23.3.3 An Infinite Line of Charge

What if the rod or wire becomes very long, becoming a line of charge, while the linear charge density $λ$ remains constant? To answer this question, we can rewrite the expression for $E_{rod}$ by factoring ${(L / 2)}^{2}$ out of the denominator:

E_{rod} = \frac{1}{4 π ϵ_{0}} \frac{| Q |}{r \cdot L / 2} \frac{1}{\sqrt{1 + 4 r^{2} / L^{2}}} = \frac{1}{4 π ϵ_{0}} \frac{2 | λ |}{r} \frac{1}{\sqrt{1 + 4 r^{2} / L^{2}}}

where $| λ | = | Q | / L$ is the magnitude of the linear charge density. If we now let $L \to \infty$ , the last term becomes simply 1 and we’re left with

{\vec{E}}_{line} = (\frac{1}{4 π ϵ_{0}} \frac{2 | λ |}{r}, {\begin{array}{l} away from line if charge + \\ toward line if charge - \end{array}) (line of charge)

(23.17)

where we’ve now included the field’s direction. FIGURE 23.13 shows the electric field vectors of an infinite line of positive charge. The vectors would point inward for a negative line of charge.

The figure illustrates the electric field vectors on an infinitely long rod.

FIGURE 23.13 Full Alternative Text

NOTE Unlike a point charge, for which the field decreases as $1 / r^{2}$ , the field of an infinitely long charged wire decreases more slowly—as only $1 / r$ .

The infinite line of charge is the second of our important electric field models. Although no real wire is infinitely long, the field of a realistic finite-length wire is well approximated by Equation 23.17 except at points near the end of the wire.