Coulomb's Law — Question 9

Question 9

The largest force on a symmetry axis occurs at a finite distance

Use $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$; treat the specified charges as points in vacuum. Two equal positive sources $Q=3.00$ nC are fixed at $(\pm a,0)$ with $a=.100$ m. A positive test charge $q=2.00$ nC is placed at $(0,y)$, where $y>0$.

1. Use symmetry and component projection to derive the net force as a function of $y$.

2. Find the height of maximum force and the maximum magnitude. Check the force as $y\to0$ and as $y\to\infty$.

Question 9 - Solution

The largest force on a symmetry axis occurs at a finite distance

(a) Method and calculation

$$r=\sqrt{a^2+y^2},\quad F_x=0,\quad F_y=2\left(\frac{kQq}{r^2}\right)\frac yr=\frac{2kQqy}{(a^2+y^2)^{3/2}}.$$$$\frac{dF_y}{dy}=\frac{2kQq(a^2-2y^2)}{(a^2+y^2)^{5/2}}.$$

(b) Interpretation and further calculation

$$y_*=a/\sqrt2\approx.0707107\,\mathrm m,\quad F_{\max}=\frac{4kQq}{3\sqrt3a^2}\approx4.15230\times10^{-6}\,\mathrm N.$$$$y\to0:\ F_y\to0;\qquad y\to\infty:\ F_y\sim2kQq/y^2\to0.$$The derivative changes from positive to negative at the stated height, proving a maximum.

Physical check

Near the midpoint, large individual forces almost cancel; maximum resultant need not occur at the shortest distance to either source.