Electric Field Lines

Faraday's brilliant invention: a way to see an invisible field. Field lines turn abstract vectors into a picture you can sketch with chalk.

Concept

An electric field line is a curve such that the tangent at every point gives the direction of $\vec E$ there.

Properties:

1. Field lines start on positive charges and end on negative charges (or extend to infinity).
2. Field lines never cross. A crossing would mean two different field directions at one point, impossible for a vector field.
3. The density of lines (lines per unit cross-sectional area) is proportional to the magnitude of the field. Close lines = strong field, sparse lines = weak field.
4. They are continuous curves without breaks — no isolated start/end except at charges.
5. Lines are normal to the surface of a conductor in electrostatic equilibrium.
6. They do not form closed loops (electrostatic fields are conservative; magnetic field lines, by contrast, do form loops).

Derivation

Why can't field lines cross? Suppose two lines crossed at $P$. Each line's tangent at $P$ gives a direction of $\vec E$. Two different tangents would mean two different field directions — but the field is single-valued. Contradiction.

Why is line density proportional to $|\vec E|$? Faraday's convention assigns $N$ lines per coulomb of source charge. If you draw a sphere of radius $r$ around a $+q$ source, the surface area is $4\pi r^2$, and $Nq$ lines pierce it. So lines per unit area $= Nq/(4\pi r^2) \propto 1/r^2$, the same scaling as $E$. The convention exactly matches the inverse-square law.

Drawing Rules — Practical

• Single positive charge: straight lines radiating outward like spokes.
• Single negative charge: straight lines pointing inward.
• Dipole (+ and -): lines start on $+$, curve through space, and end on $-$. The line through the midpoint along the perpendicular bisector is perpendicular to the dipole axis.
• Two equal positives: lines push each other apart; between the charges there is a neutral point where $\vec E = 0$.

Worked Example

Sketch field lines for a charge $+2q$ near a charge $-q$.

Twice as many lines start on $+2q$ as end on $-q$. So $1/3$ of the lines from $+2q$ terminate on $-q$, while $2/3$ extend to infinity. The neutral point lies on the line joining them, on the side of the smaller charge ($-q$), at the distance $d$ from $-q$ satisfying $\frac{2q}{(D+d)^2} = \frac{q}{d^2},$ giving $d = D/(\sqrt 2 - 1) \approx 2.41\,D$ where $D$ is the inter-charge distance.

Common Confusions

• Field lines are not paths of moving charges. A free charge follows the field tangentially only at the instant of release; thereafter it has inertia and curves.
• Lines do not show field strength by themselves — only their density does.
• Inside a conductor in equilibrium, there are no field lines. The interior field is zero.
• Magnetic field lines can be closed loops; electric (electrostatic) field lines cannot.

Key Takeaways

• Lines start on $+$ charges, end on $-$ charges or at infinity.
• Tangent direction = direction of $\vec E$; density $\propto |\vec E|$.
• Lines never intersect.
• Lines meet conductors at right angles.
• Electrostatic field lines never form closed loops.