Question

An infinite number of charge, each of charge $1\, \mu C$ are placed on the x-axis with coordinates $x=1, 2, 4, 8, ....... \infty.$ If a charge of $1\,C$ is kept at the origin, then what is the net force acting on $1\,C$ charge?

• A. $9000\,N$
• B. $12000\,N$
• C. $24000\,N$
• D. $36000\,N$

Answer 1

Answer: (B)

The schematic diagram of distribution of charges on x-axis is shown in figure

Total force acting on 1 C charge is given by
$F=\frac{1}{4\pi\varepsilon_{0}}\left[\frac{1\times1\times 10^{-6}}{\left(1\right)^{2}}+\frac{1\times 1\times 10^{-6}}{\left(2\right)^{2}}+\frac{1\times 1\times 10^{-6}}{\left(4\right)^{2}}+\frac{1\times 1\times 10^{-6}}{\left(8\right)^{2}}+...\infty\right]$
$=\frac{10^{-6}}{4\pi\varepsilon_{0}}\left[\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+...\infty\right]$
$=9\times10^{-6}\times10^{9}\left(\frac{1}{1-\frac{1}{4}}\right)$
$=9\times10^{9}\times10^{-6}\times\frac{4}{3}=12000\,N$