The Electric Field Due to a Line of Charge — Question 4

Question 4

An off-center observation point retains both rod-field components

A uniform positive line density $\lambda$ occupies $0\le x\le L$. Observe at P$=(0,y)$, $y>0$. Use k for the Coulomb constant.

1. Integrate both field components.

2. For $L=y$, find the angle of the field left of the upward vertical and the component ratio.

Question 4 - Solution

An off-center observation point retains both rod-field components

(a) Method and calculation

$$E_x=-k\lambda\int_0^L\frac{x\,dx}{(x^2+y^2)^{3/2}}=-k\lambda\left(\frac1y-\frac1{\sqrt{L^2+y^2}}\right).$$$$E_y=k\lambda y\int_0^L\frac{dx}{(x^2+y^2)^{3/2}}=\frac{k\lambda L}{y\sqrt{L^2+y^2}}.$$

(b) Interpretation and further calculation

$$L=y:\quad\frac{|E_x|}{E_y}=\sqrt2-1.$$$$\alpha=\tan^{-1}(\sqrt2-1)=22.5^\circ.$$Every element pushes P upward and leftward, so neither component cancels.

Physical check

Using the midpoint formula here would incorrectly impose a symmetry the source does not possess.