Electric Field Lines

A vector field is a monstrous object to communicate: three numbers at every one of infinitely many points. Yet Michael Faraday — who could not have written down a vector if you had paid him — invented a picture that compresses all of it into something you can sketch on an envelope and reason with. Thread curves through space so that each curve runs along the local field direction, launch them in fixed quota from every charge, and two miracles follow. The tangent of a line tells you the field's direction at a glance. And the crowding of the lines tells you its strength — not roughly, but exactly, because a fixed number of lines fanning over spheres of growing area $4\pi r^2$ reproduces the inverse-square law with no algebra at all. That second miracle is why this article is more than a drawing lesson. Field lines are conserved in empty space: they begin on positive charge, end on negative charge, and in between they neither appear nor vanish. Chase that single statement and you will find it is Gauss's law, wearing picture clothes — count the net lines poking out of any closed surface and you have measured the charge inside. The next three articles turn this counting into flux integrals and the crown jewel of electrostatics; this one builds the pictures those integrals formalize, and builds them carefully, because a field-line diagram read correctly is a quantitative instrument, and read carelessly is a factory of myths. You will also learn what the pictures cannot say — why lines never cross, why they are not particle trajectories, and why the density rule silently fails in flat drawings.

Key takeaways

A field line is an integral curve of $\mathbf{E}$ : at every point its tangent is the field direction, so the pattern maps direction everywhere at once.
Lines begin on positive charges, end on negative charges, and are conserved in empty space; the number launched is proportional to the charge, $N\propto q$ .
Density carries the magnitude: lines per unit perpendicular area is proportional to $|\mathbf{E}|$ — and conservation over spheres of area $4\pi r^2$ then forces the Coulomb $1/r^2$ law.
Counting net lines through a surface is measuring flux; counting through a closed surface reads off the enclosed charge — Gauss's law in picture form.
Lines never cross or touch (the field has one direction per point), and static lines never close into loops (a loop would do net work on a round trip).
They are not trajectories: the field sets acceleration, not velocity — a moving charge leaves its field line immediately unless the line is straight.
The density rule is three-dimensional; flat drawings dilute lines as $1/r$ and quietly lie about magnitudes.
Key idea: field lines are tangent curves whose 3-D areal density equals field strength, so counting lines through surfaces is doing flux integrals by eye.

The core insight

Field lines are the integral curves of $\mathbf{E}$ : tangent to the field direction, with areal density proportional to field strength, so that counting lines through a surface measures electric flux and embodies Gauss's law.

Where this lives today

Field-line tracing is production technology across physics. Solar physicists reconstruct coronal magnetic field lines from surface magnetograms to predict which loops will snap and hurl coronal mass ejections at Earth — the reconnection of field lines is the space-weather event. Tokamak designers live and die by line topology: fusion confinement means keeping magnetic field lines wrapped on nested toroidal surfaces, and the "separatrix" and "X-point" vocabulary of divertor design is exactly the null-point line-crossing geometry of this article. In the electrostatics of engineering, every high-voltage insulator and switchgear design is reviewed as a field-line-and-equipotential plot, because designers read line crowding as breakdown risk at a glance — software computes $\mathbf{E}$ numerically, then integrates equation (1) to draw the lines, precisely because the human eye extracts gradients from line spacing faster than from color maps. And in the visualization community, the streamline algorithms that render airflow over simulated aircraft are the same integral-curve machinery, seeded and spaced by rules descended directly from "density proportional to strength."

The physical picture

Picture the electric field as the velocity field of an imaginary incompressible fluid. Positive charges are springs where the fluid gushes out; negative charges are drains where it disappears; empty space is pipework — the fluid streams through without being created or destroyed. Field lines are the streamlines of this flow. Everything on this page follows from the incompressibility: lines start only at springs, end only at drains, and in between their number through any tube of flow is fixed. Where the tube narrows, the lines crowd and the flow is fast — strong field; where it widens, they thin — weak field.

