Question

An infinite number of point charges, each carrying $1\mu \text{C}$ charge, are placed along the $y-$axis at $y=1\text{m},2\text{m},4\text{m},8\text{m}.......$ so on. The total force on a $1\text{C}$ point charge, placed at the origin, is $x\times {10}^3\text{N}$. The value of $x$, to the nearest integer, is ______.

$(\frac{1}{4\pi {ϵ}_0}=9\times {10}^9{\text{Nm}}^2{\text{/C}}^2)$

Answer 1

$F_{\text{total}}=\frac{k{q}_0{q}_1}{r_1^2}+\frac{k{q}_0{q}_2}{r_2^2}+\frac{k{q}_0{q}_3}{r_3^2}+.......$

$=9\times {10}^9\times 1\times {10}^{-6}[1+\left(\frac{1}{2^2}\right)+\left(\frac{1}{4^2}\right)+\left(\frac{1}{8^2}\right)+.......]$

$=9\times {10}^3[1+\left(\frac{1}{2^2}\right)+\left(\frac{1}{2^4}\right)+\left(\frac{1}{2^6}\right)+.......]$

$=9\times {10}^3\left[\frac{1}{1-\frac{1}{4}}\right]$

$[\because S_{\infty }=\frac{a}{1-r}\text{for infinite geometric progression}]$

$=12\times {10}^3\text{N}$