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 $\dfrac{\lambda}{2\pi\varepsilon_0}$.

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).

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.

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:

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:

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.