Figure

The grammar of field lines — out of positive, into negative, tangent everywhere, denser where stronger

Two counting habits turn the picture into an instrument. First, the launch quota: draw lines in proportion to charge — eight from $+q$ means sixteen from $+2q$ , and a $-q$ nearby must swallow exactly eight. Second, the window count: hold up an imaginary window and count lines stabbing through it. Face-on, the count is maximal; edge-on, zero — the count depends on orientation exactly as the dot product $\mathbf{E}\cdot\mathrm{d}\mathbf{a}$ does, which is the flux integral of the next article being performed by eye. Close the window into a box and the net count (out minus in) no longer cares about the box's shape or size at all: lines are conserved in empty space, so the net outflow is fixed by the springs and drains inside. That shape-indifference, which you can see in any diagram, is Gauss's law.

Figure

Why crowding equals strength — a fixed quota of lines dilutes over growing spheres, and the dilution is exactly one over r squared

The rules, and why each one is forced

A field line is defined by a differential equation: follow the field's direction, step by step. Parametrized by arc length $s$ ,

\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s} = \frac{\mathbf{E}(\mathbf{r})}{|\mathbf{E}(\mathbf{r})|} \tag{1}

where: the right side is the local unit vector of the field, so the curve threads through space always tangent to $\mathbf{E}$ . In components, the same statement reads $\mathrm{d}x/E_x = \mathrm{d}y/E_y = \mathrm{d}z/E_z$ — the form you integrate when you want a line's shape analytically, and the recipe every plotting routine implements numerically (take a small step along $\hat{\mathbf{E}}$ , re-evaluate, repeat). Equation (1) is also the source of the no-crossing rule: at any point where $\mathbf{E}\neq\mathbf{0}$ the field has exactly one direction, so exactly one line passes through. Two crossing lines would demand two directions at the crossing — forbidden (at a point where $\mathbf{E}=\mathbf{0}$ the direction is undefined, and there — only there — separatrix lines may meet.) The density rule is a convention with teeth: draw (or imagine) lines launched isotropically from each charge with quota $N\propto q$ , and define the drawn field strength as lines per unit perpendicular area. For a single point charge the $N$ lines cross every concentric sphere, so the density is $N/4\pi r^2 \propto 1/r^2$ — precisely Coulomb's law, derived by counting (the first derivation below sharpens this into an argument, not a coincidence). Conservation then upgrades window counts into physics:

\Phi_E = \int_S\mathbf{E}\cdot\mathrm{d}\mathbf{a} \;\propto\; \text{net number of lines through } S \tag{2}

where: "net" means outward-crossing minus inward-crossing, and the proportionality constant is just the flux each drawn line represents ( $q/\varepsilon_0 N$ for quota $N$ on charge $q$ ). Equation (2) is the bridge the next article will cross carefully; here it earns its keep as an instant-read tool — worked example 1 calibrates a real diagram with it. Finally, statics forbids closed lines. A field line that closed into a loop would push a positive test charge all the way around with the field at its back, delivering net work every lap — a perpetual-motion machine. Electrostatic fields conserve energy (their curl vanishes, as the end of this chapter proves), so no static line ever bites its own tail. Every line either runs from a positive charge to a negative charge or escapes to infinity. File the loophole for later: when magnetic fields change in time, curling electric fields with genuinely closed lines appear — that is Faraday induction, half a track away.

rule	statement	forced by
tangency	line direction = field direction	definition, eq. (1)
origin & end	begin on $+$ , end on $-$ , or run to infinity	conservation in empty space
quota	$N \propto q$	makes counting linear, matches superposition
density	lines / perpendicular area $\propto \lvert\mathbf{E}\rvert$	consistency of quota + conservation
no crossing	one line per point ( $\mathbf{E}\neq0$ )	single-valued field direction
no static loops	no closed lines in electrostatics	energy conservation ( $\nabla\times\mathbf{E}=\mathbf{0}$ )
conductors	lines meet conductor surfaces at right angles	tangential field would drive surface currents

Derivation ACoulomb's inverse square from conservation of lines

We run Faraday's picture in reverse: assume only that a point charge emits a fixed quota of field lines isotropically and that lines are neither created nor destroyed in empty space, and show these two assumptions force the field to fall as $1/r^2$ . The strategy is pure bookkeeping over nested spheres. The point of the exercise is to expose what the inverse square is: not an arbitrary experimental exponent but the unique falloff compatible with conservation in three-dimensional space — the same dilution-over-spheres geometry that Gauss's law will industrialize. Watch the dimension-counting in the last step; it is the punchline.

