Field of infinite line charge

Intermediate

Falls as 1/r.

Key relationE = λ/2πε₀r

By Gauss with a cylinder: E = λ/(2πε₀r), pointing radially, falling as 1/r.

Worked example

λ = 1 nC/m at 0.5 m: 36 N/C.

Common mistake

Valid for lines much longer than the distance.

In more detail

An infinite line of charge with linear density λ produces a field E = λ/(2πε0 r), falling off as 1/r. The field points radially away from the line.

Modelling long charged wires and cables.
Designing electrostatic lenses and particle beams.
Engineers designing coaxial cables and wire insulators estimate the field near a long, thin charged wire.

Why does it fall as 1/r?

Because the field spreads over a cylinder whose area grows with r.

Is an infinite line realistic?

As an approximation, it works for lines much longer than the distance of interest.

Why does the field of a line fall off as one over r?

The charge spreads along the infinite line, so the field decreases more slowly than for a point charge.