Counting the lines: the field from density

The five rules of field lines become quantitative with a simple choice of convention: we establish that each field line “represents” the same amount of charge. Once this value is fixed, the field strength can be found by counting the lines, without having to solve any integral. This is the leap that turns an intuitive picture into a calculation tool.

Principle — Calculating $|\vec{E}|$ from the line density

Choose at a point a small surface $S_\perp$ perpendicular to the field lines. Let $N$ be the equivalent charge (in coulombs) of the lines that cross it. Then: $\ev{|\vec{E}| = 4\pi k\,\frac{N}{S_\perp}}$

The heart of the idea is that the field strength coincides with the line density: the more crowded the lines are as they cross a given surface, the stronger the field is at that point. The surface must be perpendicular to the lines precisely so that only the lines that truly “pierce” it are counted, not the ones that graze it edge-on.

Key formula

Field lines: $|\vec{E}| = 4\pi k\,\frac{N}{S_\perp}$ where $N$ is the equivalent charge of the lines crossing $S_\perp$, a surface perpendicular to the field.

Remember: $4\pi k = 1/\varepsilon_0$.

This formula is the operational method of Faraday’s paradigm: there is no need to know either the charge $Q$ of the sources or the distance $r$ — it is enough to count the lines crossing a perpendicular surface. The constant $4\pi k$, which we can also write as $1/\varepsilon_0$, bridges the geometry of the lines with the physical units of the field.

Example — Verification on a point charge

From a charge $Q$ there emerge $N = Q$ coulombs of lines (by rule 5). At distance $r$, these lines spread uniformly over a sphere of area $S_\perp = 4\pi r^2$. Hence: $|\vec{E}| = 4\pi k\,\frac{Q}{4\pi r^2} = k\,\frac{Q}{r^2}$ We recover exactly the field of the point charge: the counting formula is consistent with Coulomb’s law.

This neat consistency is no coincidence: spherical symmetry ensures that all lines cross the sphere perpendicularly and uniformly, so the density $N/S_\perp$ is the same at every point on the surface and simplifies cleanly to the expected $1/r^2$.

The perpendicular-surface method: we choose $S_\perp$ orthogonal to the field lines (in green) and count how many lines pierce it ($N = 7$ in this example). The field is $|\vec{E}| = 4\pi k\, N/S_\perp$. Where the lines are denser, $N/S_\perp$ is larger, so the field is stronger.

Collegamenti

Argomenti: Campo elettrico e potenziale · Elettrostatica Concetti: Campo elettrico Competenze: Principio di sovrapposizione

Esercizi collegati: Shield and charge in a hollow conductor · Reading the field lines · Dipole and test charge