Step 1. Launch the quota. Surround a point charge $q$ with a sphere of radius $r$ . By assumption, $N$ lines (with $N\propto q$ ) cross this sphere, launched isotropically so they cross uniformly over its surface. Why isotropic? A lone point charge defines no preferred direction — any anisotropy in the launch pattern would have nothing in the physics to cause it.

Step 2. Invoke conservation. The same $N$ lines cross the sphere of radius $2r$ , and $3r$ , and every larger sphere: in the empty space between spheres no line begins or ends. The count through concentric spheres is an invariant of the charge alone.

Step 3. Convert count to density. The density of line-crossings on the sphere of radius $r$ is $N/4\pi r^2$ . By the density rule this is the field strength up to the fixed flux-per-line constant:

E(r) \;\propto\; \frac{N}{4\pi r^2} \;\propto\; \frac{q}{r^2}

Why per perpendicular area? Lines cross a concentric sphere at normal incidence, so no obliquity factor intrudes — the clean case. Tilted windows pick up the $\cos\theta$ of the flux dot product.

Step 4. Read the punchline dimensionally. The exponent 2 is the dimension of a sphere's surface in three-dimensional space. In a world of $d$ spatial dimensions the same conservation argument gives $E\propto 1/r^{d-1}$ — and precisely this reasoning is used in reverse by modern experiments: sub-millimetre tests of the inverse-square law (for gravity) and the Cavendish-type null tests (for electricity) are checks that space has no hidden extra dimensions and the photon no mass at those scales. The line picture is not a cartoon of Coulomb's law; the two are the same statement about geometry. The general- $d$ scaling follows the solid-angle generalization $\Omega_{d-1}r^{d-1}$ ; for $d=3$ this gives $E = q/4\pi\varepsilon_0 r^2$ exactly, fixing the flux carried by each drawn line as $q/\varepsilon_0N$ .

Go deeper · Derivation B · 2 minReading an unequal pair quantitatively — which lines escape, and at what angle

Derivation BReading an unequal pair quantitatively — which lines escape, and at what angle

Field-line diagrams of unequal charges hide a sharp quantitative result, and extracting it teaches the counting method better than any drill. The setup: charges $+2q$ and $-q$ sit a finite distance apart. Since line quota is proportional to charge, only half the lines leaving $+2q$ can terminate on $-q$ ; the rest escape to infinity, where the system — net charge $+q$ — must look like a single point charge with a radially uniform fan. The question with a crisp answer: which lines escape? We find the critical launch angle at $+2q$ separating captured lines from escaping ones, using nothing but solid-angle bookkeeping. This is the skill of turning a sketch into numbers — the whole point of Faraday's instrument.

Step 1. Count the captives. Quotas: $2N$ lines leave $+2q$ ; $N$ lines must land on $-q$ (its quota, with all of them arriving since no other negative charge exists). So exactly $N$ of the $2N$ — one half — terminate; the other half escape to infinity.

Step 2. Argue contiguity. Lines cannot cross, so the captured bundle and the escaping bundle cannot interleave: captured lines launch from the hemisphere-side of $+2q$ facing $-q$ , escaping lines from the far side, separated by one critical launch cone at polar angle $\theta_c$ (measured at $+2q$ from the axis pointing toward $-q$ ). Why does no-crossing enforce this? If an escaping line launched inside the captured cone, it would have to thread between captured lines that start on either side of it and converge to $-q$ — forcing a crossing somewhere. Topology does the sorting.

Step 3. Convert the fraction to an angle with solid angles. Isotropic launch means the fraction of lines inside a cone of half-angle $\theta$ is the fractional solid angle

f(\theta) = \frac{2\pi(1-\cos\theta)}{4\pi} = \frac{1-\cos\theta}{2}

Setting the captured fraction to one half: $\tfrac{1-\cos\theta_c}{2} = \tfrac12$ , so $\cos\theta_c = 0$ and

\theta_c = 90°

Exactly the hemisphere facing $-q$ is captured — a clean, checkable prediction visible in every correctly drawn diagram of this pair.

