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 $\frac{q}{n \pi \varepsilon_{0}} .$ Find $n$.

Answer 1

Electric field,
$E =\frac{1}{4 \pi \varepsilon_{0}}\left[\frac{q}{1}+\frac{q}{2^{2}}+\frac{q}{4^{2}}+\frac{q}{8^{2}}+\ldots\right] $
$=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{\left(1-\frac{1}{4}\right)}=\frac{q}{3 \pi \varepsilon_{0}}$