The Electric Field Due to a Line of Charge — Question 2

Question 2

A long-line approximation requires enough charged length on both sides

For a uniform rod of half-length L, $E_f=2k\lambda L/[y\sqrt{L^2+y^2}]$ at its perpendicular bisector. The infinite-line approximation is $E_\infty=2k\lambda/y$. Define the fractional deficit relative to the infinite-line value.

1. Derive the deficit and find the minimum $L/y$ for it to be no greater than 1 percent.

2. Evaluate the deficit when $L=3y$ and state whether increasing line density improves the approximation.

Question 2 - Solution

A long-line approximation requires enough charged length on both sides

(a) Method and calculation

$$\delta=\frac{E_\infty-E_f}{E_\infty}=1-\frac{L}{\sqrt{L^2+y^2}}.$$$$\delta\le.01\Rightarrow L/y\ge\frac{.99}{\sqrt{1-.99^2}}\approx7.01792.$$

(b) Interpretation and further calculation

$$L=3y:\quad\delta=1-3/\sqrt{10}\approx.0513167.$$$$E_f/E_\infty\ \text{is independent of }\lambda.$$More charge scales both fields equally; only the length-to-distance ratio controls this relative geometric error.

Physical check

L is the half-length; the required full length is approximately $14.0358y$.