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 $1C$ is kept at the origin, then what is the net force acting on $1C$ charge?

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

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
$F=\frac{1}{4\pi {\epsilon }_0}\left[\frac{1\times 1\times 10^{-6}}{{\left(1\right)}^2}+\frac{1\times 1\times 10^{-6}}{{\left(2\right)}^2}+\frac{1\times 1\times 10^{-6}}{{\left(4\right)}^2}+\frac{1\times 1\times 10^{-6}}{{\left(8\right)}^2}+...\infty \right]$
$=\frac{10^{-6}}{4\pi {\epsilon }_0}\left[\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+...\infty \right]$
$=9\times 10^{-6}\times 10^9\left(\frac{1}{1-\frac{1}{4}}\right)$
$=9\times 10^9\times 10^{-6}\times \frac{4}{3}=12000N$