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Physics Question on Electric Charge

An infinite number of charge, each of charge 1 μC1\, \mu C are placed on the x-axis with coordinates x=1,2,4,8,.......∞.x=1, 2, 4, 8, ....... \infty. If a charge of 1 C1\,C is kept at the origin, then what is the net force acting on 1 C1\,C charge?

A

9000 N9000\,N

B

12000 N12000\,N

C

24000 N24000\,N

D

36000 N36000\,N

Answer

12000 N12000\,N

Explanation

Solution

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=14πε0[1×1×10−6(1)2+1×1×10−6(2)2+1×1×10−6(4)2+1×1×10−6(8)2+...∞]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]
=10−64πε0[11+14+116+164+...∞]=\frac{10^{-6}}{4\pi\varepsilon_{0}}\left[\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+...\infty\right]
=9×10−6×109(11−14)=9\times10^{-6}\times10^{9}\left(\frac{1}{1-\frac{1}{4}}\right)
=9×109×10−6×43=12000 N=9\times10^{9}\times10^{-6}\times\frac{4}{3}=12000\,N