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

Question

Aicharya · Accepted Answer

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

Total force acting on $1 \mathrm{C}$ charge is given by
$\begin{aligned} F=\frac{1}{4 \pi \varepsilon_{0}} &\left[\frac{1 \times 1 \times 10^{-6}}{(1)^{2}}+\frac{1 \times 1 \times 10^{-6}}{(2)^{2}}\right.\left.\frac{1 \times 1 \times 10^{-6}}{(4)^{2}}+\frac{1 \times 1 \times 10^{-6}}{(8)^{2}}+\ldots . \infty\right] \end{aligned}$
$=\frac{10^{-6}}{4 \pi \varepsilon_{0}}\left(\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\ldots . \infty\right)$
$\quad=9 \times 10^{9} \times 10^{-6}\left(\frac{1}{1-\frac{1}{4}}\right)$
$=9 \times 10^{9} \times 10^{-6} \times \frac{4}{3}=9 \times \frac{4}{3} \times 10^{3}=12000 \mathrm{~N}=12 \mathrm{kN}$

An infinite number of charge, each of charge \(1 \mu \mathrm{C}\), are placed on the \(x\)-axis with…

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