Step 4. Generalize the formula. For $+Q$ and $-q'$ (with $q' < Q$ ), the captured fraction is $q'/Q$ , so $\cos\theta_c = 1-2q'/Q$ . Worked example 2 runs the numbers for a $+3q/-q$ pair. Far from the pair, the $Q-q'$ escaping lines fan out isotropically — the far field of the net charge, another counting statement you can verify on the sketch. The fraction formula integrates the solid-angle element $2\pi\sin\theta\,\mathrm{d}\theta/4\pi$ correctly; the limits confirm the result — $q'\to Q$ gives $\theta_c\to180°$ (dipole: all lines captured), and $q'\to0$ gives $\theta_c\to0$ (no negative charge: none captured).

Figure

The unequal pair, read quantitatively — the hemisphere of +2q facing −q is captured; the far half escapes to a net-charge fan

Limiting cases

case	pattern	meaning
far from any bounded distribution	uniform radial fan of $N_{\text{net}}$ lines	monopole limit: only net charge visible from afar
equal and opposite pair (dipole)	every line from $+$ lands on $-$ ; none escape	zero net charge — no far radial fan; the residual pattern is the dipole field
two equal positive charges	lines repel; a null point midway with separatrix lines meeting there	crossings are legal only where $\mathbf{E}=\mathbf{0}$
statics	no closed lines anywhere	curl-free field; closed loops would violate energy conservation
time-varying magnetic field	closed electric-field loops appear	Faraday induction — the static rule's advertised loophole
conductor surface	lines arrive perpendicular, none inside	tangential field would move surface charge until gone

Common trapField lines are the paths a released charge follows — right?

Wrong, and the error is subtle enough to survive into graduate school. The field line's tangent gives the direction of the force, hence of the acceleration — but a trajectory follows the velocity, and velocity remembers history. A charge released from rest does start off along its field line; the moment the line curves, inertia carries the charge off it, like a car whose steering follows the road's curvature too late. Concretely: near a dipole, field lines arc from $+$ to $-$ , but a charge released from rest performs a complicated dive that leaves its starting line almost immediately. The two coincide only when lines are straight (uniform field, or the axis of a symmetric system) or in the fictional limit of fully overdamped motion (a charged dust grain in honey follows lines beautifully). Related myths in the same family: lines are not physical filaments (nothing material occupies them), the number drawn is a bookkeeping choice (only ratios and topology are meaningful), and 2-D drawings understate the crowding falloff ( $1/r$ on paper vs the true $1/r^2$ in space).

Worked examples

1 · Calibrating a diagram (confidence builder)

A textbook diagram represents a $+6.0$ nC point charge by 24 field lines. (a) What flux does each drawn line carry? (b) What line density corresponds to the actual field at $r = 0.50$ m — and is the diagram self-consistent?

Step 1. Total flux out of any closed surface around the charge (Gauss's law, previewed by line conservation): $\Phi = q/\varepsilon_0 = (6.0\times10^{-9})/(8.854\times10^{-12}) = 678\ \mathrm{V\,m}$ . Spread over 24 lines: each line carries $28.2\ \mathrm{V\,m}$ of flux.

Step 2. Line density on the sphere $r = 0.50$ m: $24$ lines through area $4\pi(0.50)^2 = 3.14\ \mathrm{m^2}$ gives $7.64\ \mathrm{lines/m^2}$ .

Step 3. Convert density back to field: $E = (7.64\ \mathrm{lines/m^2})\times(28.2\ \mathrm{V\,m/line}) = 215\ \mathrm{V/m}$ .

Step 4. Cross-check with Coulomb: $E = k_e q/r^2 = (8.99\times10^9)(6.0\times10^{-9})/(0.25) = 216\ \mathrm{N/C}$ . ✓ The picture and the formula agree to rounding — the density rule is not decoration; it is a unit system.

Answer: 28.2 V·m per line; density $7.6\ \mathrm{lines/m^2}$ at 0.5 m reproduces $E \approx 216$ N/C exactly — a correctly drawn diagram is quantitatively calibrated.

2 · The escape cone of a +3q/−q pair (exam level)

Charges $+3q$ and $-q$ are separated by a finite distance. Find (a) the fraction of lines from $+3q$ that terminate on $-q$ , (b) the critical launch angle at $+3q$ , and (c) the far-field appearance.

Step 1. Quota bookkeeping: $3N$ lines leave $+3q$ ; the $-q$ swallows its quota $N$ . Captured fraction: $N/3N = 1/3$ .

