An infinite number of charges, each of charge 1 $\mathrm{\mu }$C, are placed on the x-axis with co-ordinates (in m) x = 1,2, 4, 8,…..$\infty$ . 1f a charge of 1 C is kept at the origin, then what is the net force acting on 1 C charge

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

Total force acting on 1 C charge is given by

$F=\frac{1}{4\pi s_0}\left[\frac{1\times 1\times {10}^{-6}}{(1{)}^2}+\frac{1\times 1\times {10}^{-6}}{(2{)}^2}+\frac{1\times 1\times {10}^{-6}}{(4{)}^2}+\right.$$\left.\frac{1\times 1\times {10}^{-6}}{(8{)}^2}+\dots \mathrm{\infty }\right]$

$=\frac{{10}^{-6}}{4\pi {\epsilon }_0}\left(\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\dots .\mathrm{\infty }\right)$

$=9\times {10}^9\times {10}^{-6}\left(\frac{1}{1-\frac{1}{4}}\right)$

$9\times {10}^9\times {10}^{-6}\times \frac{4}{3}=9\times {10}^3\times \frac{4}{3}=$$12000\mathrm{N}$