figures in this repo) - **`matplotlib` mathtext has NO `\begin{cases}` / `\dfrac` array layout.** Those are LaTeX-only. Stack piecewise lines manually with `\n` and `linespacing`. When a line contains a tall fraction (`\dfrac`), add an extra blank line (`"\n\n"`) and bump `linespacing` to ~2.0, or the fraction collides with the line below. See `src/charged_cylindrical_shell.py` for the workaround. - **mathtext unsupported LaTeX commands** (raise `ParseSyntaxException`): `\!` (negative thin space), `\tfrac`, `\textstyle`, `\substack`. Use `\frac` (not `\tfrac`), drop `\!`, and `\dfrac` *is* OK in mathtext. If a formula throws a parse error, suspect these first. - **Piecewise at a surface → E jumps.** If your shell/cylinder E curve looks smooth across $r=R$, you've silently joined the segments — that's physically wrong. Plot inside and outside as separate arrays. - **Line-charge $V$ goes negative** for $\rho>\rho_0$. Set the V y-axis to include negatives, and draw a $V=0$ reference line. Don't clip it to $\geq 0$. - **`twinx()` colour binding**: the left spine stays the left axis's colour, the right spine the right axis's. Set both explicitly or the axes look uncoloured. - **3D model panel beside an E/V curve panel**: if the left model panel needs 3D (e.g. line charge, where E is perpendicular to the wire in 3D), use `projection="3d"` and the `Arrow3D` subclass (see `src/infinite_line_charge.py`, `src/magnetic_field_loop.py`). The right E/V panel stays 2D. ## Model-panel hints specific to E/V geometries These are the model-panel conventions that recur across E/V figures (the general skill covers the rest): - **2D cross-section** is correct for spherical shell, solid sphere, concentric shells, cylindrical shell — E lies in the plane of the section. - **3D** is needed when E is perpendicular to a line (infinite line charge): the wire runs along $z$, E arrows radiate horizontally. A 2D side-view would misrepresent the field as parallel to the wire. - **Arrow length encodes field magnitude**: radial arrows get shorter with distance ($\propto 1/r^n$). Inside a conductor/shell interior, draw **no** arrows ($E=0$) — optionally mark a few `×` or an "E=0" label. - **Reference circle**: dashed circle at the reference radius, labelled `ρ` or `R`, to anchor the distance variable visually. - **Inset cylinder** (cylindrical shell only): a small 3D cylinder in the corner with an arrow to the main circle clarifies "this is a cross-section". See `src/charged_cylindrical_shell.py` lines ~98–126; the arrow direction is controlled by an `ANGLE` constant there. ## Reference implementations Closest match per geometry — copy as a starting point: | Target geometry | Copy from | |---|---| | Point charge | `src/point_charge_field.py` (single shared axis, no twin) | | Spherical shell | `src/charged_sphere.py` | | Solid sphere | `src/charged_solid_sphere.py` (note V(0)=3kQ/2R peak) | | Concentric shells | `src/concentric_shells.py` (3-piece piecewise) | | Infinite line charge | `src/infinite_line_charge.py` (3D model, V<0 region) | | Cylindrical shell | `src/charged_cylindrical_shell.py` (2D + cylinder inset) | To adapt one: swap the **constants block**, the **piecewise arrays in `draw_curves`**, and the **region boundaries** (vertical lines + labels). The dual-axis scaffolding, annotation style, and formula-box layout stay the same.