An infinite number of point charges,
Question:

An infinite number of point charges, each carrying $1 \mu \mathrm{C}$ charge, are placed along the $y$-axis at $\mathrm{y}=1 \mathrm{~m}, 2 \mathrm{~m}, 4 \mathrm{~m}, 8 \mathrm{~m}$……………

The total force on a $1 \mathrm{C}$ point charge, placed at the origin, is $x \times 10^{3} \mathrm{~N}$. The value of $x$, to the nearest integer, is________.

[Take $\left.\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\right]$

Solution:

$\mathrm{F}=\mathrm{k}(1 \mathrm{C})(1 \mu \mathrm{C})\left[1+\frac{1}{2^{2}}+\frac{1}{4^{2}}+\frac{1}{8^{2}}+\ldots\right]$

$=9 \times 10^{3}\left[\frac{1}{1-\frac{1}{4}}\right]=12 \times 10^{3} \mathrm{~N}$