Question

An infinite number of charges, each equal to $q$, are placed along the $x$-axis at $x=1,x=2,x=4,x=8$, etc. The electric field at the point $x=0$ due to the set of charges is $\frac{q}{n\pi {\epsilon }_0}.$ Find $n$.

Answer 1

Answer: 3

Electric field,
$E=\frac{1}{4\pi {\epsilon }_0}\left[\frac{q}{1}+\frac{q}{2^2}+\frac{q}{4^2}+\frac{q}{8^2}+\dots \right]$
$=\frac{1}{4\pi {\epsilon }_0}\frac{q}{\left(1-\frac{1}{4}\right)}=\frac{q}{3\pi {\epsilon }_0}$