Field lines: the five rules

Field lines are oriented curves that represent the vector $\vec{E}$ point by point: they are Faraday’s way of drawing the field, making visible a quantity that would otherwise be abstract. Their full meaning is captured in five rules, which allow us to read direction, orientation and strength of the field directly from a diagram.

Principle — The five rules of field lines

1. Origin and arrival. Lines emerge from positive charges and enter negative charges. If the system as a whole is not neutral, some lines arrive from, or escape towards, infinity.
2. Tangency. At every point in space the field line is tangent to the vector $\vec{E}$ at that point.
3. No crossing. Two field lines never cross, because the field has a single well-defined direction at every point.
4. Density and strength. The density of lines (number of lines per unit area perpendicular to them) is proportional to $|\vec{E}|$: where the lines are denser the field is stronger.
5. Number and charge. The number of lines emerging from a charge $Q$ is proportional to $|Q|$. If one charge has twice as many lines as another, it has twice the charge.

These rules are not arbitrary: tangency (rule 2) and no crossing (rule 3) express the fact that $\vec{E}$ is a vector field with a single value at every point; density (rule 4) visually translates the field strength; and proportionality with charge (rule 5) is the key that makes the paradigm quantitative, allowing us to count lines instead of solving integrals.

Electrostatic shielding

A system is said to be shielded if there are no field lines arriving from or escaping to infinity: all lines connect charges internal to the system. The conductor in equilibrium is the most important example (Faraday cage): inside a conducting shell the electric field is always zero, regardless of the external field.

The fundamental configurations

In the figures below we see the field lines for the fundamental configurations: the isolated charge, two equal charges, the dipole and the infinite plane.

Field lines of a positive (left) and negative (right) point charge. The lines have spherical symmetry: they emerge/enter radially and thin out with distance from the charge ($E \propto 1/r^2$).

Field lines for two equal charges $+Q/+Q$ (repulsion). No line connects the two charges: all escape towards infinity. The neutral point (P.N.) at the centre is where the two fields cancel out; the lines avoid it.

Field lines of the electric dipole $+Q/-Q$. The lines emerge from the positive pole and enter the negative one. The lines closest to the axis connect the two poles directly; the outer ones make a wide arc. No line goes to infinity (fully shielded system).

Field lines of an infinite uniformly charged plane (surface density $\sigma > 0$). The lines emerge perpendicular to the plane on both sides, are parallel and equally spaced: the field is uniform and does not depend on the distance from the plane.

Collegamenti

Argomenti: Campo elettrico e potenziale · Elettrostatica Concetti: Campo elettrico · Dipolo elettrico

Esercizi collegati: Dipole and test charge · Worked exercise — electric dipole on the axis · Dipole in a uniform field