Chapter 21, Problem 21.97CP

CALC Two thin rods of length L lie along the x-axis, one between $\text{x}=\frac{1}{2}\text{a}$ and $\text{x}=\frac{1}{2}\text{a + L}$ and the other between $\text{x}=-\frac{1}{2}\text{a}$ and $\text{x}=\frac{1}{2}\text{a }-\text{ L}$. Each rod has positive charge Q distributed uniformly along its length. (a) Calculate the electric field produced by the second rod at points along the positive x-axis. (b) Show that the magnitude of the force that one rod exerts on the other is

$\text{F = }\frac{{\text{Q}}^2}{4\pi {\in }_0{\text{L}}^2}\ln \left[\frac{{(\text{a}+\text{L})}^2}{\text{a}{(\text{a}+2\text{L})}^2}\right]$

(c) Show that if a >> L, the magnitude of this force reduces to F = Q²/4π∈₀a². (Hint: Use the expansion $\text{z = }-\frac{1}{2}{\text{z}}^2+\frac{1}{3}{\text{z}}^3-\cdots$, valid for |z| << 1. Carry all expansions to at least order L²/a².) Interpret this result.