The Electric Field Due to a Line of Charge — Question 1

Question 1

A finite straight line approaches different limits near and far from its midpoint

A thin insulating rod occupies $-L\le x\le L$ and carries uniform positive line density $\lambda$. Find the field at $(0,y)$, $y>0$, using Coulomb constant k.

1. Integrate the surviving component and derive the exact field.

2. Obtain the limits $y\ll L$ and $y\gg L$, identifying the total charge in the latter.

Question 1 - Solution

A finite straight line approaches different limits near and far from its midpoint

(a) Method and calculation

$$dq=\lambda\,dx,\quad dE_y=\frac{k\lambda y\,dx}{(x^2+y^2)^{3/2}},\quad E_x=0.$$$$E_y=k\lambda y\left[\frac{x}{y^2\sqrt{x^2+y^2}}\right]_{-L}^L=\frac{2k\lambda L}{y\sqrt{L^2+y^2}}.$$

(b) Interpretation and further calculation

$$y\ll L:\ E_y\simeq2k\lambda/y.$$$$y\gg L:\ E_y\simeq2k\lambda L/y^2=kQ/y^2,\quad Q=2\lambda L.$$The nearby field sees a long line; the distant field sees a compact total charge.

Physical check

The symmetry cancels horizontal contributions before integration; it does not justify adding their magnitudes.