Electric field lines

The field E already exists at every point; a line is a way of walking through it. Start anywhere, read E, take a short step along it, read E again at the new point, and repeat. The curve that comes out satisfies dy/dx = E_y/Eₓ everywhere along itself, and its arrowhead points the way E points, which is the direction of the force on a positive test charge and the opposite of the force on a negative one. Two things follow at once. Through a given point there is exactly one such curve, because there is exactly one field vector there — which is what makes the whole picture well defined. And the curve is not a thing: start from a neighbouring point and you get a neighbouring line, and the field in the gap between two drawn lines is every bit as real as the field on them.

Fix a scale: n lines per coulomb, so a charge q carries N = n|q| of them. Around an isolated charge every one of those lines pierces every sphere drawn about it, so the count per unit area at radius r is n|q|/(4πr²) while the field is |q|/(4πε₀r²). Both carry the same 1/r², so eliminating q leaves N/A⊥ = n ε₀ E: the crowding is the field strength, in a fixed ratio. Take q = 3.0 nC drawn at 6 lines per nanocoulomb, so N = 18. At r = 0.15 m the sphere has area 0.2827 m², giving 63.7 lines m⁻², and E = 1.20 × 10³ N C⁻¹. Move out to r = 0.30 m: the area is four times larger, the density falls to 15.9 lines m⁻², and E falls to 300 N C⁻¹. The match is not luck. Only an inverse-square law lets unbroken lines carry the field outward without any of them having to stop in empty space.

Because the count belongs to the charge, one n must serve every charge in a diagram, and then the counts are in proportion to the charges. At 4 lines per nanocoulomb, 20 lines leave a +5.0 nC charge and 12 arrive at a −3.0 nC charge nearby. Twelve of the twenty find the negative charge; the remaining eight have nowhere to land and run out of the picture. That leftover is set by the net charge alone, N_∞ = n|q₁ + q₂| = 4 × 2.0 = 8, which is why a pair seen from far away looks like a single charge of +2.0 nC. The same bookkeeping works for any closed surface you care to draw: lines out minus lines in equals n q_enc, since nothing begins or ends in empty space. That statement becomes Gauss's law once the counting is replaced by an integral.

At a crossing, a curve would have two tangents, so E would have two directions at the same point. Superposition forbids it: the contributions from every charge add to one vector, with one magnitude and one direction. Note what is being added — the field vectors, not the pictures. You cannot lay one charge's line pattern over another's and call the overlay a field map. The single exception is a point where the vector sum is zero, because a zero vector has no direction. For +4q and +q a distance d apart, that point sits on the line between them at 2d/3 from the larger charge, since 4/x² = 1/(d − x)² gives 2(d − x) = x. Lines approaching it swerve aside, and the boundary lines that divide the pattern — one charge's territory from the other's — do meet there, at a definite angle. It is the one place in the picture where the no-crossing rule has nothing to say.

In electrostatic equilibrium the field inside conducting material is zero, so no line is drawn in the metal at all. At the surface the parallel component must vanish too: any Eₜ would push the free surface charge sideways, and the charge keeps moving until it has arranged itself so that Eₜ = 0. What survives is perpendicular, so external lines strike a conductor at right angles and terminate on the surface charge they induce. The surface count follows the charge: n|σ| lines leave or land per square metre. Set that against N/A⊥ = n ε₀ E, which applies just outside where the lines run perpendicular, and n cancels to leave E = σ/ε₀ immediately outside any conductor. On a pear-shaped conductor, σ is largest at the sharp end, so that is where the lines crowd and where the outside field is strongest.

A picture will honestly give you signs from the arrowheads, a charge ratio from the counts, the direction of E at a point from the tangent, and a ranking of strengths from the crowding. It will not give you a number for E, because n was your choice. It also lies about one thing quietly. A printed pattern is a plane slice through a three-dimensional one, and lines drawn radially on paper thin out as 1/r along a circle rather than 1/r², so the page always understates how fast the field really dies away. Use the drawing for structure and ranking, and the algebra for values. In the laboratory the order is reversed: a probe maps the equipotential contours, and the field lines are then drawn crossing them at right angles, pointing towards lower potential. Even measured, a field line is inferred and never photographed.