Coulomb's Law — Question 1

Question 1

A measured attractive force determines a signed unknown charge

Use $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$; treat the specified charges as points in vacuum. A known charge $+3.00$ nC and an unknown charge are separated by .100 m. Their attractive force has magnitude $2.697\times10^{-5}$ N.

1. Infer the sign and magnitude of the unknown charge.

2. Find the force magnitude if the separation becomes .250 m. Determine the separation needed to reduce the original force to one ninth.

Question 1 - Solution

A measured attractive force determines a signed unknown charge

(a) Method and calculation

$$F=k\frac{|q_1q_2|}{r^2},\quad|q_2|=\frac{Fr^2}{k|q_1|}=\frac{(2.697\times10^{-5})(.100)^2}{(8.99\times10^9)(3.00\times10^{-9})}.$$$$q_2=-1.00\times10^{-8}\,\mathrm C=-10.0\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$F\prime=F(.100/.250)^2=4.3152\times10^{-6}\,\mathrm N.$$$$F\prime/F=1/9=(r/r\prime)^2\Rightarrow r\prime=3r=.300\,\mathrm m.$$The negative sign comes from attraction to the positive known charge, not from the nonnegative force magnitude.

Physical check

Inverse-square scaling applies to center-to-center point-charge separations, not the gap between finite conducting surfaces.