Step 2. Critical angle from the solid-angle formula of Derivation B: the captured bundle fills the cone (about the axis toward $-q$ ) with $f = (1-\cos\theta_c)/2 = 1/3$ , so $\cos\theta_c = 1-\tfrac23 = \tfrac13$ and $\theta_c = \arccos(1/3) = 70.5°$ .

Step 3. Interpret: lines launched within $70.5°$ of the $-q$ direction are captured; the remaining $2/3$ of the sphere's launch directions feed escapers. Note $\theta_c < 90°$ : the weaker the negative charge relative to the positive, the tighter the capture cone hugs the axis — check the limits, $\cos\theta_c = 1-2q'/Q$ gives $\theta_c\to0$ as $q'\to0$ (nothing captured) and $180°$ for the dipole $q'=Q$ (everything captured). ✓

Step 4. Far field: $3N - N = 2N$ lines escape, and at distances large compared with the separation they fan out isotropically — the pattern of a single point charge $+2q$ . A quick numerical audit: at $r = 20$ separations, the pair's field differs from the point- $2q$ field by terms of order (separation)/ $r = 5\%$ — the dipole correction — so the isotropic fan is already an excellent description.

Answer: one third of the lines are captured, inside a cone of half-angle $70.5°$ about the axis; the escaping $2N$ lines form the far-field fan of the net charge $+2q$ .

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Cambridge and the Cavendish Laboratory

Maxwell's first great electromagnetic paper, written at 24, was an act of translation: "On Faraday's Lines of Force" (1855–56) took the sketches of a self-taught experimentalist — then widely patronized as a brilliant man's private mnemonics — and showed they were exact mathematics in disguise. Maxwell modeled the lines as streamlines of an incompressible fluid, made "density equals strength" a theorem about flux tubes, and demonstrated that Faraday's pictures encoded the same content as the continental action-at-a-distance formulas, while generalizing more gracefully. Faraday, then in his sixties, wrote to him in gratitude and mild astonishment: "I was at first almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well." Two decades of that translation program later, the Treatise delivered the field equations that bear Maxwell's name — but he always insisted the geometry of lines came first, and taught electromagnetic theory lines-first until his death, at 48, as the first director of the Cavendish Laboratory.

History

1837

— Faraday, explaining dielectrics, begins treating "lines of inductive force" as physically real structures in space

1852

— Faraday's "On the Physical Character of Lines of Magnetic Force" argues lines are more than bookkeeping — the field concept in embryo

1855

— Maxwell's "On Faraday's Lines of Force" gives the picture exact fluid-flow mathematics: density = strength becomes a flux-tube theorem

1873

— Maxwell's Treatise organizes electromagnetism around fields and their geometry; line diagrams become standard scientific graphics

1931

— Dirac's magnetic-monopole analysis makes line topology a tool of quantum field theory — where lines end became a deep question again

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §2.2.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §1.7 · Jackson, Classical Electrodynamics, 3rd ed., §1.5 · Zangwill, Modern Electrodynamics, §3.3 · Feynman Lectures on Physics, Vol. II, Ch. 4 Key relations used throughout: point-charge field $E=q/(4\pi\varepsilon_0 r^2)$ ; closed-surface flux $\oint\mathbf{E}\cdot\mathrm{d}\mathbf{a}=q/\varepsilon_0$ ; no-loop condition $\nabla\times\mathbf{E}=0$ (via Stokes' theorem); $\varepsilon_0=8.854\times10^{-12}\,\mathrm{F/m}$ . Worked numbers: 678 V·m total flux, 28.2 V·m per line, density $7.64\ \mathrm{lines/m^2}$ matching $E=216$ N/C; capture fraction $1/3$ and $\theta_c=70.5°$ from $\cos\theta_c=1-2q'/Q$ .

Builds on: The Electric Field · Coulomb's Law · Superposition Principle for Electric Fields Leads to: Electric Flux · Gauss's Law in Integral Form · The Curl of the Electrostatic Field · Electrostatic Potential

Are field lines physically real, or just a drawing aid?

The field is real (it carries energy and momentum); the lines are a representation — space is not actually threaded by filaments, and between any two drawn lines the field is just as present as on them. But the representation is faithful in a strong sense: the topology (where lines start, end, and how they connect) and the ratios (quota per charge, density ratios) encode exact physics, which is why arguments made with lines — like both derivations on this page — produce correct equations. Faraday believed the lines were physically real; Maxwell showed it doesn't matter, because the mathematics is identical either way. Modern physics sides with Maxwell on ontology and Faraday on usefulness.

How many lines am I supposed to draw? Textbook figures all differ.

The absolute number is a free choice — a resolution setting, like pixels. What must be respected is proportionality: choose "lines per coulomb" once, then every charge in the same figure gets its proportional quota, and each line implicitly carries flux $q/(\varepsilon_0 N)$ . Worked example 1 shows the calibration explicitly. Comparisons between figures with different conventions are meaningless; comparisons within one correct figure are quantitative. Software makes the same choice when it decides seeding density for streamlines — too few lines miss structure, too many saturate the image, and neither changes the physics.

If lines can't cross, what happens at the point exactly between two equal positive charges?

That midpoint is a field null — $\mathbf{E}=\mathbf{0}$ — and nulls are the licensed exception. The no-crossing theorem rests on the field having a single direction at the point; where the field vanishes, no direction is defined, and special lines called separatrices may meet there, dividing space into cells of distinct line topology (each cell's lines belong to one charge). For two equal positive charges, four separatrix branches meet at the midpoint in an X shape. These X-points are not exotica: in solar and fusion plasmas, magnetic X-points are where field lines reconnect and release energy — entire subfields study exactly the structure your sketch shows at that innocent midpoint.

Do field lines ever start or end in empty space?

Never, in electrostatics — that is the pictorial form of Gauss's law, and it is exact. A line ending in vacuum would mean flux disappearing through nested surfaces with no charge to account for it. Two apparent exceptions dissolve on inspection: lines "ending" at the edge of a figure merely leave the drawing, and lines running to infinity end nowhere at all (their flux is accounted on the sphere at infinity, which is how the far-field fan counts net charge). The rule is also the sharp diagnostic for reading student sketches: any line born or dying in empty space is an error you can circle without knowing anything else about the problem.

Why must field lines meet a conductor's surface at right angles?

In electrostatic equilibrium the conductor's mobile charges have finished moving, which requires zero force along the surface — any tangential field component would still be driving surface currents. So the field at the surface is purely perpendicular, and lines arrive normal to it, ending on the induced surface charge (lines cannot continue inside, where the field is zero). The corollary is the geometry of shielding: conductor surfaces are equipotentials, lines leave them perpendicularly, and crowd toward sharp points — which is why lightning rods are pointed and why high-voltage hardware is rounded. The full story, with the induced charge densities, is the conductors article of the next tier.

In diagrams the lines get farther apart as they leave a charge — but E along the axis between two charges can grow. How does spacing encode that?

Spacing encodes strength only through perpendicular density in three dimensions, and reading it requires imagining the flux tube — the bundle of lines around your location — rather than the two lines nearest you on paper. Between unlike charges the tube converges toward the negative charge, so the density and field rise even though a flat drawing may not make it obvious. The 2-D projection also dilutes crowding as $1/r$ instead of $1/r^2$ , so absolute falloffs read from paper are systematically wrong. Sound practice: use drawings for topology, direction, and comparisons along one flux tube; use formulas for magnitudes. Both derivations on this page respect exactly that division of labor.

Do genuinely closed electric field lines ever exist?

Yes — the static prohibition has a famous expiry date. The no-loop rule follows from $\nabla\times\mathbf{E}=\mathbf{0}$ , which holds only for fields sourced by stationary charges. When a magnetic field changes in time, Faraday's law $\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$ generates electric fields whose lines close on themselves, encircling the changing flux — no charges at either end, because there are no ends. Every transformer and every induction cooktop runs on such closed-loop electric fields, and the betatron accelerates electrons by letting them surf one. Within electrostatics, though, the rule is airtight, and "could this line pattern do net work on a loop?" remains one of the fastest sanity checks on any sketch.

Electric Field Lines

Where this lives today

The physical picture

The rules, and why each one is forced

Limiting cases

Worked examples

1 · Calibrating a diagram (confidence builder)

2 · The escape cone of a +3q/−q pair (exam level)

Going further

Before this

Where this leads